Linear K-Power Preservers and Trace of Power-Product Preservers

*Huajun Huang and Ming-Cheng Tsai*

## **Abstract**

Let *V* be the set of *n* � *n* complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix *k*∈ nf g 0, 1 . We characterize linear maps *ψ* : *<sup>V</sup>* ! *<sup>V</sup>* that satisfy *<sup>ψ</sup> Ak* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* on an open neighborhood *<sup>S</sup>* of *In* in *<sup>V</sup>*. The *<sup>k</sup>*-power preservers are necessarily *k*-potent preservers, and *k* ¼ 2 corresponds to Jordan homomorphisms. Applying the results, we characterize maps *ϕ*, *ψ* : *V* ! *V* that satisfy "*tr <sup>ϕ</sup>*ð Þ *<sup>A</sup> <sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>k</sup>* <sup>¼</sup> *tr ABk* for all *<sup>A</sup>* <sup>∈</sup>*V*, *<sup>B</sup>*<sup>∈</sup> *<sup>S</sup>*, and *<sup>ψ</sup>* is linear" or "*tr <sup>ϕ</sup>*ð Þ *<sup>A</sup> <sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>k</sup>* <sup>¼</sup> *tr ABk* for all *<sup>A</sup>*, *<sup>B</sup>*<sup>∈</sup> *<sup>S</sup>* and both *<sup>ϕ</sup>* and *<sup>ψ</sup>* are linear." The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.

**Keywords:** k-power, power preserver, trace preserver, power series of matrices

## **1. Introduction**

Preserver problem is one of the most active research areas in matrix theory (e.g. [1–4]). Researchers would like to characterize the maps on a given space of matrices preserving certain subsets, functions or relations. One of the preserver problems concerns maps *ψ* on some sets *V* of matrices which preserves *k*-power for a fixed integer *k*≥2, that is, *<sup>ψ</sup> Ak* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* for any *<sup>A</sup>* <sup>∈</sup>*<sup>V</sup>* (e.g. [3, 5, 6]). The *<sup>k</sup>*-power preservers form a special class of polynomial preservers. One important reason of this problem lies on the fact that the case *k* ¼ 2 corresponds to Jordan homomorphisms. Moreover, every *k*-power preserver is also a *<sup>k</sup>*-potent preserver, that is, *<sup>A</sup><sup>k</sup>* <sup>¼</sup> *<sup>A</sup>* imply that *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* for any *<sup>A</sup>* <sup>∈</sup>*V*. Some researches on *k*-potent preservers can be found in [6–8].

Given a field , let M*n*ð Þ , S*n*ð Þ , D*n*ð Þ , N *<sup>n</sup>*ð Þ , and T *<sup>n</sup>*ð Þ denote the set of *n* � *n* general, symmetric, diagonal, strictly upper triangular, and upper triangular matrices over , respectively. When is the complex field , we may write M*<sup>n</sup>* instead of M*n*ð Þ , and so on. Let H*n*, P*n*, and P*<sup>n</sup>* denote the set of complex Hermitian, positive definite, and positive semidefinite matrices, and H*n*ð Þ¼ S*n*ð Þ , P*n*ð Þ , and P*n*ð Þ the corresponding set of real matrices, respectively. A matrix space is a subspace of <sup>M</sup>*<sup>m</sup>*,*<sup>n</sup>*ð Þ for certain *<sup>m</sup>*, *<sup>n</sup>* <sup>∈</sup>þ. Let *<sup>A</sup><sup>t</sup>* (resp. *<sup>A</sup>*<sup>∗</sup> ) denote the transpose (resp. conjugate transpose) of a matrix *A*.

In 1951, Kadison [9] showed that a Jordan ∗ -isomorphism on M*n*, namely, a bijective linear map with *<sup>ψ</sup> <sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* <sup>2</sup> and *<sup>ψ</sup> <sup>A</sup>*<sup>∗</sup> ð Þ¼ *<sup>ψ</sup>*ð Þ *<sup>A</sup>* <sup>∗</sup> for all *<sup>A</sup>* <sup>∈</sup>M*n*, is the direct sum of a <sup>∗</sup> -isomorphism and a <sup>∗</sup> -anti-isomorphism. Hence *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> UAU* <sup>∗</sup> for all *<sup>A</sup>* <sup>∈</sup>M*<sup>n</sup>* or *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> UATU* <sup>∗</sup> for all *<sup>A</sup>* <sup>∈</sup>M*<sup>n</sup>* by [[3], Theorem A.8]. Let *<sup>k</sup>*≥2 be a fixed integer. In 1992, Chan and Lim ([5]) determined a nonzero linear operator *ψ* : <sup>M</sup>*n*ð Þ! <sup>M</sup>*n*ð Þ (resp. *<sup>ψ</sup>* : <sup>S</sup>*n*ð Þ! <sup>S</sup>*n*ð Þ ) such that *<sup>ψ</sup> <sup>A</sup><sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* for all *A* ∈M*n*ð Þ (resp. S*n*ð Þ ) (See Theorems 3.1 and 5.1). In 1998, Brešar, Martindale, and Miers considered additive maps of general prime rings to solve an analogous problem by using the deep algebraic techniques ([10]). Monlár [[3], P6] described a particular case of their result which extends Theorem 3.1 to surjective linear operators on B Hð Þ. In 2004, Cao and Zhang determined additive *k*-power preserver on M*n*ð Þ and S*n*ð Þ ([11]). They also characterized injective additive *k*-power preserver on T *<sup>n</sup>*ð Þ ([12] or [[6], Theorem 6.5.2]), which leads to injective linear *k*power preserver on T *<sup>n</sup>*ð Þ (see Theorem 8.1). In 2006, Cao and Zhang also characterized linear *k*-power preservers from M*n*ð Þ to M*m*ð Þ and from S*n*ð Þ to M*m*ð Þ (resp. S*m*ð Þ ) [8].

In this article, given an integer *k*∈nf g 0, 1 , we show that a unital linear map *ψ* : *V* ! *W* between matrix spaces preserving *k*-powers on a neighborhood of identity must preserve all integer powers (Theorem 2.1). Then we characterize, for ¼ and , linear operators on sets *V* ¼ M*n*ð Þ , H*n*, S*n*ð Þ , P*n*, P*n*ð Þ , D*n*ð Þ , and T *<sup>n</sup>*ð Þ that satisfy *<sup>ψ</sup> Ak* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* on an open neighborhood of *In* in *<sup>V</sup>*. In the following descriptions, *P*∈M*n*ð Þ is invertible, *U* ∈M*n*ð Þ is unitary, *O* ∈M*n*ð Þ is orthogonal, and *<sup>λ</sup>*∈ satisfies that *<sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> 1.


Our results on M*n*ð Þ and S*n*ð Þ extend Chan and Lim's results in Theorems 3.1 and 5.1, and result on T *<sup>n</sup>*ð Þ extend Cao and Zhang's linear version result in [12].

Another topic is the study of a linear map *ϕ* from a matrix set *S* to another matrix set *T* preserving trace equation. In 1931, Wigner's unitary-antiunitary theorem [[3], p. 12] says that if *ϕ* is a bijective map defined on the set of all rank one projections on a Hilbert space *H* satisfying

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

$$\text{tr}(\phi(A)\phi(B)) = \text{tr}(AB),\tag{1}$$

then there is an either unitary or antiunitary operator *U* on *H* such that *ϕ*ð Þ¼ *P <sup>U</sup>* <sup>∗</sup> *PU* or *<sup>ϕ</sup>*ð Þ¼ *<sup>P</sup> <sup>U</sup>* <sup>∗</sup> *Pt U* for all rank one projections *P*. In 1963, Uhlhorn generalized Wigner's theorem to show that the same conclusion holds if the equality trð Þ¼ *ϕ*ð Þ *P ϕ*ð Þ *Q* trð Þ *PQ* is replaced by trð Þ¼ *ϕ*ð Þ *P ϕ*ð Þ *Q* 0⇔trð Þ¼ *PQ* 0 (see [13]).

In 2002, Molnár (in the proof of [[14], Theorem 1]) showed that maps *ϕ* on the space of all bounded linear operators on a Banach space *B X*ð Þ satisfying (1) for *A* ∈*B X*ð Þ, rank one operator *B* ∈*B X*ð Þ are linear. In 2012, Li, Plevnik, and Šemrl [15] characterized bijective maps *ϕ* : *S* ! *S* satisfying trð Þ¼ *ϕ*ð Þ *A ϕ*ð Þ *B c*⇔trð Þ¼ *AB c* for a given real number *c*, where *S* is H*n*, S*n*ð Þ , or the set of rank one projections.

In [[16], Lemma 3.6], Huang et al. showed that the following statements are equivalent for a unital map *ϕ* on P*n*:

$$\begin{aligned} \text{1.}tr(\phi(A)\phi(B)) &= tr(AB) \text{ for } A, B \in \mathcal{P}\_n; \\\\ \text{2.}tr\left(\phi(A)\phi(B)^{-1}\right) &= tr(AB^{-1}) \text{ for } A, B \in \mathcal{P}\_n; \end{aligned}$$

3.*ϕ*ð Þ¼ *<sup>A</sup> <sup>U</sup>* <sup>∗</sup> *AU* or *<sup>U</sup>* <sup>∗</sup> *At U* for a unitary matrix *U*.

The authors also determined the cases if *ϕ* is not assuming unital, the set P*<sup>n</sup>* is replaced by another set like M*n*, S*n*, T *<sup>n</sup>*, or D*n*. In [[17], Theorem 3.8], Leung, Ng, and Wong considered the relation (1) on infinite dimensional space.

Let h i*S* denote the subspace spanned by a subset *S* of a vector space. Recently, Huang and Tsai studied two maps preserving trace of product [18]. Suppose two maps *ϕ* : *V*<sup>1</sup> ! *W*<sup>1</sup> and *ψ* : *V*<sup>2</sup> ! *W*<sup>2</sup> between subsets of matrix spaces over a field under some conditions satisfy

$$\operatorname{tr}(\phi(A)\psi(B)) = \operatorname{tr}(AB) \tag{2}$$

for all *A* ∈*V*1, *B*∈*V*2. The authors showed that these two maps can be extended to bijective linear maps *<sup>ϕ</sup>*<sup>~</sup> : h i *<sup>V</sup>*<sup>1</sup> ! h i *<sup>W</sup>*<sup>1</sup> and *<sup>ψ</sup>*<sup>~</sup> : h i *<sup>V</sup>*<sup>2</sup> ! h i *<sup>W</sup>*<sup>2</sup> that satisfy tr *<sup>ϕ</sup>*~ð Þ *<sup>A</sup> <sup>ψ</sup>*~ð Þ *<sup>B</sup>* <sup>¼</sup> trð Þ *AB* for all *<sup>A</sup>* <sup>∈</sup>h i *<sup>V</sup>*<sup>1</sup> , *<sup>B</sup>* <sup>∈</sup>h i *<sup>V</sup>*<sup>2</sup> (see Theorem 2.2). Hence when a matrix space *V* is closed under conjugate transpose, every linear bijection *ϕ* : *V* ! *V* corresponds to a unique linear bijection *ψ* : *V* ! *V* that makes (2) hold (see Corollary 2.3). Therefore, each of *ϕ* and *ψ* has no specific form.

One natural question is to ask when the following equality holds for a fixed *k*∈nf g 0, 1 :

$$\text{tr}\left(\phi(A)\psi(B)^k\right) = \text{tr}\left(AB^k\right). \tag{3}$$

The second major work of this paper is to use our descriptions of linear *k*-power preservers on an open neighborhood *S* of *In* in *V* to characterize maps *ϕ*, *ψ* : *V* ! *V* under one of the assumptions:

1.equality (3) holds for all *A* ∈*V*, *B*∈ *S*, and *ψ* is linear, or

2.equality (3) holds for all *A*, *B* ∈*S* and both *ϕ* and *ψ* are linear,

for the sets *V* ¼ M*n*, H*n*, P*n*, S*n*, D*n*, T *<sup>n</sup>*, and their real counterparts. These results, together with Theorem 2.2 and the characterizations of maps *ϕ*1, ⋯, *ϕ<sup>m</sup>* : *V* ! *V* (*m* ≥ 3) that satisfy *tr*ð*ϕ*1ð Þ *A*<sup>1</sup> ⋯*ϕm*ð Þ *Am* Þ ¼ *tr A*ð Þ <sup>1</sup>⋯*Am* in [18], make a comprehensive picture of the preservers of trace of matrix products in the related matrix spaces and sets.

In the following characterizations, ¼ or , *P*, *Q* ∈ *Mn*ð Þ are invertible, *U* ∈ *Mn*ð Þ is unitary, *O* ∈ *Mn*ð Þ is orthogonal, and *c*∈nf g0 .

1.*V* ¼ M*n*ð Þ (Theorem 3.5):


2.*V* ¼ H*<sup>n</sup>* (Theorem 4.2):


$$\textbf{3.}\,V = \mathcal{S}\_{\pi}(\mathbb{F}) \text{ (Theorem 5.3):}$$


The sets M*n*, H*n*, P*n*, S*n*, D*n*, and their real counterparts are closed under conjugate transpose. In these sets, *tr AB* ð Þ¼ *<sup>A</sup>*<sup>∗</sup> h i , *<sup>B</sup>* for the standard inner product. Our trace of product preservers can also be interpreted as inner product preservers, which have wide applications in research areas like quantum information theory.

## **2. Preliminary**

### **2.1 Linear operators preserving powers**

We show below that: given *k*∈nf g 0, 1 , a unital linear map *ψ* : *V* ! *W* between matrix spaces preserving *k*-powers on a neighborhood of identity in *V* must preserve all

integer powers. Let <sup>þ</sup> (resp. �) denote the set of all positive (resp. negative) integers. Theorem 2.1. *Let* ¼ *or . Let V* ⊆M*p*ð Þ *and W* ⊆M*q*ð Þ *be matrix spaces. Fix k*∈nf g 0, 1 *.*

1.*Suppose the identity matrix Ip* ∈*V and A<sup>k</sup>* ∈*V for all matrices A in an open neighborhood* S*<sup>V</sup> of Ip in V consisting of invertible matrices. Then*

$$\{AB + BA : A, B \in V\} \subseteq V,\tag{4}$$

$$\left\{ A^{-1} : A \in V \text{ is } invertible \right\} \subseteq V. \tag{5}$$

In particular,

$$\{A^r: A \in V\} \subseteq V, \quad r \in \mathbb{Z}\_+, \quad \text{and} \tag{6}$$

$$\{A^r: A \in V \text{ is } invertible\} \subseteq V, \quad r \in \mathbb{Z}\_-. \tag{7}$$

2. *Suppose Ip* ∈*V, Iq* ∈*W, and Ak* ∈*V for all matrices A in an open neighborhood* S*<sup>V</sup> of Ip in V consisting of invertible matrices. Suppose ψ* : *V* ! *W is a linear map that satisfies the following conditions:*

$$
\psi\left(I\_p\right) = I\_q,\tag{8}
$$

$$
\psi\left(A^k\right) = \psi\left(A\right)^k, \quad A \in \mathcal{S}\_V. \tag{9}
$$

Then

$$
\psi(AB+BA) = \psi(A)\psi(B) + \psi(B)\psi(A), \quad A, B \in V,\tag{10}
$$

$$
\psi\left(A^{-1}\right) = \psi\left(A\right)^{-1}, \quad \text{invertible } A \in \mathcal{V}. \tag{11}
$$

In particular,

$$
\psi(A^r) = \psi(A)^r, \quad A \in V, \ r \in \mathbb{Z}\_+, \quad \text{and} \tag{12}
$$

$$\psi(A^r) = \psi(A)^r, \quad \text{invertible } A \in V, \ r \in \mathbb{Z}\_-. \tag{13}$$

*Proof.* We prove the complex case. The real case is done similarly.

1. For each *A* ∈*V*nf g0 , there is ϵ> 0 such that *Ip* þ *xA* ∈S*<sup>V</sup>* for all *x*∈ with ∣*x*∣< min ϵ, <sup>1</sup> ∥*A*∥ n o. Thus

$$\left(I\_p + \varkappa A\right)^k = I\_p + \varkappa k A + \varkappa^2 \frac{k(k-1)}{2} A^2 + \dots \in V. \tag{14}$$

The second derivative

$$\left. \frac{d^2}{d\mathbf{x}^2} \left( I\_p + \varkappa A \right)^k \right|\_{\mathbf{x}=\mathbf{0}} = k(k-1)A^2 \in \mathbf{V}.\tag{15}$$

Since *k* �<sup>∈</sup> f g 0, 1 , we have *<sup>A</sup>*<sup>2</sup> <sup>∈</sup>*<sup>V</sup>* for all *<sup>A</sup>* <sup>∈</sup>*V*. Therefore, for *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup>*V*,

$$AB + BA = \left(A + B\right)^2 - A^2 - B^2 \in V. \tag{16}$$

In particular, *<sup>A</sup>* <sup>∈</sup>*<sup>V</sup>* implies that *<sup>A</sup><sup>r</sup>* <sup>∈</sup>*<sup>V</sup>* for all *<sup>r</sup>*∈þ.

Cayley-Hamilton theorem implies that every invertible matrix *A* satisfies that *<sup>A</sup>*�<sup>1</sup> <sup>¼</sup> *f A*ð Þ for a certain polynomial *f x*ð Þ∈½ � *<sup>x</sup>* . Therefore, *<sup>A</sup>*�<sup>1</sup> <sup>∈</sup>*V*, so that *<sup>A</sup><sup>r</sup>* <sup>∈</sup>*<sup>V</sup>* for all *r*∈�.

2.Now suppose (8) and (9) hold. The proof is proceeded similarly to the proof of part (1). For every *A* ∈*V*, there is ϵ>0 such that for all *x*∈ with ∣*x*∣< min ϵ, <sup>1</sup> <sup>∥</sup>*A*<sup>∥</sup> , <sup>1</sup> ∥*ψ*ð Þ *A* ∥ n o,

$$\left(\left(\wp\left(I\_p + \varkappa A\right)\right)^k = I\_q + \varkappa k \wp(A) + \varkappa^2 \frac{k(k-1)}{2} \wp(A)^2 + \dots \in W,\tag{17}$$

$$\Psi\left(\left(I\_p + \varkappa A\right)^k\right) = I\_q + \varkappa k \psi(A) + \varkappa^2 \frac{k(k-1)}{2} \psi\left(A^2\right) + \dots \in W. \tag{18}$$

since (17) and (18) equal, we have

$$\left(\psi(A)\right)^{2} = \psi\left(A^{2}\right), \quad A \in V. \tag{19}$$

Therefore, for *A*, *B*∈*V*,

$$
\psi\left(\left(A+B\right)^2\right) = \psi\left(A+B\right)^2\tag{20}
$$

We get (10): *<sup>ψ</sup>*ð Þ¼ *AB* <sup>þ</sup> *BA <sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>ψ</sup>*ð Þþ *<sup>B</sup> <sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>ψ</sup>*ð Þ *<sup>A</sup>* . In particular *<sup>ψ</sup> Ar* ð Þ¼ *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>r</sup>* for all *A* ∈*V* and *r*∈þ.

Every invertible *<sup>A</sup>* <sup>∈</sup>*<sup>V</sup>* can be expressed as *<sup>A</sup>*�<sup>1</sup> <sup>¼</sup> *f A*ð Þ for a certain polynomial *f x*ð Þ<sup>∈</sup> ½ � *<sup>x</sup>* . Then *<sup>ψ</sup> <sup>A</sup>*�<sup>1</sup> � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ¼ *f A*ð Þ *<sup>f</sup>*ð Þ *<sup>ψ</sup>*ð Þ *<sup>A</sup>* is commuting with *<sup>ψ</sup>*ð Þ *<sup>A</sup>* . Hence

$$2\nu\left(A^{-1}\right)\nu(A) = \nu\left(A^{-1}\right)\nu(A) + \nu(A)\nu\left(A^{-1}\right) = \nu\left(A^{-1}A + AA^{-1}\right) = 2I\_q.\tag{21}$$

We get *<sup>ψ</sup> <sup>A</sup>*�<sup>1</sup> � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* �<sup>1</sup> . Therefore, *<sup>ψ</sup> Ar* ð Þ¼ *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>r</sup>* for all *<sup>r</sup>*∈�.

Theorem 2.1 is powerful in exploring *k*-power preservers in matrix spaces. Note that every *k*-power preserver is a *k*-potent preserver. Theorem 2.1 can also be used to investigate *k*-potent preservers in matrix spaces.

#### **2.2 Two maps preserving trace of product**

We recall two results about two maps preserving trace of product in [18]. They are handy in proving linear bijectivity of maps preserving trace of products. Recall that if *S* is a subset of a vector space, then h i*S* denotes the subspace spanned by *S*.

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

Theorem 2.2 (Huang, Tsai [18]). *Let ϕ* : *V*<sup>1</sup> ! *W*<sup>1</sup> *and ψ* : *V*<sup>2</sup> ! *W*<sup>2</sup> *be two maps between subsets of matrix spaces over a field such that:*

$$
\dim \langle V\_1 \rangle = \dim \langle V\_2 \rangle \ge \max \left\{ \dim \langle W\_1 \rangle, \dim \langle W\_2 \rangle \right\}.
$$

1.*AB are well-defined square matrices for A*ð Þ , *B* ∈ð Þ *V*<sup>1</sup> � *V*<sup>2</sup> ∪ ð Þ *W*<sup>1</sup> � *W*<sup>2</sup> *.*

2.*If A* ∈h i *V*<sup>1</sup> *satisfies that* trð Þ¼ *AB* 0 *for all B*∈h i *V*<sup>2</sup> *, then A* ¼ 0*.*

3.*ϕ and ψ satisfy that*

$$\operatorname{tr}(\phi(A)\psi(B)) = \operatorname{tr}(AB), \quad A \in V\_1, \ B \in V\_2. \tag{22}$$

Then dimh i *V*<sup>1</sup> ¼ dimh i *V*<sup>2</sup> ¼ dimh i *W*<sup>1</sup> ¼ dimh i *W*<sup>2</sup> and *ϕ* and *ψ* can be extended to bijective linear map *<sup>ϕ</sup>*<sup>~</sup> : h i *<sup>V</sup>*<sup>1</sup> ! h i *<sup>W</sup>*<sup>1</sup> and *<sup>ψ</sup>*<sup>~</sup> : h i *<sup>V</sup>*<sup>2</sup> ! h i *<sup>W</sup>*<sup>2</sup> , respectively, such that

$$\text{tr}\left(\tilde{\phi}(A)\tilde{\varphi}(B)\right) = \text{tr}(AB), \quad A \in \langle V\_1 \rangle, \ B \in \langle V\_2 \rangle. \tag{23}$$

A subset *<sup>V</sup>* of <sup>M</sup>*<sup>n</sup>* is closed under conjugate transpose if *<sup>A</sup>*<sup>∗</sup> f g : *<sup>A</sup>* <sup>∈</sup>*<sup>V</sup>* <sup>⊆</sup>*V*. A real or complex matrix space *V* is closed under conjugate transpose if and only if *V* equals the direct sum of its subspace of Hermitian matrices and its subspace of skew-Hermitian matrices.

Corollary 2.3 (Huang, Tsai [18]). *Let V be a subset of* M*<sup>n</sup> closed under conjugate transpose. Suppose two maps ϕ*, *ψ* : *V* ! *V satisfy that*

$$\text{tr}(\phi(A)\psi(B)) = \text{tr}(AB), \quad A, B \in V. \tag{24}$$

*Then ϕ and ψ can be extended to linear bijections on V*h i*. Moreover, when V is a vector space, every linear bijection ϕ* : *V* ! *V corresponds to a unique linear bijection ψ* : *V* ! *V such that* (24) *holds. Explicitly, given an orthonormal basis A*f g 1, … , *A*<sup>ℓ</sup> *of V with respect to the inner product A*h i , *<sup>B</sup>* <sup>¼</sup> tr *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>B</sup> , <sup>ψ</sup> is defined by <sup>ψ</sup>*ð Þ¼ *Ai Bi in which B*f g 1, … , *<sup>B</sup>*<sup>ℓ</sup> *is a basis of V with* tr *ϕ A*<sup>∗</sup> *i B <sup>j</sup>* <sup>¼</sup> *<sup>δ</sup><sup>i</sup>*,*<sup>j</sup> for all i*, *<sup>j</sup>*<sup>∈</sup> f g 1, … , <sup>ℓ</sup> *.*

Corollary 2.3 shows that when a matrix space *V* is closed under conjugate transpose, every linear bijection *ϕ* : *V* ! *V* corresponds to a unique linear bijection *ψ* : *V* ! *V* that makes (24) hold. The next natural thing is to determine *ϕ* and *ψ* that satisfy *tr <sup>ϕ</sup>*ð Þ *<sup>A</sup> <sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>k</sup>* <sup>¼</sup> *tr ABk* for a fixed *<sup>k</sup>*∈nf g 0, 1 *:*

From now on, we focus on the fields ¼ or .

## **3.** *k***-power linear preservers and trace of power-product preservers on** M*<sup>n</sup>* **and** M*n*ð Þ

## **3.1** *k***-power preservers on** M*<sup>n</sup>* **and** M*n*ð Þ

Chan and Lim described the linear *k*-power preservers on M*<sup>n</sup>* and M*n*ð Þ for *k*≥2 in [7, Theorem 1] as follows.

Theorem 3.1. *(Chan, Lim* [5]*) Let an integer k*≥2*. Let be a field with char*ð Þ¼ 0 *or char*ð Þ >*k. Suppose that ψ* : M*n*ð Þ! M*n*ð Þ *is a nonzero linear operator such that <sup>ψ</sup> Ak* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A* <sup>∈</sup>M*n*ð Þ *. Then there exist <sup>λ</sup>*<sup>∈</sup> *with <sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> *and an invertible matrix P*∈M*n*ð Þ *such that*

$$
\psi(A) = \lambda P A P^{-1}, \quad A \in \mathcal{M}\_n(\mathbb{F}), \quad or \tag{25}
$$

$$
\psi(A) = \lambda P A^t P^{-1}, \quad A \in \mathcal{M}\_n(\mathbb{F}).\tag{26}
$$

(25) and (26) need not hold if *ψ* is zero or is a map on a subspace of M*n*ð Þ . The following are two examples. Another example can be found in maps on D*n*ð Þ (Theorem 7.1).

Example 3.2. *The zero map <sup>ψ</sup>*ð Þ� *<sup>A</sup>* <sup>0</sup> *clearly satisfies <sup>ψ</sup> Ak* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A* <sup>∈</sup>M*<sup>n</sup> but they are not of the form* (25) *or* (26)*.*

Example 3.3. *Let n* ¼ *k* þ *m, k*, *m* ≥2*, and consider the operator ψ on the subspace <sup>W</sup>* <sup>¼</sup> <sup>M</sup>*k*⊕M*<sup>m</sup> of* <sup>M</sup>*<sup>n</sup> defined by <sup>ψ</sup>*ð Þ¼ *<sup>A</sup>*⊕*<sup>B</sup> <sup>A</sup>*⊕*B<sup>t</sup> for A* <sup>∈</sup>M*<sup>k</sup> and B* <sup>∈</sup>M*m: Then <sup>ψ</sup> Ak* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A* <sup>∈</sup> *W and k*<sup>∈</sup> þ*, but <sup>ψ</sup> is not of the form* (25) *or* (26)*.*

We now generalize Theorem 3.1 to include negative integers *k* and to assume the *k*power preserving condition *<sup>ψ</sup> Ak* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* only on matrices nearby the identity.

Theorem 3.4. *Let* ¼ *or . Let an integer k*∈nf g 0, 1 *. Suppose that ψ* :

<sup>M</sup>*n*ð Þ! <sup>M</sup>*n*ð Þ *is a nonzero linear map such that <sup>ψ</sup> <sup>A</sup><sup>k</sup>* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A in an open neighborhood of In consisting of invertible matrices. Then there exist <sup>λ</sup>*∈ *with <sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> *and an invertible matrix P*∈M*n*ð Þ *such that*

$$\psi(A) = \lambda P A P^{-1}, \quad A \in \mathcal{M}\_n(\mathbb{F}), \quad or \tag{27}$$

$$
\psi(A) = \lambda P A^t P^{-1}, \quad A \in \mathcal{M}\_n(\mathbb{F}).\tag{28}
$$

Proof. We prove for the case ¼ . The case ¼ can be done similarly. Obviously, *ψ*ð Þ¼ *In ψ I k n* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *In <sup>k</sup>* .

1.First suppose *k*≥2. For each *A* ∈M*n*, there exists ϵ>0 such that for all *x*∈ with ∣*x*∣ < ϵ, the following two power series converge and equal:

$$\begin{aligned} \left(\boldsymbol{\Psi}(\boldsymbol{I}\_{n}+\boldsymbol{\chi}\boldsymbol{A})\right)^{k} &= \boldsymbol{\Psi}(\boldsymbol{I}\_{n}) + \boldsymbol{\chi}\left(\sum\_{i=0}^{k-1} \boldsymbol{\psi}(\boldsymbol{I}\_{n})^{i} \boldsymbol{\varmu}(\boldsymbol{A}) \boldsymbol{\upnu}(\boldsymbol{I}\_{n})^{k-1-i}\right) \\ &+ \boldsymbol{\varmu}^{2}\left(\sum\_{i=0}^{k-2} \sum\_{j=0}^{k-2-i} \boldsymbol{\varmu}(\boldsymbol{I}\_{n})^{i} \boldsymbol{\upmu}(\boldsymbol{A}) \boldsymbol{\upnu}(\boldsymbol{I}\_{n})^{j} \boldsymbol{\upnu}(\boldsymbol{A}) \boldsymbol{\upnu}(\boldsymbol{I}\_{n})^{k-2-i-j}\right) + \cdots \\ &\quad \boldsymbol{\upnu}\left(\left(\boldsymbol{I}\_{n}+\boldsymbol{\upmu}\boldsymbol{A}\right)^{k}\right) = \boldsymbol{\varmu}(\boldsymbol{I}\_{n}) + \boldsymbol{\upnu}\boldsymbol{\upmu}(\boldsymbol{A}) + \boldsymbol{\upnu}^{2}\frac{k(k-1)}{2}\boldsymbol{\upnu}(\boldsymbol{A}^{2}) + \cdots \end{aligned} \tag{30}$$

Equating degree one terms above, we get

$$k\psi(A) = \sum\_{i=0}^{k-1} \psi(I\_n)^i \psi(A)\psi(I\_n)^{k-1-i}.\tag{31}$$

Applying (31), we have

$$k\mu\left(I\_n\right)\left\mu\left(A\right) - k\mu\left(A\right)\mu\left(I\_n\right) = \left\mu\left(I\_n\right)^k\mu\left(A\right) - \left\mu\left(A\right)\mu\left(I\_n\right)^k = \left\mu\left(I\_n\right)\mu\left(A\right) - \left\mu\left(A\right)\mu\left(I\_n\right). \tag{32}$$

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

Hence *ψ*ð Þ *In ψ*ð Þ¼ *A ψ*ð Þ *A ψ*ð Þ *In* for *A* ∈M*n*, that is, *ψ*ð Þ *In* commutes with the range of *ψ*.

Now equating degree two terms of (29) and (30) and taking into account that *k* �∈f g 0, 1 , we have

$$
\left(\varphi(I\_n)\right)^{k-2}\varphi(A)^2 = \varphi\left(A^2\right). \tag{33}
$$

Define *<sup>ψ</sup>*1ð Þ¼ *<sup>A</sup> <sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* for *<sup>A</sup>* <sup>∈</sup>M*n*. Then *<sup>ψ</sup>*<sup>1</sup> *<sup>A</sup>*<sup>2</sup> � � <sup>¼</sup> ð Þ *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>2</sup> for all *<sup>A</sup>* <sup>∈</sup>M*n*. (31) and the assumption that *ψ* is nonzero imply that *ψ*ð Þ *In* 6¼ 0. So *ψ*1ð Þ *In ψ*ð Þ¼ *In <sup>ψ</sup>*ð Þ *In <sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *In* 6¼ 0. Thus *<sup>ψ</sup>*1ð Þ *In* 6¼ 0 and *<sup>ψ</sup>*<sup>1</sup> is nonzero. By Theorem 3.1, there exists an invertible *<sup>P</sup>*∈M*<sup>n</sup>* such that *<sup>ψ</sup>*1ð Þ¼ *<sup>A</sup> PAP*�<sup>1</sup> for *<sup>A</sup>* <sup>∈</sup>M*<sup>n</sup>* or *<sup>ψ</sup>*1ð Þ¼ *<sup>A</sup> PA<sup>t</sup> P*�<sup>1</sup> for *A* ∈M*n*. Moreover, *ψ*ð Þ *In* commutes with all *ψ*1ð Þ *A* , so that *ψ*ð Þ¼ *In λIn* for a *λ*∈ . By *In* <sup>¼</sup> *<sup>ψ</sup>*1ð Þ¼ *In <sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>1</sup> , we get *<sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> 1. Therefore, *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> λψ*1ð Þ *<sup>A</sup>* . We get (27) and (28).

1.Next Suppose *k*< 0. For every *A* ∈M*n*, the power series expansions of ð Þ *<sup>ψ</sup>*ð Þ *In* <sup>þ</sup> *xA* �*<sup>k</sup>* and *<sup>ψ</sup>* ð Þ *In* <sup>þ</sup> *xA <sup>k</sup>* � ��<sup>1</sup> are equal when ∣*x*∣ is sufficiently small:

$$\left(\left(\boldsymbol{\nu}(I\_n + \boldsymbol{x}A)\right)^{-k} = \boldsymbol{\nu}(I\_n)^{-1} + \mathbf{x}\left(\sum\_{i=0}^{-k-1} \boldsymbol{\nu}(I\_n)^i \boldsymbol{\nu}(A) \boldsymbol{\nu}(I\_n)^{-k-1-i}\right) + \cdots \tag{34}$$

$$\left(\psi\left(\left(I\_n + \varkappa A\right)^k\right)^{-1} = \psi(I\_n)^{-1} - \varkappa k \psi(I\_n)^{-1} \psi(A) \psi(I\_n)^{-1} + \cdots \tag{35}$$

Equating degree one terms of (34) and (35), we get

$$-k\boldsymbol{\eta}(I\_n)^{-1}\boldsymbol{\eta}(A)\boldsymbol{\eta}(I\_n)^{-1} = \sum\_{i=0}^{-k-1} \boldsymbol{\eta}(I\_n)^i \boldsymbol{\eta}(A)\boldsymbol{\eta}(I\_n)^{-k-1-i}.\tag{36}$$

Therefore,

$$\begin{split} & -k \left( \boldsymbol{\nu}(\boldsymbol{A}) \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-1} - \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-1} \boldsymbol{\nu}(\boldsymbol{A}) \right) \\ &= \boldsymbol{\nu}(\boldsymbol{I\_{n}}) \left( \sum\_{i=0}^{-k-1} \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{i} \boldsymbol{\nu}(\boldsymbol{A}) \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-k-1-i} \right) - \left( \sum\_{i=0}^{-k-1} \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{i} \boldsymbol{\nu}(\boldsymbol{A}) \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-k-1-i} \right) \boldsymbol{\nu}(\boldsymbol{I\_{n}}) \\ &= \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-k} \boldsymbol{\nu}(\boldsymbol{A}) - \boldsymbol{\nu}(\boldsymbol{A}) \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-k} = \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-1} \boldsymbol{\nu}(\boldsymbol{A}) - \boldsymbol{\nu}(\boldsymbol{A}) \boldsymbol{\nu}(\boldsymbol{I\_{n}})^{-1}. \end{split} \tag{37}$$

We get *<sup>ψ</sup>*ð Þ *In* �<sup>1</sup> *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> <sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>ψ</sup>*ð Þ *In* �<sup>1</sup> for *<sup>A</sup>* <sup>∈</sup>M*n*. So *<sup>ψ</sup>*ð Þ *In* �<sup>1</sup> and *<sup>ψ</sup>*ð Þ *In* commute with the range of *ψ*. The following power series are equal for every *A* ∈M*<sup>n</sup>* when ∣*x*∣ is sufficiently small:

$$\left(\left(\boldsymbol{\nu}(I\_{n}+\boldsymbol{\chi}A)\right)^{k} = \boldsymbol{\nu}(I\_{n}) + \boldsymbol{\chi}k\boldsymbol{\nu}(I\_{n})^{k-1}\boldsymbol{\nu}(A) + \boldsymbol{\nu}^{2}\frac{k(k-1)}{2}\boldsymbol{\nu}(I\_{n})^{k-2}\boldsymbol{\nu}(A)^{2} + \cdots \tag{38}$$

$$\left(\psi\left(\left(I\_n + \varkappa A\right)^k\right) = \psi\left(I\_n\right) + \varkappa k \psi\left(A\right) + \varkappa^2 \frac{k(k-1)}{2} \psi\left(A^2\right) + \cdots \tag{39}$$

Equating degree two terms of (38) and (39), we get *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* <sup>2</sup> <sup>¼</sup> *<sup>ψ</sup> <sup>A</sup>*<sup>2</sup> � �*:* Let *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> <sup>ψ</sup>*ð Þ *In* �<sup>1</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* . Then *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>2</sup> <sup>¼</sup> *<sup>ψ</sup>*<sup>1</sup> *<sup>A</sup>*<sup>2</sup> � � and *<sup>ψ</sup>*<sup>1</sup> is nonzero. Using Theorem 3.1, we can get (27) and (39).

## **3.2 Trace of power-product preserers on** M*<sup>n</sup>* **and** M*n*ð Þ

Corollary 2.3 shows that every linear bijection *ϕ* : M*n*ð Þ! M*n*ð Þ corresponds to another linear bijection *ψ* : M*n*ð Þ! M*n*ð Þ such that *tr*ð Þ¼ *ϕ*ð Þ *A ψ*ð Þ *B tr AB* ð Þ for all *A*, *B*∈M*n*ð Þ . When *m* ≥3, maps *ϕ*1, ⋯, *ϕ<sup>m</sup>* on M*n*ð Þ that satisfy *tr*ð*ϕ*1ð Þ *A*<sup>1</sup> ⋯*ϕm*ð Þ *Am* Þ ¼ *tr A*ð Þ <sup>1</sup>⋯*Am* for *A*1, … , *Am* ∈M*n*ð Þ are determined in [18].

If two maps on M*n*ð Þ satisfy the following trace condition about *k*-powers, then they have specific forms.

Theorem 3.5. *Let* ¼ *or . Let k*∈nf g 0, 1 *. Let S be an open neighborhood of In consisting of invertible matrices. Then two maps ϕ*, *ψ* : M*n*ð Þ! M*n*ð Þ *satisfy that*

$$\text{tr}\left(\phi(A)\psi(B)^k\right) = \text{tr}\left(AB^k\right),\tag{40}$$

1.*for all A* ∈M*n*ð Þ , *B*∈ *S, and ψ is linear, or*

2.*for all A*, *B*∈ *S and both ϕ and ψ are linear,*

if and only if *ϕ* and *ψ* take the following forms:

*a. When k* ¼ �1*, there exist invertible matrices P*, *Q* ∈M*n*ð Þ *such that*

$$\begin{pmatrix} \phi(A) = PAQ \\ \varphi(B) = PBQ \end{pmatrix} \quad or \quad \begin{pmatrix} \phi(A) = PA^t Q \\ \varphi(B) = PB^t Q \end{pmatrix} \quad A, B \in \mathcal{M}\_n(\mathbb{F}). \tag{41}$$

*b. When k*∈n �f g 1, 0, 1 *, there exist c*∈ nf g0 *and an invertible matrix P*∈M*n*ð Þ *such that*

$$\begin{pmatrix} \phi(A) = c^{-k} P A P^{-1} \\ \Psi(B) = c P B P^{-1} \end{pmatrix} \quad \text{or} \begin{pmatrix} \phi(A) = c^{-k} P A^t P^{-1} \\ \Psi(B) = c P B^t P^{-1} \end{pmatrix} \quad A, B \in \mathcal{M}\_n(\mathbb{F}). \tag{42}$$

*Proof.* We prove the case ¼ ; the case ¼ can be done similarly.

Suppose assumption (2) holds. Then for every *A* ∈M*n*ð Þ , there exists *c*∈ nf g0 such that *In* � *cA* ∈*S*, so that for all *B* ∈*S*:

$$\operatorname{tr}\left(\boldsymbol{B}^{k}\right) = \operatorname{tr}\left(\left(\phi(I\_{n} - c\boldsymbol{A}) + c\phi(\boldsymbol{A})\right)\boldsymbol{\upmu}(\boldsymbol{B})^{k}\right) = \operatorname{tr}\left(\left(I\_{n} - c\boldsymbol{A}\right)\boldsymbol{B}^{k}\right) + c\operatorname{tr}\left(\phi(\boldsymbol{A})\boldsymbol{\upmu}(\boldsymbol{B})^{k}\right). \tag{43}$$

Thus tr *<sup>ϕ</sup>*ð Þ *<sup>A</sup> <sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>k</sup>* � � <sup>¼</sup> tr *ABk* � � for *<sup>A</sup>* <sup>∈</sup>M*n*ð Þ and *<sup>B</sup>*∈*S*, which leads to assumption (1).

Now we prove the theorem under assumption (1), that is, (40) holds for all *A* ∈M*n*ð Þ and *B*∈*S*, and *ψ* is linear. Only the necessary part is needed to prove.

Let *<sup>S</sup>*<sup>0</sup> <sup>¼</sup> *<sup>B</sup>*∈P*<sup>n</sup>* : *<sup>B</sup>*<sup>1</sup>*=<sup>k</sup>* <sup>∈</sup>*<sup>S</sup>* � �, which is an open neighborhood of *In* in <sup>P</sup>*n*. Define *<sup>ψ</sup>*<sup>~</sup> : *<sup>S</sup>*<sup>0</sup> ! <sup>M</sup>*<sup>n</sup>* such that *<sup>ψ</sup>*~ð Þ¼ *<sup>B</sup> <sup>ψ</sup> <sup>B</sup>*<sup>1</sup>*=<sup>k</sup>* � �*<sup>k</sup>* . Then (40) implies that

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

$$\operatorname{tr}(\phi(A)\bar{\varphi}(B)) = \operatorname{tr}\left(\phi(A)\varphi\left(B^{1/k}\right)^k\right) = \operatorname{tr}(AB), \quad A \in \mathcal{M}\_n, \ B \in \mathbb{S}'.\tag{44}$$

The complex span of *S*<sup>0</sup> is M*n*. By Theorem 2.2, *ϕ* is bijective linear, and *ψ*~ can be extended to a linear bijection on M*n*.

The linearity of *ψ* and (40) imply that for every *B*∈M*n*, there exists ϵ>0 such that *In* <sup>þ</sup> *xB*∈*<sup>S</sup>* and the power series of ð Þ *In* <sup>þ</sup> *xB <sup>k</sup>* converges whenever <sup>∣</sup>*x*<sup>∣</sup> <sup>&</sup>lt;ϵ. Then

$$\text{tr}\left(\phi(A)(\psi(I\_n) + \mathbf{x}\psi(B))^k\right) = \text{tr}\left(A(I\_n + \mathbf{x}B)^k\right), \quad A \in \mathcal{M}\_n, \ |\mathbf{x}| \ll \epsilon. \tag{45}$$

1. First suppose *k*≥ 2. Equating degree one terms and degree ð Þ *k* � 1 terms on both sides of (45) respectively, we get the following identities for *A*, *B*∈M*n*:

$$\operatorname{tr}\left(\phi(A)\left(\sum\_{i=0}^{k-1}\psi(I\_n)^{k-1-i}\psi(B)\psi(I\_n)^i\right)\right) = \operatorname{tr}(kAB),\tag{46}$$

$$\operatorname{tr}\left(\phi(A)\left(\sum\_{i=0}^{k-1}\psi(B)^{i}\psi(I\_{n})\psi(B)^{k-1-i}\right)\right) = \operatorname{tr}\left(kAB^{k-1}\right).\tag{47}$$

Let *Ci* : *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*<sup>2</sup> � � be a basis of projection matrices (i.e. *<sup>C</sup>*<sup>2</sup> *<sup>i</sup>* ¼ *Ci*) in M*n*. For example, we may choose the following basis of rank 1 projections:

$$\{E\_{ii} : 1 \le i \le n\} \cup \left\{ \frac{1}{\sqrt{2}} \left( E\_{ii} + E\_{\vec{\eta}} + \delta E\_{\vec{\eta}} + \overline{\delta} E\_{\vec{\eta}} \right) : 1 \le i < j \le n, \delta \in \{1, \mathbf{i}\} \right\}.\tag{48}$$

By (40) and (47), for *<sup>A</sup>* <sup>∈</sup>M*<sup>n</sup>* and *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*2,

$$\operatorname{tr}\left(k\phi(A)\varphi(\mathbf{C}\_i)^k\right) = \operatorname{tr}(kAC\_i) = \operatorname{tr}\left(\phi(A)\left(\sum\_{j=0}^{k-1}\varphi(\mathbf{C}\_i)^j\varphi(I\_n)\varphi(\mathbf{C}\_i)^{k-1-j}\right)\right). \tag{49}$$

By the bijectivity of *ϕ*,

$$k\mu \left(\mathbf{C}\_{i}\right)^{k} = \sum\_{j=0}^{k-1} \mu \left(\mathbf{C}\_{i}\right)^{j} \mu \left(I\right) \mu \left(\mathbf{C}\_{i}\right)^{k-1-j}.\tag{50}$$

Therefore, for *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*2,

$$\begin{split} 0 &= \boldsymbol{\nu}(\mathbf{C}\_{i}) \left( \sum\_{j=0}^{k-1} \boldsymbol{\nu}(\mathbf{C}\_{i})^{j} \boldsymbol{\nu}(I\_{n}) \boldsymbol{\nu}(\mathbf{C}\_{i})^{k-1-j} \right) - \left( \sum\_{j=0}^{k-1} \boldsymbol{\nu}(\mathbf{C}\_{i})^{j} \boldsymbol{\nu}(I\_{n}) \boldsymbol{\nu}(\mathbf{C}\_{i})^{k-1-j} \right) \boldsymbol{\nu}(\mathbf{C}\_{i}) \\ &= \boldsymbol{\nu}(\mathbf{C}\_{i})^{k} \boldsymbol{\nu}(I\_{n}) - \boldsymbol{\nu}(I\_{n}) \boldsymbol{\nu}(\mathbf{C}\_{i})^{k} . \end{split} \tag{51}$$

Since

$$\text{tr}\left(A\boldsymbol{\uprho}(\mathbf{C}\_{i})^{k}\right) = \text{tr}\left(\boldsymbol{\uprho}^{-1}(A)\mathbf{C}\_{i}^{k}\right) = \text{tr}\left(\boldsymbol{\uprho}^{-1}(A)\mathbf{C}\_{i}\right), \quad A \in \mathcal{M}\_{n}, \ i = 1, \ldots, n^{2},\tag{52}$$

the only matrix *A* ∈M*<sup>n</sup>* such that tr *Aψ*ð Þ *Ci <sup>k</sup>* � � <sup>¼</sup> 0 for all *<sup>i</sup>*<sup>∈</sup> 1, … , *<sup>n</sup>*<sup>2</sup> � � is the zero matrix. So *ψ*ð Þ *Ci <sup>k</sup>* : *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*<sup>2</sup> n o is a basis of <sup>M</sup>*n*. (51) implies that *<sup>ψ</sup>*ð Þ¼ *In cIn* for certain *c*∈ nf g0 .

(46) shows that

$$\varepsilon^{k-1}\text{tr}(\phi(A)\varphi(B)) = \text{tr}(AB), \quad A, B \in \mathcal{M}\_n. \tag{53}$$

Therefore,

$$\mathcal{L}^{k-1}\text{tr}\left(\phi(A)\psi\left(B^k\right)\right) = \text{tr}\left(AB^k\right) = \text{tr}\left(\phi(A)\psi\left(B\right)^k\right), \quad A \in \mathcal{M}\_n, \ B \in \mathbb{S}.\tag{54}$$

The bijectivity of *<sup>ϕ</sup>* shows that *ck*�<sup>1</sup>*<sup>ψ</sup> Bk* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>k</sup>* for *<sup>B</sup>* <sup>∈</sup>*S*, that is,

$$\mathcal{L}^{-1}\psi\left(\mathcal{B}^k\right) = \left[\mathcal{c}^{-1}\psi(\mathcal{B})\right]^k, \quad \mathcal{B} \in \mathcal{S}.\tag{55}$$

Notice that *<sup>c</sup>*�<sup>1</sup>*ψ*ð Þ¼ *In In*. By Theorem 3.4, there is an invertible *<sup>P</sup>*∈M*<sup>n</sup>* such that *<sup>ψ</sup>* is of the form *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> cPBP*�<sup>1</sup> or *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> cPBt P*�<sup>1</sup> for *B* ∈M*n*. Consequently, we get (42).

2. Now suppose *k*<0. Then *ψ*ð Þ *In* is invertible. For every *B*∈M*<sup>n</sup>* and sufficiently small *x*, we have the power series expansion:

$$\begin{split} & \quad \left(\boldsymbol{\Psi}(I\_{n}) + \mathbf{x}\boldsymbol{\eta}(B)\right)^{k} \\ &= \left[ \left(I\_{n} + \mathbf{x}\boldsymbol{\eta}(I\_{n})^{-1}\boldsymbol{\eta}(B)\right)^{-1}\boldsymbol{\eta}(I\_{n})^{-1} \right]^{|k|} \\ &= \left[ \left(I\_{n} - \mathbf{x}\boldsymbol{\eta}(I\_{n})^{-1}\boldsymbol{\eta}(B) + \mathbf{x}^{2}\boldsymbol{\eta}(I\_{n})^{-1}\boldsymbol{\eta}(B)\boldsymbol{\upnu}(I\_{n})^{-1}\boldsymbol{\nu}(B) + \cdots \right) \boldsymbol{\upnu}(I\_{n})^{-1} \right]^{|k|} \\ &= \boldsymbol{\upnu}(I\_{n})^{k} - \mathbf{x}\left(\sum\_{i=1}^{|k|} \boldsymbol{\upnu}(I\_{n})^{-i}\boldsymbol{\upnu}(B)\boldsymbol{\upnu}(I\_{n})^{k-1+i} \right) \\ & \quad + \mathbf{x}^{2} \left(\sum\_{i=1}^{|k|} \sum\_{j=1}^{|k|+1-i} \boldsymbol{\upnu}(I\_{n})^{-i}\boldsymbol{\upnu}(B)\boldsymbol{\upnu}(I\_{n})^{-j}\boldsymbol{\upnu}(B)\boldsymbol{\upnu}(I\_{n})^{k-2+i+j} \right) + \cdots \end{split} \tag{56}$$

Equating degree one terms and degree two terms of (45) respectively and using (56), we get the following identities for *A*, *B*∈M*n*:

$$\operatorname{tr}\left(\phi(A)\left(\sum\_{i=1}^{|k|}\psi(I\_n)^{-i}\psi(B)\psi(I\_n)^{k-1+i}\right)\right) = \operatorname{tr}(|k|AB),\tag{57}$$

$$\operatorname{tr}\left(\phi(A)\left(\sum\_{i=1}^{|k|}\sum\_{j=1}^{|k|+1-i}\psi(I\_n)^{-i}\psi(B)\psi(I\_n)^{-j}\psi(B)\psi(I\_n)^{k-2+i+j}\right)\right) = \operatorname{tr}\left(\frac{k(k-1)}{2}AB^2\right). \tag{58}$$

(57) and (40) imply that

$$\sum\_{i=1}^{|k|} \psi(I\_n)^{-i} \psi\left(B^k\right) \psi\left(I\_n\right)^{k-1+i} = \ |k| \psi\left(B\right)^k, \quad B \in \mathbb{S}.\tag{59}$$

Let *Fr*ð Þ *<sup>B</sup>* denote the degree *<sup>r</sup>* coefficient in the power series of ð Þ *<sup>ψ</sup>*ð Þþ *In <sup>x</sup>ψ*ð Þ *<sup>B</sup> <sup>k</sup>* . Then (57) and (58) show that:

$$\frac{k-1}{2}F\_1(\mathcal{B}^2) = F\_2(\mathcal{B}), \quad \mathcal{B} \in \mathcal{M}\_n. \tag{60}$$

Denote *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>ψ</sup>*ð Þ *In* �<sup>1</sup> . We discuss the cases *k* ¼ �1 and *k* 6¼ �1.

a. When *k* ¼ �1, (60) leads to

$$
\psi\left(B^2\right) = \psi(B)\psi\left(I\_n\right)^{-1}\psi\left(B\right), \quad B \in \mathcal{M}\_n. \tag{61}
$$

So *<sup>ψ</sup>*<sup>1</sup> *<sup>B</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>2</sup> for *<sup>B</sup>*∈M*n*. Note that *<sup>ψ</sup>*1ð Þ¼ *In In*. By Theorem 3.4, there exists an invertible *<sup>P</sup>*∈M*<sup>n</sup>* such that *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> PBP*�<sup>1</sup> or *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> PBt P*�<sup>1</sup> for *B*∈M*n*. Let *Q* ≔ *P*�<sup>1</sup> *<sup>ψ</sup>*ð Þ *In* . Then *<sup>Q</sup>* is invertible, and *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> PBQ* or *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> PB<sup>t</sup> Q* for *B* ∈M*n*. Using (40), we get (41).

b. Suppose the integer *k*< � 1. Then (60) implies that

$$\frac{k-1}{2}\left(\wp(I\_n)^{-1}F\_1(B^2) - F\_2(B^2)\wp(I\_n)^{-1}\right) = \wp(I\_n)^{-1}F\_2(B) - F\_2(B)\wp(I\_n)^{-1},\tag{62}$$

which gives

$$\begin{split} &\frac{1-k}{2}\left(\boldsymbol{\wp}(\boldsymbol{I}\_{n})^{k}\boldsymbol{\wp}\left(\boldsymbol{B}^{2}\right)-\boldsymbol{\wp}\left(\boldsymbol{B}^{2}\right)\boldsymbol{\wp}\left(\boldsymbol{I}\_{n}\right)^{k}\right) \\ &=\boldsymbol{\wp}\left(\boldsymbol{I}\_{n}\right)^{k}\boldsymbol{\wp}\left(\boldsymbol{B}\right)\boldsymbol{\wp}\left(\boldsymbol{I}\_{n}\right)^{-1}\boldsymbol{\wp}\left(\boldsymbol{B}\right)-\boldsymbol{\wp}\left(\boldsymbol{B}\right)\boldsymbol{\wp}\left(\boldsymbol{I}\_{n}\right)^{-1}\boldsymbol{\wp}\left(\boldsymbol{B}\right)\boldsymbol{\wp}\left(\boldsymbol{I}\_{n}\right)^{k}. \end{split} \tag{63}$$

In other words, for *B*∈M*n*:

$$\left(\boldsymbol{\nu}(I\_{\boldsymbol{n}})^{k}\left(\frac{1-k}{2}\boldsymbol{\nu}\_{1}(\boldsymbol{B}^{2})-\boldsymbol{\nu}\_{1}(\boldsymbol{B})^{2}\right) + \left(\frac{1-k}{2}\boldsymbol{\nu}\_{1}(\boldsymbol{B}^{2})-\boldsymbol{\nu}\_{1}(\boldsymbol{B})^{2}\right)\boldsymbol{\nu}(I\_{\boldsymbol{n}})^{k}.\tag{64}$$

Let *B* ¼ *In* þ *xE* for an arbitrary matrix *E* ∈M*n*. Then (64) becomes

$$\begin{split} & \left[ \boldsymbol{\nu}(I\_{n})^{k} \left[ \boldsymbol{\kappa}(-1-k)\boldsymbol{\nu}\_{1}(E) + \boldsymbol{\kappa}^{2} \left( \frac{1-k}{2}\boldsymbol{\nu}\_{1}(E^{2}) - \boldsymbol{\nu}\_{1}(E)^{2} \right) \right] \right] \\ &= \left[ \boldsymbol{\kappa}(-1-k)\boldsymbol{\nu}\_{1}(E) + \boldsymbol{\kappa}^{2} \left( \frac{1-k}{2}\boldsymbol{\nu}\_{1}(E^{2}) - \boldsymbol{\nu}\_{1}(E)^{2} \right) \right] \boldsymbol{\nu}(I\_{n})^{k}. \end{split} \tag{65}$$

The equality on degree one terms shows that *<sup>ψ</sup>*ð Þ *In <sup>k</sup>* commutes with all *<sup>ψ</sup>*1ð Þ *<sup>E</sup>* . Hence *<sup>ψ</sup>*ð Þ *In <sup>k</sup>* commutes with the range of *<sup>ψ</sup>*. (??) can be rewritten as

$$\text{tr}\left(\left(\sum\_{i=1}^{|k|}\psi(I\_n)^{k-1+i}\phi(A)\psi(I\_n)^{-i}\right)\psi(B)\right) = \text{tr}(|k|AB), \quad A, B \in \mathcal{M}\_n. \tag{66}$$

By Theorem 2.2, *<sup>ψ</sup>* is a linear bijection and its range is <sup>M</sup>*n*. So *<sup>ψ</sup>*ð Þ *In <sup>k</sup>* <sup>¼</sup> *<sup>μ</sup>In* for certain *μ*∈ .

Now by (59), for *B*∈*S*:

$$\begin{aligned} &|k|\boldsymbol{\mu}(I\_{n})\boldsymbol{\nu}(\boldsymbol{B})^{k} - |k|\boldsymbol{\nu}(\boldsymbol{B})^{k}\boldsymbol{\nu}(I\_{n}) \\ &= \boldsymbol{\nu}(I\_{n})\left(\sum\_{i=1}^{|k|}\boldsymbol{\nu}(I\_{n})^{-i}\boldsymbol{\nu}(\boldsymbol{B}^{k})\boldsymbol{\nu}(I\_{n})^{k-1+i}\right) - \left(\sum\_{i=1}^{|k|}\boldsymbol{\nu}(I\_{n})^{-i}\boldsymbol{\nu}(\boldsymbol{B}^{k})\boldsymbol{\nu}(I\_{n})^{k-1+i}\right)\boldsymbol{\nu}(I\_{n}) \\ &= \boldsymbol{\nu}(\boldsymbol{B}^{k})\boldsymbol{\nu}(I\_{n})^{k} - \boldsymbol{\nu}(I\_{n})^{k}\boldsymbol{\nu}\left(\boldsymbol{B}^{k}\right) = \mathbf{0} \end{aligned} \tag{67}$$

So *<sup>ψ</sup>*ð Þ *In* commutes with *<sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>k</sup>* for *<sup>B</sup>*∈*S*. In particular, *<sup>ψ</sup>*ð Þ *In* commutes with *<sup>ψ</sup>*~ð Þ¼ *<sup>B</sup> <sup>ψ</sup> <sup>B</sup>*1*=<sup>k</sup>* � �*<sup>k</sup>* for *<sup>B</sup>*∈*S*<sup>0</sup> . The complex span of *S*<sup>0</sup> is M*n*, and *ψ*~ can be extended to a linear bijection on M*n*. Hence *ψ*ð Þ¼ *In cIn* for certain *c*∈ nf g0 . By (59), we get *<sup>ψ</sup>*<sup>1</sup> *Bk* � � <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>B</sup> <sup>k</sup>* for *<sup>B</sup>* <sup>∈</sup>*S*. Note that *<sup>ψ</sup>*1ð Þ¼ *In In*. By Theorem 3.4, there is an invertible *<sup>P</sup>*∈M*<sup>n</sup>* such that *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> PBP*�<sup>1</sup> or *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> PB<sup>t</sup> P*�<sup>1</sup> . Then *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> cPBP*�<sup>1</sup> or *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> cPBt P*�<sup>1</sup> . Using (40), we get (42).

Remark 3.6 *The following modifications could be applied to the proof of Theorem 3.5 for* ¼ *:*


$$\begin{aligned} \{E\_{ii} : \mathbf{1} \le i \le n\} \cup \left\{ \frac{\mathbf{1}}{\sqrt{2}} \left( E\_{ii} + E\_{\vec{j}\bar{l}} + E\_{\vec{i}\bar{l}} + E\_{\vec{j}\bar{l}} \right) : \mathbf{1} \le i < j \le n \right\} \\ \cup \left\{ \alpha\_1 E\_{ii} + \alpha\_2 E\_{\vec{j}\bar{l}} + E\_{\vec{i}\bar{j}} - E\_{\vec{j}\bar{l}} : \mathbf{1} \le i < j \le n \right\}, \end{aligned} \tag{68}$$

in which *<sup>ω</sup>*1,*ω*<sup>2</sup> are distinct roots of *<sup>x</sup>*<sup>2</sup> � *<sup>x</sup>* � <sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*.

The arguments in the above proof will be applied analogously to maps on the other sets discussed in this paper.

## **4.** *k***-power linear preservers and trace of power-product preservers on** H*<sup>n</sup>*

### **4.1** *k***-power linear preservers on** H*<sup>n</sup>*

We give a result that determine linear operators on <sup>H</sup>*<sup>n</sup>* that satisfy *<sup>ψ</sup> Ak* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* on a neighborhood of *In* in H*<sup>n</sup>* for certain *k*∈nf g 0, 1

Theorem 4.1. *Fix k*∈nf g 0, 1 *. A nonzero linear map ψ* : H*<sup>n</sup>* ! H*<sup>n</sup> satisfies that*

$$
\psi\left(A^k\right) = \psi\left(A\right)^k\tag{69}
$$

on an open neighborhood of *In* consisting of invertible matrices if and only if *ψ* is of the following forms for certain unitary matrix *U* ∈M*n*:

1.*When k is even,*

*<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> <sup>U</sup>* <sup>∗</sup> *AU*, *<sup>A</sup>* <sup>∈</sup> <sup>H</sup>*n*; *or <sup>ψ</sup>*ð Þ¼ *<sup>A</sup> <sup>U</sup>* <sup>∗</sup> *<sup>A</sup><sup>t</sup> U*, *A* ∈ H*n:* (70) 2.*When k is odd,*

$$
\psi(A) = \pm U^\* A U, \quad A \in \mathcal{H}\_n; \quad \text{or} \quad \psi(A) = \pm U^\* A^t U, \quad A \in \mathcal{H}\_n. \tag{71}
$$

*Proof.* It suffices to prove the necessary part. Suppose (69) holds on an open neighborhood *S* of *In* in H*n*.

1. First assume *k*≥ 2. Replacing M*<sup>n</sup>* by H*<sup>n</sup>* in part (1) of the proof of Theorem 3.4 up to (33), we can prove that *ψ*ð Þ *In* commutes with the range of *ψ*, and *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* is a nonzero linear map that satisfies *<sup>ψ</sup>*<sup>1</sup> *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>2</sup> for *A* ∈ H*n*.

Every matrix in M*<sup>n</sup>* can be uniquely expressed as *A* þ **i***B* for *A*, *B* ∈ H*n*. Extend *ψ*<sup>1</sup> to a map *ψ*~ : M*<sup>n</sup>* ! M*<sup>n</sup>* such that

$$
\bar{\boldsymbol{\varphi}}(\boldsymbol{A} + \mathbf{i}\mathbf{B}) = \boldsymbol{\varphi}\_1(\boldsymbol{A}) + \mathbf{i}\boldsymbol{\varphi}\_1(\boldsymbol{B}), \quad \boldsymbol{A}, \boldsymbol{B} \in \mathcal{H}\_n. \tag{72}
$$

It is straightforward to check that *ψ*~ is a complex linear bijection. Moreover, for *A*, *B*∈ H*n*,

$$\begin{aligned} \left(\boldsymbol{\nu}\_{1}(\boldsymbol{AB}+\boldsymbol{BA}) &= \boldsymbol{\nu}\_{1}\Big(\left(\boldsymbol{A}+\boldsymbol{B}\right)^{2}\Big) - \boldsymbol{\nu}\_{1}\Big(\boldsymbol{A}^{2}\Big) - \boldsymbol{\nu}\_{1}\Big(\boldsymbol{B}^{2}\Big) \\ &= \boldsymbol{\nu}\_{1}(\boldsymbol{A}+\boldsymbol{B})^{2} - \boldsymbol{\nu}\_{1}(\boldsymbol{A})^{2} - \boldsymbol{\nu}\_{1}(\boldsymbol{B})^{2} \\ &= \boldsymbol{\nu}\_{1}(\boldsymbol{A})\boldsymbol{\nu}\_{1}(\boldsymbol{B}) + \boldsymbol{\nu}\_{1}(\boldsymbol{B})\boldsymbol{\nu}\_{1}(\boldsymbol{A}). \end{aligned}$$

It implies that

$$
\tilde{\boldsymbol{\psi}} \left( (\boldsymbol{A} + \mathbf{i}\boldsymbol{B})^2 \right) = \tilde{\boldsymbol{\psi}} (\boldsymbol{A} + \mathbf{i}\boldsymbol{B})^2, \quad \boldsymbol{A}, \boldsymbol{B} \in \mathcal{H}\_n.
$$

By Theorem 3.1, there is an invertible matrix *U* ∈M*<sup>n</sup>* such that

a. *<sup>ψ</sup>*~ð Þ¼ *<sup>A</sup> UAU*�<sup>1</sup> for all *<sup>A</sup>* <sup>∈</sup>M*n*, or

b. *<sup>ψ</sup>*~ð Þ¼ *<sup>A</sup> UA<sup>t</sup> U*�<sup>1</sup> for all *A* ∈M*n*.

First suppose *<sup>ψ</sup>*~ð Þ¼ *<sup>A</sup> UAU*�<sup>1</sup> . The restriction of *ψ*~ on H*<sup>n</sup>* is *ψ*<sup>1</sup> : H*<sup>n</sup>* ! H*n*. Hence for *<sup>A</sup>* <sup>∈</sup> <sup>H</sup>*n*, we have *UAU*�<sup>1</sup> <sup>¼</sup> *UAU*�<sup>1</sup> <sup>∗</sup> <sup>¼</sup> *<sup>U</sup>*� <sup>∗</sup> *AU* <sup>∗</sup> ; then *<sup>U</sup>* <sup>∗</sup> *UA* <sup>¼</sup> *AU* <sup>∗</sup> *<sup>U</sup>* for all *<sup>A</sup>* <sup>∈</sup> <sup>H</sup>*n*, which shows that *<sup>U</sup>* <sup>∗</sup> *<sup>U</sup>* <sup>¼</sup> *cIn* for certain *<sup>c</sup>*<sup>∈</sup> þ*:* By adjusting a scalar if necessary, we may assume that *<sup>U</sup>* is unitary. So *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> UAU* <sup>∗</sup> . Then *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>1</sup> <sup>¼</sup> *In*, so that *<sup>ψ</sup>*ð Þ¼ *In In* when *<sup>k</sup>* is even and *<sup>ψ</sup>*ð Þ *In* <sup>∈</sup>f g *In*, �*In* when *<sup>k</sup>* is odd. Thus *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> UAU* <sup>∗</sup> when *<sup>k</sup>* is even and *<sup>ψ</sup>*ð Þ¼� *<sup>A</sup> UAU* <sup>∗</sup> when *<sup>k</sup>* is odd. Similarly for the case *<sup>ψ</sup>*~ð Þ¼ *<sup>A</sup> UA<sup>t</sup> U*�<sup>1</sup> . Therefore, (70) or (71) holds.

2.Now assume that *k*<0. Replacing M*<sup>n</sup>* by H*<sup>n</sup>* in part (2) of the proof of Theorem 3.4, we can show that *ψ*ð Þ *In* commutes with the range of *ψ*, and furthermore the nonzero linear map *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *In* �<sup>1</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup>* satisfies that *<sup>ψ</sup>*<sup>1</sup> *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>2</sup> . By arguments in the preceding paragraphs, we get (70) or (71).

## **4.2 Trace of power-product preservers on** H*<sup>n</sup>*

By Corollary 2.3, every linear bijection *ϕ* : H*<sup>n</sup>* ! H*<sup>n</sup>* corresponds to another linear bijection *ψ* : H*<sup>n</sup>* ! H*<sup>n</sup>* such that trð Þ¼ *ϕ*ð Þ *A ψ*ð Þ *B* trð Þ *AB* for all *A*, *B* ∈ H*n*. When *m* ≥3, linear maps *ϕ*1, ⋯, *ϕ<sup>m</sup>* : H*<sup>n</sup>* ! H*<sup>n</sup>* that satisfy trð*ϕ*1ð Þ *A*<sup>1</sup> ⋯*ϕm*ð Þ *Am* Þ ¼ trð Þ *A*1⋯*Am* are characterized in [18].

Theorem 4.2. *Let k*∈nf g 0, 1 *. Let S be an open neighborhood of In in* H*<sup>n</sup> consisting of invertible Hermitian matrices. Then two maps ϕ*, *ψ* : H*<sup>n</sup>* ! H*<sup>n</sup> satisfy that*

$$\text{tr}\left(\phi(A)\psi(B)^k\right) = \text{tr}\left(AB^k\right),\tag{73}$$

1.*for all A* ∈ H*n*, *B* ∈*S, and ψ is linear, or*

2.*for all A*, *B*∈ *S and both ϕ and ψ are linear,*

if and only if *ϕ* and *ψ* take the following forms:

a. *When k* ¼ �1*, there exist an invertible matrix P*∈M*<sup>n</sup> and c*∈f g 1, �1 *such that*

$$\begin{pmatrix} \phi(A) = cP^\*AP \\ \psi(B) = cP^\*BP \end{pmatrix} \quad A, B \in \mathcal{H}\_n; \quad \text{or} \quad \begin{pmatrix} \phi(A) = cP^\*A^tP \\ \psi(B) = cP^\*B^tP \end{pmatrix} \quad A, B \in \mathcal{H}\_n. \tag{74}$$

b. *When k*∈n �f g 1, 0, 1 *, there exist a unitary matrix U* ∈M*<sup>n</sup> and c*∈ nf g0 *such that*

$$\begin{cases} \phi(A) = c^{-k} U^\* A U \\ \psi(B) = c U^\* B U \end{cases} \text{ A, B \in \mathcal{H}\_n;} \quad \text{or} \quad \begin{pmatrix} \phi(A) = c^{-k} U^\* A^t U \\ \psi(B) = c U^\* B^t U \end{pmatrix} \quad \text{A, B \in \mathcal{H}\_n.} \tag{75}$$

*Proof.* Assumption (2) leads to assumption (1) (cf. the proof of Theorem 3.5). We prove the theorem under assumption (1). It suffices to prove the necessary part.


For the case *k* ¼ �1, the equality corresponding to (60) can be simplified as

$$
\psi\left(B^2\right) = \psi(B)\psi\left(I\_n\right)^{-1}\psi\left(B\right), \quad B \in \mathcal{H}\_n. \tag{76}
$$

Let *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *In* �<sup>1</sup> *ψ*ð Þ *B* . Then *ψ*<sup>1</sup> : H*<sup>n</sup>* ! M*<sup>n</sup>* is a nonzero real linear map that satisfies *<sup>ψ</sup>*<sup>1</sup> *<sup>B</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>2</sup> for *<sup>B</sup>*<sup>∈</sup> <sup>H</sup>*n*. Extend *<sup>ψ</sup>*<sup>1</sup> to a complex linear map *<sup>ψ</sup>*<sup>~</sup> : <sup>M</sup>*<sup>n</sup>* ! M*<sup>n</sup>* such that

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

$$
\bar{\boldsymbol{\varphi}}(\boldsymbol{A} + \mathbf{i}B) \coloneqq \boldsymbol{\varphi}\_1(\boldsymbol{A}) + \mathbf{i}\boldsymbol{\varphi}\_1(B), \quad \boldsymbol{A}, B \in \mathcal{H}\_n. \tag{77}
$$

Similarly to the arguments in part (1) of the proof of Theorem 4.1, we have *<sup>ψ</sup>*<sup>~</sup> ð Þ *<sup>A</sup>* <sup>þ</sup> **<sup>i</sup>***<sup>B</sup>* <sup>2</sup> � � <sup>¼</sup> ð Þ *<sup>ψ</sup>*~ð Þ *<sup>A</sup>* <sup>þ</sup> **<sup>i</sup>***<sup>B</sup>* <sup>2</sup> for all *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>H</sup>*n*. Using Theorem 3.4 and the fact that *ψ*~ð Þ¼ *In ψ*1ð Þ¼ *In In*, we can prove that there is an invertible *P*∈M*<sup>n</sup>* such that for all *<sup>B</sup>*<sup>∈</sup> <sup>H</sup>*n*, either *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> <sup>P</sup>*�<sup>1</sup> *BP* or *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> <sup>P</sup>*�<sup>1</sup> *Bt P*. So

$$
\psi(B) = \psi(I\_n) P^{-1} B P, \quad B \in \mathcal{H}\_n; \quad \text{or} \tag{78}
$$

$$
\psi(B) = \psi(I\_n) P^{-1} B^t P, \quad B \in \mathcal{H}\_n. \tag{79}
$$

If *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> <sup>ψ</sup>*ð Þ *In <sup>P</sup>*�<sup>1</sup> *BP* for *<sup>B</sup>* <sup>∈</sup> <sup>H</sup>*n*, then *<sup>ψ</sup>*ð Þ *In <sup>P</sup>*�<sup>1</sup> *BP* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *In <sup>P</sup>*�<sup>1</sup> *BP* � � <sup>∗</sup> <sup>¼</sup> *<sup>P</sup>*<sup>∗</sup> *BP*� <sup>∗</sup> *<sup>ψ</sup>*ð Þ *In* , which gives

$$B\left(P^{-\*}\,\varphi(I\_n)P^{-1}\right)B = B\left(P^{-\*}\,\varphi(I\_n)P^{-1}\right), \quad B \in \mathcal{H}\_n. \tag{80}$$

Hence *<sup>P</sup>*� <sup>∗</sup> *<sup>ψ</sup>*ð Þ *In <sup>P</sup>*�<sup>1</sup> <sup>¼</sup> *cIn* for certain *<sup>c</sup>*<sup>∈</sup> nf g<sup>0</sup> *:* We have *<sup>ψ</sup>*ð Þ¼ *In cP*<sup>∗</sup> *<sup>P</sup>* so that *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> cP*<sup>∗</sup> *BP* for *<sup>B</sup>*<sup>∈</sup> <sup>H</sup>*n*. Similarly for the case *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> <sup>ψ</sup>*ð Þ *In <sup>P</sup>*�<sup>1</sup> *Bt P*. Adjusting *c* and *P* by scalar factors simultaneously, we may assume that *c*∈ f g 1, �1 . It implies (74).

Remark 4.3. *Theorem 4.2 does not hold if ψ is not assumed to be linear. Let k be a positive even integer. Let ψ*~ : H*<sup>n</sup>* ! H*<sup>n</sup> be any bijective linear map such that ψ*~ P*<sup>n</sup>* � �⊆P*n. For example, ψ*~ *may be a completely positive map of the form <sup>ψ</sup>*~ð Þ¼ *<sup>B</sup>* <sup>P</sup>*<sup>r</sup> <sup>i</sup>*¼<sup>1</sup>*<sup>N</sup>* <sup>∗</sup> *<sup>i</sup> BNi for r*≥2*, N*1, … , *Nr* ∈M*<sup>n</sup> linearly independent, and at least one of N*1, … , *Nr is invertible. By Corollary 2.3, there is a linear bijection ϕ* : H*<sup>n</sup>* ! H*<sup>n</sup> such that* trð Þ¼ *ϕ*ð Þ *A ψ*~ð Þ *B* trð Þ *AB for all A*, *B* ∈ H*n. Let ψ* : H*<sup>n</sup>* ! H*<sup>n</sup> be defined by <sup>ψ</sup>*ð Þ¼ *<sup>B</sup> <sup>ψ</sup>*<sup>~</sup> *Bk* � �<sup>1</sup>*=<sup>k</sup> : Then*

$$\operatorname{tr}\left(\phi(A)\psi(B)^k\right) = \operatorname{tr}\left(\phi(A)\check{\psi}\left(B^k\right)\right) = \operatorname{tr}\left(AB^k\right), \quad A, B \in \mathcal{H}\_n.$$

Obviously, *ψ* may be non-linear, and the choices of pairs ð Þ *ϕ*, *ψ* are much more than those in (74) and (75).

## **5.** *k***-power linear preservers and trace of power-product preservers on** S*<sup>n</sup>* **and** S*n*ð Þ

## **5.1** *k***-power linear preservers on** S*<sup>n</sup>* **and** S*n*ð Þ

Chan and Lim described the linear *k*-power preservers on S*n*ð Þ for *k*≥2 in [7, Theorem 2] as follows.

Theorem 5.1. *(Chan, Lim* [5]*) Let an integer k*≥2*. Let be an algebraic closed field with char*ð Þ¼ 0 *or char*ð Þ >*k. Suppose that ψ* : S*n*ð Þ! S*n*ð Þ *is a nonzero linear operator such that <sup>ψ</sup> Ak* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A* <sup>∈</sup>S*n*ð Þ *. Then there exist <sup>λ</sup>*<sup>∈</sup> *with <sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> *and an orthogonal matrix O* ∈M*n*ð Þ *such that*

$$
\psi(A) = \lambda OAO^t, \quad A \in \mathcal{S}\_n. \tag{81}
$$

We generalize Theorem 5.1 to include the case S*n*ð Þ , to include negative integers *k*, and to assume the *k*-power preserving condition only on matrices nearby the identity.

Theorem 5.2. *Let k*∈nf g 0, 1 *. Let* ¼ *or . Suppose that ψ* : S*n*ð Þ! S*n*ð Þ *is a nonzero linear map such that <sup>ψ</sup> <sup>A</sup><sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A in an open neighborhood of In in* <sup>S</sup>*n*ð Þ *consisting of invertible matrices. Then there exist <sup>λ</sup>*∈ *with <sup>λ</sup>k*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> *and an orthogonal matrix O* ∈M*n*ð Þ *such that*

$$
\psi(A) = \lambda OAO^t, \quad A \in \mathcal{S}\_n(\mathbb{F}).\tag{82}
$$

*Proof.* It suffices to prove the necessary part. In both *k*≥2 and *k*< 0 cases, using analogous arguments as parts (1) and (2) of the proof of Theorem 3.4, we get that *<sup>ψ</sup>*ð Þ *In* commutes with the range of *<sup>ψ</sup>*, and the nonzero map *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> *ψ*ð Þ *A* satisfies that *<sup>ψ</sup>*<sup>1</sup> *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>2</sup> for *<sup>A</sup>* <sup>∈</sup>S*n*ð Þ . Then

$$\begin{split} \left(\boldsymbol{\nu}\_{1}(\boldsymbol{A})\boldsymbol{\nu}\_{1}(\boldsymbol{B}) + \boldsymbol{\nu}\_{1}(\boldsymbol{B})\boldsymbol{\nu}\_{1}(\boldsymbol{A}) = \boldsymbol{\nu}\_{1}(\boldsymbol{A}+\boldsymbol{B})^{2} - \boldsymbol{\nu}\_{1}(\boldsymbol{A})^{2} - \boldsymbol{\nu}\_{1}(\boldsymbol{B})^{2} \\ = \boldsymbol{\nu}\_{1}\left(\left(\boldsymbol{A}+\boldsymbol{B}\right)^{2}\right) - \boldsymbol{\nu}\_{1}\left(\boldsymbol{A}^{2}\right) - \boldsymbol{\nu}\_{1}\left(\boldsymbol{B}^{2}\right) \\ = \boldsymbol{\nu}\_{1}(\boldsymbol{A}\boldsymbol{B}+\boldsymbol{B}\boldsymbol{A}). \end{split} \tag{83}$$

In particular, *<sup>ψ</sup>*1ð Þ *<sup>A</sup> <sup>ψ</sup>*<sup>1</sup> *Ar* ð Þþ *<sup>ψ</sup>*<sup>1</sup> *Ar* ð Þ*ψ*1ð Þ¼ *<sup>A</sup>* <sup>2</sup>*ψ*<sup>1</sup> *Ar*þ<sup>1</sup> for *<sup>r</sup>*<sup>∈</sup> þ. Using induction, we get *<sup>ψ</sup>*<sup>1</sup> *<sup>A</sup>*<sup>ℓ</sup> <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>ℓ</sup> for all *<sup>A</sup>* <sup>∈</sup>S*n*ð Þ and <sup>ℓ</sup>∈þ. By [26, Corollary 6.5.4], there is an orthogonal matrix *<sup>O</sup>* <sup>∈</sup>M*n*ð Þ such that *<sup>ψ</sup>*1ð Þ¼ *<sup>A</sup> OAO<sup>t</sup>* . Since *ψ*ð Þ *In* commutes with the range of *<sup>ψ</sup>*1, we have *<sup>ψ</sup>*ð Þ¼ *In <sup>λ</sup>In* for certain *<sup>λ</sup>*∈ in which *<sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> 1. So *<sup>ψ</sup>*ð Þ¼ *<sup>A</sup> λOAOt* as in (82).

Obviously, in ¼ case, (82) has *λ* ¼ 1 when *k* is even and *λ*∈f g 1, �1 when *k* is odd.

## **5.2 Trace of power-product preservers on** S*<sup>n</sup>* **and** S*n*ð Þ

Corollary 2.3 shows that every linear bijection *ϕ* : S*n*ð Þ! S*n*ð Þ corresponds to another linear bijection *ψ* : S*n*ð Þ! S*n*ð Þ such that trð Þ¼ *ϕ*ð Þ *A ψ*ð Þ *B* trð Þ *AB* for all *A*, *B*∈ S*n*ð Þ . When *m* ≥3, maps *ϕ*1, ⋯, *ϕ<sup>m</sup>* : S*n*ð Þ! S*n*ð Þ that satisfy trð*ϕ*1ð Þ *A*<sup>1</sup> ⋯*ϕm*ð Þ *Am* Þ ¼ trð Þ *A*1⋯*Am* are determined in [18].

We characterize the trace of power-product preserver for S*n*ð Þ here.

Theorem 5.3. *Let* ¼ *or . Let k*∈nf g 0, 1 *. Let S be an open neighborhood of In in* S*n*ð Þ *consisting of invertible matrices. Then two maps ϕ*, *ψ* : S*n*ð Þ! S*n*ð Þ *satisfy that*

$$\text{tr}\left(\phi(A)\psi(B)^k\right) = \text{tr}\left(AB^k\right),\tag{84}$$

1.*for all A* ∈S*n*ð Þ , *B*∈ *S, and ψ is linear, or*

### 2.*for all A*, *B*∈ *S and both ϕ and ψ are linear,*

if and only if *ϕ* and *ψ* take the following forms:

a. *When k* ¼ �1*, there exist an invertible matrix P*∈M*n*ð Þ *and c*∈nf g0 *such that*

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

$$
\phi(A) = cPAP^\sharp, \quad \psi(B) = cPBP^\sharp, \quad A, B \in \mathcal{S}\_n(\mathbb{F}).\tag{85}
$$

We may choose *c* ¼ 1 for ¼ and *c*∈f g 1, �1 for ¼ .

b. *When k*∈n �f g 1, 0, 1 *, there exist c*∈nf g0 *and an orthogonal matrix O* ∈M*n*ð Þ *such that*

$$
\phi(A) = c^{-k} OAO^{\ell}, \quad \psi(B) = c OBO^{\ell}, \quad A, B \in \mathcal{S}\_n(\mathbb{F}).\tag{86}
$$

*Proof.* Assumption (2) leads to assumption (1) (cf. the proof of Theorem 3.5). We prove the theorem under assumption (1). It suffices to prove the necessary part.

Obviously, S*n*∩H*<sup>n</sup>* ¼ S*n*ð Þ and S*n*∩P*<sup>n</sup>* ¼ P*n*ð Þ . Let *S*<sup>0</sup> <sup>≔</sup> *<sup>B</sup>* <sup>∈</sup>P*n*ð Þ : *<sup>B</sup>*1*=<sup>k</sup>* <sup>∈</sup>*<sup>S</sup>* � �, which is an open neighborhood of *In* in P*n*ð Þ and whose real (resp. complex) span is S*n*ð Þ (resp. S*n*). Using an analogous argument of the proof of Theorem 3.5, and replacing M*<sup>n</sup>* by S*n*ð Þ , replacing the basis (48) of M*<sup>n</sup>* by the following basis of rank 1 projections in S*n*ð Þ :

$$\{E\_{ii} : \mathbf{1} \le i \le n\} \cup \left\{ \frac{\mathbf{1}}{\sqrt{2}} \left( E\_{ii} + E\_{\vec{\eta}} + E\_{\vec{\eta}} + E\_{\vec{\mu}} \right) : \mathbf{1} \le i < j \le n \right\},\tag{87}$$

and replacing the usage of Theorem 3.4 by that of Theorem 5.2, we can prove the case *k*≥2, and for *k*< 0, we can get the corresponding equalities up to (60).

Define a linear map *<sup>ψ</sup>*<sup>1</sup> : <sup>S</sup>*n*ð Þ! <sup>M</sup>*n*ð Þ by *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>ψ</sup>*ð Þ *In* �<sup>1</sup>

When *<sup>k</sup>* ¼ �1, we get the corresponding equality of (61), so that *<sup>ψ</sup>*<sup>1</sup> *<sup>B</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>2</sup> for *<sup>B</sup>*<sup>∈</sup> <sup>S</sup>*n*ð Þ . Similar to the proof of Theorem 5.2, we get *<sup>ψ</sup>*<sup>1</sup> *Br* ð Þ¼ *<sup>ψ</sup>*1ð Þ *<sup>B</sup> <sup>r</sup>* for all *r*∈þ. By [26, Theorem 6.5.3], there is an invertible matrix *P*∈M*n*ð Þ such that *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> PBP*�<sup>1</sup> , so that *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> PBP*�<sup>1</sup> *<sup>ψ</sup>*ð Þ *In* for *<sup>B</sup>*<sup>∈</sup> <sup>S</sup>*n*ð Þ . Since *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> <sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>t</sup>* , we get

$$B\left(P^{-1}\varphi(I\_n)P^{-t}\right)B = B\left(P^{-1}\varphi(I\_n)P^{-t}\right), \quad B \in \mathcal{S}\_n(\mathbb{F}).\tag{88}$$

.

Therefore, *P*�<sup>1</sup> *<sup>ψ</sup>*ð Þ *In <sup>P</sup>*�*<sup>t</sup>* <sup>¼</sup> *cIn* for certain *<sup>c</sup>*∈nf g<sup>0</sup> , so that *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> cPBP<sup>t</sup>* for all *B*∈ S*n*ð Þ . Consequently, we get (85). The remaining claims are obvious.

When *k*< � 1, using analogous argument as in the proof of *k*< � 1 case of Theorem 3.5 and applying Theorem 5.2, we can get (86).

## **6.** *k***-power linear preservers and trace of power-product preservers on** P*<sup>n</sup>* **and** P*n*ð Þ

In this section, we will determine *k*-power linear preservers and trace of powerproduct preservers on maps P*<sup>n</sup>* ! P*<sup>n</sup>* (resp. P*n*ð Þ! P*n*ð Þ ). Properties of such maps can be applied to maps P*<sup>n</sup>* ! P*<sup>n</sup>* and P*<sup>n</sup>* ! P*<sup>n</sup>* (resp. P*n*ð Þ! P*n*ð Þ and P*n*ð Þ! P*n*ð Þ ).

## **6.1** *k***-power linear preservers on** P*<sup>n</sup>* **and** P*n*ð Þ

Theorem 6.1. *Fix k*∈nf g 0, 1 *. A nonzero linear map ψ* : P*<sup>n</sup>* ! P*<sup>n</sup> (resp. ψ* : P*n*ð Þ! P*n*ð Þ *) satisfies that*

$$
\psi\left(A^k\right) = \psi\left(A\right)^k\tag{89}
$$

on an open neighborhood of *In* in P*<sup>n</sup>* (resp. P*n*ð Þ ) if and only if there is a unitary (resp. real orthogonal) matrix *U* ∈ *Mn* such that

$$
\psi(A) = U^\* A U, \quad A \in \mathcal{P}\_n; \quad \text{or} \quad \psi(A) = U^\* A^t U, \quad A \in \mathcal{P}\_n. \tag{90}
$$

*Proof.* We prove the case *ψ* : P*<sup>n</sup>* ! P*n*. The sufficient part is obvious. About the necessary part, the nonzero linear map *ψ* : P*<sup>n</sup>* ! P*<sup>n</sup>* can be easily extended to a linear map *<sup>ψ</sup>*<sup>~</sup> : <sup>H</sup>*<sup>n</sup>* ! <sup>H</sup>*<sup>n</sup>* that satisfies *<sup>ψ</sup>*<sup>~</sup> *<sup>A</sup><sup>k</sup>* � � <sup>¼</sup> *<sup>ψ</sup>*~ð Þ *<sup>A</sup> <sup>k</sup>* on an open neighborhood of *In*. By Theorem 4.1, we immediately get (90).

The case *ψ* : P*n*ð Þ! P*n*ð Þ can be similarly proved using Theorem 5.2.

## **6.2 Trace of powered product preservers on** P*<sup>n</sup>* **and** P*n*ð Þ

Now consider the maps P*<sup>n</sup>* ! P*<sup>n</sup>* (resp. P*n*ð Þ! P*n*ð Þ ) that preserve trace of powered products. Unlike M*<sup>n</sup>* and H*n*, the set P*<sup>n</sup>* (resp. P*n*ð Þ ) is not a vector space. The trace of powered product preservers of two maps have the following forms.

Theorem 6.2 (Huang, Tsai [18]). *Let a*, *b*,*c*, *d*∈ nf g0 *. Two maps ϕ*, *ψ* : P*<sup>n</sup>* ! P*<sup>n</sup> satisfy*

$$\operatorname{tr}\left(\phi(A)^{a}\,\mu(B)^{b}\right) = \operatorname{tr}\left(A^{c}B^{d}\right), \quad A, B \in \mathcal{P}\_{n},\tag{91}$$

if and only if there exists an invertible *P*∈M*<sup>n</sup>* such that

$$\begin{pmatrix} \phi(A) = (P^\* A^c P)^{1/a} \\ \nu(B) = \left(P^{-1} B^d P^{-\*}\right)^{1/b} \end{pmatrix} \begin{pmatrix} \phi(A) = \left[P^\* \left(A^t\right)^c P\right]^{1/a} \\ \nu(B) = \left[P^{-1} (B^t)^d P^{-\*}\right]^{1/b} \end{pmatrix} \quad A, B \in \mathcal{P}\_n. \tag{92}$$

Theorem 6.3 (Huang, Tsai [18]). *Given an integer m* ≥3 *and real numbers α*1, … , *αm*, *β*1, … , *β<sup>m</sup>* ∈ nf g0 *, maps ϕ<sup>i</sup>* : P*<sup>n</sup>* ! P*<sup>n</sup> (i* ¼ 1, … , *m) satisfy that*

$$\operatorname{tr}(\phi\_1(A\_1)^{a\_1}\cdots\phi\_m(A\_m)^{a\_m}) = \operatorname{tr}\left(A\_1^{\beta\_1}\cdots A\_m^{\beta\_1}\right), \quad A\_1,\ldots,A\_m \in \mathcal{P}\_n,\tag{93}$$

if and only if they have the following forms for certain *c*1, … ,*cm* ∈ <sup>þ</sup> with *c*1⋯*cm* ¼ 1:

1.*When m is odd, there exists a unitary matrix U* ∈M*<sup>n</sup> such that for i* ¼ 1, … , *m:*

$$\phi\_i(A) = c\_i^{1/a\_i} U^\* A^{\beta\_i/a\_i} U, \quad A \in \mathcal{P}\_n. \tag{94}$$

2.*When m is even, there exists an invertible M* ∈M*<sup>n</sup> such that for i* ¼ 1, … , *m:*

$$\phi\_i(A) = \begin{pmatrix} c\_i^{1/a\_i} \left(\mathbf{M}^\* \mathbf{A}^{\beta\_i} \mathbf{M}\right)^{1/a\_i}, & i \text{ is odd},\\ c\_i^{1/a\_i} \left(\mathbf{M}^{-1} \mathbf{A}^{\beta\_i} \mathbf{M}^{-\*}\right)^{1/a\_i}, & i \text{ is even}, \end{pmatrix} \tag{95}$$

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

Both Theorems 6.2 and 6.3 can be analogously extended to maps P*n*ð Þ! P*n*ð Þ without difficulties.

Theorem 6.2 determines maps *ϕ*, *ψ* : P*<sup>n</sup>* ! P*<sup>n</sup>* that satisfy (91) throughout their domain. If we only assume the equality (91) for ð Þ *A*, *B* in certain subset of P*<sup>n</sup>* � P*<sup>n</sup>* and assume certain linearity of *ϕ* and *ψ*, then *ϕ* and *ψ* may have slightly different forms. We determine the case *a* ¼ *c* ¼ 1 and *b* ¼ *d* ¼ *k*∈nf g0 here.

Theorem 6.4. *Let k*∈nf g0 *. Let S be an open neighborhood of In in* P*n. Two maps ϕ*, *ψ* : P*<sup>n</sup>* ! P*<sup>n</sup> satisfy*

$$\text{tr}\left(\phi(A)\psi(B)^k\right) = \text{tr}\left(AB^k\right),\tag{96}$$

3.*for all A*, *B*∈P*n, or*

4.*for all A* ∈*S*, *B* ∈P*n, and ϕ is linear,*

if and only if there exists an invertible *P*∈M*<sup>n</sup>* such that

$$\begin{pmatrix} \phi(A) = P^\* A P \\\\ \nu(B) = \left( P^{-1} B^k P^{-\*} \right)^{1/k} \end{pmatrix} \begin{pmatrix} \phi(A) = P^\* A^t P \\\\ \nu(B) = \left[ P^{-1} (B^t)^k P^{-\*} \right]^{1/k} \end{pmatrix} \quad A, B \in \mathcal{P}\_n. \tag{97}$$

*The maps ϕ and ψ satisfy* (96)

1.*for all A* ∈P*n*, *B*∈*S, and ψ is linear, or*

2.*for all A*, *B*∈ *S and both ϕ and ψ are linear,*

*if and only if when k*∈ f g �1, 1 *, ϕ and ψ take the form* (97)*, and when k*∈n �f g 1, 0, 1 *, there exist a unitary matrix U* ∈M*<sup>n</sup> and c*∈ <sup>þ</sup> *such that*

$$\begin{pmatrix} \phi(A) = c^{-k} U^\* A U & \text{or} \\ \psi(B) = c U^\* B U \end{pmatrix} \quad \text{or} \begin{pmatrix} \phi(A) = c^{-k} U^\* A^t U \\ \psi(B) = c U^\* B^t U \end{pmatrix} \quad A, B \in \mathcal{P}\_n. \tag{98}$$

*Proof.* It suffices to prove the necessary part.

The case of assumption (1) has been proved by Theorem 6.2.

Similar to the proof of Theorem 3.5, assumption (2) implies assumption (1); assumption (4) implies assumption (3). It remains to prove the case with assumption (3).

When *k* ¼ 1, assumption (3) is analogous to assumption (2), and we get (97). Suppose *<sup>k</sup>*<sup>∈</sup> nf g 1, 0 . Let *<sup>ψ</sup>*<sup>1</sup> : <sup>P</sup>*<sup>n</sup>* ! <sup>P</sup>*<sup>n</sup>* be defined by *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>≔</sup> *<sup>ψ</sup> <sup>B</sup>*<sup>1</sup>*=<sup>k</sup>* � �*<sup>k</sup>* . Let S<sup>1</sup> ≔ *B* ∈P*<sup>n</sup>* : *B*<sup>1</sup>*=<sup>k</sup>* ∈ *S* � �*:* Then (97) with assumption (2) becomes

$$\text{tr}(\phi(A)\mu\_1(B)) = \text{tr}(AB), \quad A \in \mathcal{P}\_n, \ B \in \mathcal{S}\_1. \tag{99}$$

Let *ψ*~ : H*<sup>n</sup>* ! H*<sup>n</sup>* be the linear extension of *ψ*. By Theorem 2.2, *ϕ* can be extended to a linear bijection *<sup>ϕ</sup>*<sup>~</sup> : <sup>H</sup>*<sup>n</sup>* ! <sup>H</sup>*<sup>n</sup>* such that

$$\text{tr}\left(\bar{\phi}(A)\bar{\psi}(B)^k\right) = \text{tr}\left(\bar{\phi}(A)\boldsymbol{\mu}\_1(B^k)\right) = \text{tr}\left(AB^k\right), \quad A \in \mathcal{H}\_n, \ B \in \mathbb{S}.\tag{100}$$

By Theorem 4.2 and taking into account the ranges of *ϕ* and *ψ*, we see that when *k* ¼ �1, *ϕ* and *ψ* take the form of (97), and when *k*∈n �f g 1, 0, 1 , *ϕ* and *ψ* take the form of (98).

Theorem 6.4 has counterpart results for *ϕ*, *ψ* : P*n*ð Þ! P*n*ð Þ and the proof is analogous using Theorem 5.3 instead of Theorem 4.2.

## **7.** *k***-power linear preservers and trace of power-product preservers on** D*<sup>n</sup>* **and** D*n*ð Þ

Let <sup>¼</sup> or . Define the function diag : *<sup>n</sup>* ! <sup>D</sup>*n*ð Þ to be the linear bijection that sends each ð Þ *<sup>c</sup>*1, <sup>⋯</sup>,*cn <sup>t</sup>* to the diagonal matrix with *<sup>c</sup>*1, … ,*cn* (in order) as the diagonal entries. Define diag�<sup>1</sup> : <sup>D</sup>*n*ð Þ! *<sup>n</sup>* the inverse map of diag.

With the settings, every linear map *ψ* : D*n*ð Þ! D*n*ð Þ uniquely corresponds to a matrix *L<sup>ψ</sup>* ∈M*n*ð Þ such that

$$\varphi(A) = \text{diag}\left(L\_{\psi}\text{diag}^{-1}(A)\right), \quad A \in \mathcal{D}\_{\pi}(\mathbb{F}).\tag{101}$$

## **7.1** *k***-power linear preservers on** D*<sup>n</sup>* **and** D*n*ð Þ

We define the linear functionals *fi* : D*n*ð Þ! ð Þ *i* ¼ 0, 1, … , *n* , such that for each *A* ¼ diagð Þ *a*1, … , *an* ∈ D*n*ð Þ ,

$$f\_{\; \!\!\!= 0}(A) = \mathbf{0}; \quad f\_{\;\!\!=}(A) = a\_i, \quad i = \mathbf{1}, \ldots, n. \tag{102}$$

Theorem 7.1. *Let* ¼ *or . Let k*∈nf g 0, 1 *. Let S be an open neighborhood of In in* D*n*ð Þ *. A linear map ψ* : D*n*ð Þ! D*n*ð Þ *satisfies that*

$$
\psi\left(A^k\right) = \psi\left(A\right)^k, \quad A \in \mathcal{S}, \tag{103}
$$

if and only if

$$\psi(A) = \psi(I\_n) \text{diag}\left(f\_{p(1)}(A), \dots, f\_{p(n)}(A)\right), \quad A \in \mathcal{D}\_n(\mathbb{F}), \tag{104}$$

*in which <sup>ψ</sup>*ð Þ *In <sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *In and p* : f g! 1, … , *<sup>n</sup>* f g 0, 1, … , *<sup>n</sup> is a function such that p i*ð Þ 6¼ 0 *when k*<0 *for i* ¼ 1, … , *n. In particular, a linear bijection ψ* : D*n*ð Þ! D*n*ð Þ *satisfies* (103) *if and only if there is a diagonal matrix C*∈M*n*ð Þ *with C<sup>k</sup>*�<sup>1</sup> <sup>¼</sup> *In and a permutation matrix P*∈M*n*ð Þ *such that*

$$\varphi(A) = P\text{CAP}^{-1}, \quad A \in \mathcal{D}\_n(\mathbb{F}).\tag{105}$$

*Proof.* For every *A* ¼ diagð Þ *a*1, … , *an* ∈ D*n*ð Þ , when *x*∈ is sufficiently close to 0, we have *In* <sup>þ</sup> *xA* <sup>∈</sup>*<sup>S</sup>* and the power series of ð Þ *In* <sup>þ</sup> *xA <sup>k</sup>* converges, so that *<sup>ψ</sup>* ð Þ *In* <sup>þ</sup> *xA <sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *In* <sup>þ</sup> *xA <sup>k</sup>* .

$$\Psi\left(\left(I\_{\mathfrak{n}} + \mathfrak{x}A\right)^{k}\right) = \psi\left(I\_{\mathfrak{n}}\right) + \mathfrak{x}k\psi\left(A\right) + \mathfrak{x}^{2}\frac{k\left(k-1\right)}{2}\psi\left(A^{2}\right) + \cdots \tag{106}$$

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

$$\left(\boldsymbol{\varphi}(\boldsymbol{I}\_{n}+\boldsymbol{\chi}\boldsymbol{A})\right)^{k}=\boldsymbol{\varphi}(\boldsymbol{I}\_{n})+\boldsymbol{\chi}k\boldsymbol{\varphi}(\boldsymbol{I}\_{n})^{k-1}\boldsymbol{\varphi}(\boldsymbol{A})+\boldsymbol{\chi}^{2}\frac{k(k-1)}{2}\boldsymbol{\varphi}(\boldsymbol{I}\_{n})^{k-2}\boldsymbol{\varphi}(\boldsymbol{A})^{2}+\cdots \tag{107}$$

So for all *A* ∈ D*n*ð Þ :

$$
\psi(A) = \psi(I\_n)^{k-1} \psi(A),\tag{108}
$$

$$
\psi\left(A^2\right) = \psi\left(I\_n\right)^{k-2}\psi\left(A\right)^2. \tag{109}
$$

The linear map *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> *ψ*ð Þ *A* satisfies that

$$
\psi\_1(A^2) = \psi\_1(A)^2, \quad A \in \mathcal{D}\_n(\mathbb{F}).\tag{110}
$$

By (101), let *<sup>L</sup>ψ*<sup>1</sup> <sup>¼</sup> <sup>ℓ</sup>*ij* � �∈M*n*ð Þ such that diag�<sup>1</sup> ð Þ¼ *<sup>ψ</sup>*1ð Þ *<sup>A</sup> <sup>L</sup>ψ*<sup>1</sup> diag�<sup>1</sup> ð Þ *<sup>A</sup>* � � for *A* ∈ D*n*ð Þ . Then (110) implies that for all *A* ¼ diagð Þ *a*1, … , *an* ∈ D*n*ð Þ :

$$\sum\_{j=1}^{n} \ell\_{\vec{\eta}} a\_j^2 = \left(\sum\_{j=1}^{n} \ell\_{\vec{\eta}}^{\ell} a\_j\right)^2, \quad i = 1, 2, \dots, n. \tag{111}$$

Therefore, each row of *Lψ*<sup>1</sup> has at most one nonzero entry and each nonzero entry must be 1. We get

$$\varphi\_1(A) = \text{diag}\left(f\_{p(1)}(A), \dots, f\_{p(n)}(A)\right) \tag{112}$$

in which *p* : f g 1, … , *n* ! f g 0, 1, … , *n* is a function. Suppose *ψ*ð Þ¼ *In* diagð Þ *λ*1, … , *λ<sup>n</sup>* . Then (108) implies that *ψ*ð Þ¼ *A ψ*ð Þ *In ψ*1ð Þ *A* has the form (104). Obviously, *<sup>ψ</sup>*ð Þ *In <sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *In* and when *<sup>k</sup>*<0, each *p i*ð Þ 6¼ 0 for *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*. Moreover, when *<sup>ψ</sup>* is a linear bijection, (112) shows that *<sup>ψ</sup>*1ð Þ¼ *<sup>A</sup> PAP*�<sup>1</sup> for a permutation matrix *P*. (105) can be easily derived.

## **7.2 Trace of power-product preservers on** D*<sup>n</sup>* **and** D*n*ð Þ

In [18], we show that two maps *ϕ*, *ψ* : D*n*ð Þ! D*n*ð Þ satisfy trð Þ¼ *ϕ*ð Þ *A ψ*ð Þ *B* trð Þ *AB* for *A*, *B* ∈ D*n*ð Þ if and only if there exists an invertible *N* ∈M*n*ð Þ such that

$$\phi(A) = \text{diag}\left(\text{Ndiag}^{-1}(A)\right), \ y(B) = \text{diag}\left(N^{-t}\text{diag}^{-1}(B)\right), \quad A, B \in \mathcal{D}\_n(\mathbb{F}).\tag{113}$$

When *m* ≥3, the maps *ϕ*1, … , *ϕ<sup>m</sup>* : D*n*ð Þ! D*n*ð Þ satisfying

*tr*ð*ϕ*1ð Þ *A*<sup>1</sup> ⋯*ϕm*ð Þ *Am* Þ ¼ *tr A*ð Þ <sup>1</sup>⋯*Am* for *A*1, … , *Am* ∈ D*n*ð Þ are also determined in [18]. Next we consider the trace of power-product preserver on D*n*ð Þ .

Theorem 7.2. *Let* ¼ *or . Let k*∈nf g 0, 1 *. Let S be an open neighborhood of In in* D*n*ð Þ *. Two maps ϕ*, *ψ* : D*n*ð Þ! D*n*ð Þ *satisfy that*

$$\text{tr}\left(\phi(A)\psi(B)^k\right) = \text{tr}\left(AB^k\right),\tag{114}$$

1.*for all A* ∈ D*n*ð Þ , *B*∈ *S, and ψ is linear, or*

2.*for all A*, *B*∈ *S and both ϕ and ψ are linear,*

if and only if there exist an invertible diagonal matrix *C* ∈ D*n*ð Þ and a permutation matrix *P*∈M*n*ð Þ *F* such that

$$\phi(A) = P C^{-k} A P^{-1}, \quad \psi(B) = P C B P^{-1}, \quad A, B \in \mathcal{D}\_n(\mathbb{F}).\tag{115}$$

*Proof.* Assumption (2) leads to assumption (1) (cf. the proof of Theorem 3.5). We prove the theorem under assumption (1).

For every *<sup>B</sup>*<sup>∈</sup> <sup>D</sup>*n*ð Þ , *In* <sup>þ</sup> *xB*∈*<sup>S</sup>* and the power series of ð Þ *In* <sup>þ</sup> *xB <sup>k</sup>* converges when *x*∈ is sufficiently close to 0, so that

$$\operatorname{tr}\left(\phi(A)\psi(I\_n+\varkappa B)^k\right) = \operatorname{tr}\left(A(I\_n+\varkappa B)^k\right) \tag{116}$$

Comparing degree one terms and degree two terms in the power series of the above equality, respectively, we get the following equalities for *A*, *B* ∈ D*n*ð Þ :

$$\text{tr}\left(\phi(A)\psi(B)\psi(I\_n)^{k-1}\right) = \text{tr}(AB),\tag{117}$$

$$\operatorname{tr}\left(\phi(A)\psi(B)^2\psi(I\_n)^{k-2}\right) = \operatorname{tr}\left(AB^2\right). \tag{118}$$

Applying Theorem 2.2 to (117), *ψ*ð Þ *In* is invertible and both *ϕ* and *ψ* are linear bijections. (117) and (119) imply that *<sup>ψ</sup> <sup>B</sup>*<sup>2</sup> *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>1</sup> <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>B</sup>* <sup>2</sup> *<sup>ψ</sup>*ð Þ *In <sup>k</sup>*�<sup>2</sup> . Let *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>ψ</sup>*ð Þ *In* �<sup>1</sup> . Then *<sup>ψ</sup>*<sup>1</sup> *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>B</sup>* <sup>2</sup> for *<sup>B</sup>*<sup>∈</sup> <sup>D</sup>*n*ð Þ . By Theorem 7.1 and *<sup>ψ</sup>*1ð Þ¼ *In In*, there exists a permutation matrix *<sup>P</sup>*∈M*n*ð Þ *<sup>F</sup>* such that *<sup>ψ</sup>*1ð Þ¼ *<sup>B</sup> PBP*�<sup>1</sup> for *<sup>B</sup>*<sup>∈</sup> <sup>D</sup>*n*ð Þ . So *<sup>ψ</sup>*ð Þ¼ *<sup>B</sup> <sup>ψ</sup>*ð Þ *In PBP*�<sup>1</sup> <sup>¼</sup> *PCBP*�<sup>1</sup> for *<sup>C</sup>* <sup>≔</sup> *<sup>P</sup>*�<sup>1</sup> *ψ*ð Þ *In P*∈ D*n*ð Þ . Then (114) implies (115).

## **8.** *k***-power injective linear preservers and trace of power-product preservers on** T *<sup>n</sup>* **and** T *<sup>n</sup>*ð Þ

## **8.1** *k***-power preservers on** T *<sup>n</sup>*ð Þ

The characterization of injective linear *k*-power preserver on T *<sup>n</sup>*ð Þ can be derived from Cao and Zhang's characterization of injective additive *k*-power preserver on T *<sup>n</sup>*ð Þ ([12] or [[6], Theorem 6.5.2]).

Theorem 8.1 (Cao and Zhang [12]). *Let k*≥2 *and n*≥ 3*. Let be a field with char*ð Þ¼ 0 *or char*ð Þ > *k. Then ψ* : T *<sup>n</sup>*ð Þ ↦T *<sup>n</sup>*ð Þ *is an injective linear map such that <sup>ψ</sup> Ak* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A* <sup>∈</sup> <sup>T</sup> *<sup>n</sup>*ð Þ *if and only if there exists a k*ð Þ � <sup>1</sup> *th root of unity <sup>λ</sup> and an invertible matrix P*∈T *<sup>n</sup>*ð Þ *such that*

$$
\psi(A) = \lambda P A P^{-1}, \quad A \in \mathcal{T}\_n(\mathbb{F}), \quad or \tag{119}
$$

$$\psi(A) = \lambda \mathbf{P} A^{-} P^{-1}, \quad A \in \mathcal{T}\_{\pi}(\mathbb{F}), \tag{120}$$

where *A*� ¼ *an*þ1�*j*,*n*þ1�*<sup>i</sup>* if *<sup>A</sup>* <sup>¼</sup> *aij* .

$$\begin{aligned} & \text{Example 8.2.} \text{ When } n = 2 \text{, the injective linear maps that satisfy } \psi \left( \mathbf{A}^k \right) = \psi(\mathbf{A})^k \text{ for } \\ & A \in T\_2(\mathbb{F}) \text{ send } A = \begin{pmatrix} a\_{11} & a\_{12} \\ 0 & a\_{22} \end{pmatrix} \text{ to the following } \psi(A) \text{:} \end{aligned}$$

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

$$
\lambda \begin{pmatrix} a\_{11} & c a\_{12} \\ 0 & a\_{22} \end{pmatrix}, \quad \lambda \begin{pmatrix} a\_{22} & c a\_{12} \\ 0 & a\_{11} \end{pmatrix}, \tag{121}
$$

in which *<sup>λ</sup>k*�<sup>1</sup> <sup>¼</sup> 1 and *<sup>c</sup>*∈nf g<sup>0</sup> *:*.

Example 8.3. *Theorem 8.1 does not hold if ψ is not assumed to be injective. Let n* ¼ 3 *and suppose <sup>ψ</sup>* : <sup>T</sup> <sup>3</sup>ð Þ! <sup>T</sup> <sup>3</sup>ð Þ *is a linear map that sends A* <sup>¼</sup> *aij* � � <sup>3</sup>�<sup>3</sup> <sup>∈</sup><sup>T</sup> <sup>3</sup>ð Þ *to one of the following ψ*ð Þ *A (c*, *d*∈*):*

$$
\begin{pmatrix} a\_{11} & ca\_{12} & 0 \\ 0 & a\_{22} & da\_{23} \\ 0 & 0 & a\_{33} \end{pmatrix}, \begin{pmatrix} a\_{33} & 0 & 0 \\ 0 & a\_{11} & 0 \\ 0 & 0 & a\_{22} \end{pmatrix}, \begin{pmatrix} a\_{22} & 0 & ca\_{12} \\ 0 & 0 & 0 \\ 0 & 0 & a\_{11} \end{pmatrix}. \tag{122}
$$

Then each *<sup>ψ</sup>* satisfies that *<sup>ψ</sup> Ak* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* for every positive integer *<sup>k</sup>* but it is not of the forms in Theorem 8.1.

We extend Theorem 8.1 to the following result that includes negative *k*-powers and that only assumes *k*-power preserving in a neighborhood of *In*.

Theorem 8.4. *Let* ¼ *or . Let integers k* 6¼ 0, 1 *and n*≥ 3*. Suppose that ψ* : <sup>T</sup> *<sup>n</sup>*ð Þ! <sup>T</sup> *<sup>n</sup>*ð Þ *is an injective linear map such that <sup>ψ</sup> Ak* � � <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup> for all A in an open neighborhood of In in* T *<sup>n</sup>*ð Þ *consisting of invertible matrices. Then there exist λ*∈ *with <sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> *and an invertible matrix P*<sup>∈</sup> <sup>T</sup> *<sup>n</sup>*ð Þ *such that*

$$\psi(A) = \lambda P A P^{-1}, \quad A \in \mathcal{T}\_n(\mathbb{F}), \quad or \tag{123}$$

$$
\psi(A) = \lambda P A^{-} P^{-1}, \quad A \in \mathcal{T}\_n(\mathbb{F}).\tag{124}
$$

where *A*� ¼ *an*þ1�*j*,*n*þ1�*<sup>i</sup>* � � <sup>¼</sup> *JnA<sup>t</sup> Jn* if *<sup>A</sup>* <sup>¼</sup> *aij* � �, *Jn* is the anti-diagonal identity.

*Proof.* Obviously *ψ* is a linear bijection. Follow the same process in the proof of Theorem 3.4. In both *k*≥ 2 and *k*<0 cases we have *ψ*ð Þ *In* commutes with the range of *<sup>ψ</sup>*, so that *<sup>ψ</sup>*ð Þ¼ *In <sup>λ</sup>In* for *<sup>λ</sup>*∈ and *<sup>λ</sup><sup>k</sup>*�<sup>1</sup> <sup>¼</sup> <sup>1</sup>*:* Moreover, let *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>≔</sup> *<sup>ψ</sup>*ð Þ *In* �<sup>1</sup> *ψ*ð Þ *A* , then *<sup>ψ</sup>*<sup>1</sup> is injective linear and *<sup>ψ</sup>*<sup>1</sup> *<sup>A</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>ψ</sup>*1ð Þ *<sup>A</sup>* <sup>2</sup> for *<sup>A</sup>* <sup>∈</sup><sup>T</sup> *<sup>n</sup>*ð Þ *:* Theorem 8.1 shows that *<sup>ψ</sup>*1ð Þ¼ *<sup>A</sup> PAP*�<sup>1</sup> or *<sup>ψ</sup>*1ð Þ¼ *<sup>A</sup> PA*�*P*�<sup>1</sup> for certain invertible *<sup>P</sup>*∈<sup>T</sup> *<sup>n</sup>*ð Þ *:* It leads to (123) and (124).

## **8.2 Trace of power-product preservers on** T *<sup>n</sup>* **and** T *<sup>n</sup>*ð Þ

Theorem 2.2 or Corollary 2.3 does not work for maps on T *<sup>n</sup>*ð Þ . However, the following trace preserving result can be easily derived from Theorem 7.2. We have T *<sup>n</sup>*ð Þ¼ D*n*ð Þ ⊕N *<sup>n</sup>*ð Þ . Let *D A*ð Þ denote the diagonal matrix that takes the diagonal of *A* ∈T *<sup>n</sup>*ð Þ .

Theorem 8.5. *Let* ¼ *or . Let k*∈nf g 0, 1 *. Let S be an open neighborhood of In in* T *<sup>n</sup>*ð Þ *consisting of invertible matrices. Then two maps ϕ*, *ψ* : T *<sup>n</sup>*ð Þ! T *<sup>n</sup>*ð Þ *satisfy that*

$$\operatorname{tr}\left(\phi(A)\psi(B)^k\right) = \operatorname{tr}(AB^k),\tag{125}$$

1.*for all A* ∈T *<sup>n</sup>*ð Þ , *B* ∈*S, and ψ is linear, or*

2.*for all A*, *B*∈ *S and both ϕ and ψ are linear,*

*if and only if <sup>ϕ</sup> and <sup>ψ</sup> send* <sup>N</sup> *<sup>n</sup>*ð Þ *to* <sup>N</sup> *<sup>n</sup>*ð Þ *, D*ð Þj <sup>∘</sup>*<sup>ϕ</sup>* <sup>D</sup>*n*ð Þ *and D*ð Þj <sup>∘</sup>*<sup>ψ</sup>* <sup>D</sup>*n*ð Þ *are linear bijections characterized by* (115) *in Theorem 7.2, and D*∘*ϕ* ¼ *D*∘*ϕ*∘*D.*

*Proof.* The sufficient part is easy to verify. We prove the necessary part here. Let *<sup>ϕ</sup>*<sup>0</sup> <sup>≔</sup> ð Þj *<sup>D</sup>*∘*<sup>ϕ</sup>* <sup>D</sup>*n*ð Þ and *<sup>ψ</sup>*<sup>0</sup> <sup>≔</sup> ð Þj *<sup>D</sup>*∘*<sup>ψ</sup>* <sup>D</sup>*n*ð Þ . Then *<sup>ϕ</sup>*<sup>0</sup> , *ψ*<sup>0</sup> : D*n*ð Þ! D*n*ð Þ satisfy *tr ϕ*<sup>0</sup> ð Þ *A ψ*<sup>0</sup> ð Þ *<sup>B</sup> <sup>k</sup>* <sup>¼</sup> *tr AB<sup>k</sup>* for *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>D</sup>*n*ð Þ . So they are characterized by (115). The bijectivity of *ϕ*<sup>0</sup> and *ψ*<sup>0</sup> implies that *ϕ* and *ψ* must send N *<sup>n</sup>*ð Þ to N *<sup>n</sup>*ð Þ in order to satisfy (125). Moreover, *ϕ* should send matrices with same diagonal to matrices with same diagonal, which implies that *D*∘*ϕ* ¼ *D*∘*ϕ*∘*D*.

## **9. Conclusion**

We characterize linear maps *<sup>ψ</sup>* : *<sup>V</sup>* ! *<sup>V</sup>* that satisfy *<sup>ψ</sup> <sup>A</sup><sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>A</sup> <sup>k</sup>* on an open neighborhood *S* of *In* in *V*, where *k*∈nf g 0, 1 and *V* is the set of *n* � *n* general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semidefinite) matrices, diagonal matrices, or upper triangular matrices, over the complex or real field. The characterizations extend the existing results of linear *k*-power preservers on the spaces of general matrices, symmetric matrices, and upper triangular matrices.

Applying the above results, we determine the maps *ϕ*, *ψ* : *V* ! *V* on the preceding sets *<sup>V</sup>* that satisfy *tr <sup>ϕ</sup>*ð Þ *<sup>A</sup> <sup>ψ</sup>*ð Þ *<sup>B</sup> <sup>k</sup>* <sup>¼</sup> *tr AB<sup>k</sup>*

1. for all *A* ∈*V*, *B* ∈*S*, and *ψ* is linear, or

2. for all *A*, *B*∈*S* and both *ϕ* and *ψ* are linear.

These results, together with Theorem 2.2 about maps satisfying *tr*ð Þ¼ *ϕ*ð Þ *A ψ*ð Þ *B tr AB* ð Þ and the characterizations of maps *ϕ*1, ⋯, *ϕ<sup>m</sup>* : *V* ! *V* (*m* ≥3) satisfying *tr*ð*ϕ*1ð Þ *A*<sup>1</sup> ⋯*ϕm*ð Þ *Am* Þ ¼ *tr A*ð Þ <sup>1</sup>⋯*Am* in [18], make a comprehensive picture of the preservers of trace of matrix products in the related matrix spaces and sets. Our results can be interpreted as inner product preservers when *V* is close under conjugate transpose, in which wide applications are found.

There are a few prospective directions to further the researches.

First, for a polynomial or an analytic function *f x*ð Þ and a matrix set *V*, we can consider "local" linear *f*-preservers, that is, linear operators *ψ* : *V* ! *V* that satisfy *ψ*ð Þ¼ *f A*ð Þ *f*ð Þ *ψ*ð Þ *A* on an open subset *S* of *V*. A linear *f*-preserver *ψ* on *S* also preserves matrices annihilated by *f* on *S*, that is, *f A*ð Þ¼ 0 (*A* ∈ *S*) implies *f*ð Þ¼ *ψ*ð Þ *A* 0. When *S* ¼ *V* is *Mn*, *B H*ð Þ, or some operator algebras, extensive studies have been done on operators preserving elements annihilated by a polynomial *f*; for examples, the results on *Mn* by R. Howard in [19], by P. Botta, S. Pierce, and W. Watkins in [20], and by C.-K. Li and S. Pierce in [21], on *B H*ð Þ by P. Šemrl [22], on linear maps *ψ* : *B H*ð Þ! *B K*ð Þ by Z. Bai and J. Hou in [23], and on some operator algebras by J. Hou and S. Hou in [24]. We may further explore linear *f*-preservers for a multivariable function *f x*ð Þ 1, … , *xr* , that is, operator *ψ* satisfying *ψ*ð *f A*ð Þ 1, … , *Ar* Þ ¼ *f*ð Þ *ψ*ð Þ *A*<sup>1</sup> , … , *ψ*ð Þ *Ar* . The corresponding annihilator preserver problem has been studied in some special cases, for example, on *Mn* for homogeneous multilinear polynomials by A. E. Guterman and B. Kuzma in [25].

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

Second, it is interesting to further investigate maps *ϕ*, *ψ* : *V* ! *V* that satisfy *tr f* ð ð Þ *ϕ*ð Þ *A g*ð Þ *ψ*ð Þ *B* Þ ¼ *tr f A* ð Þ ð Þ*g B*ð Þ for some polynomials or analytic functions *f x*ð Þ and *g x*ð Þ. This is equivalent to the inner product preserver problem *<sup>f</sup>*ð Þ *<sup>ϕ</sup>*ð Þ *<sup>A</sup>* <sup>∗</sup> h i , *<sup>g</sup>*ð Þ *<sup>ψ</sup>*ð Þ *<sup>B</sup>* <sup>¼</sup> *f A*ð Þ <sup>∗</sup> h i , *g B*ð Þ when *<sup>V</sup>* is close under conjugate transpose. More generally, given a multivariable function *h x*ð Þ 1, … , *xm* , we can ask what combinations of linear operators *ϕ*1, … , *ϕ<sup>m</sup>* : *V* ! *V* satisfy that *tr h*ð ð*ϕ*1ð Þ *A*<sup>1</sup> , … , *ϕm*ð Þ *Am* Þ ¼ *tr h A* ð Þ ð Þ 1, … , *Am* . The research on this area seems pretty new. No much has been discovered by the authors.

## **Author details**

Huajun Huang<sup>1</sup> \* and Ming-Cheng Tsai<sup>2</sup>

1 Department of Mathematics and Statistics, Auburn University, AL, USA

2 General Education Center, Taipei University of Technology, Taipei, Taiwan

\*Address all correspondence to: huanghu@auburn.edu

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Li CK, Pierce S. Linear preserver problems. American Mathematical Monthly. 2001;**108**:591-605

[2] Li CK, Tsing NK. Linear preserver problems: A brief introduction and some special techniques. Linear Algebra and its Applications. 1992;**162–164**: 217-235

[3] Molnár L. Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Mathematics. Vol. 1895. Berlin: Springer-Verlag; 2007

[4] Pierce S et al. A survey of linear preserver problems. Linear and Multilinear Algebra. 1992;**33**:1-129

[5] Chan GH, Lim MH. Linear preservers on powers of matrices. Linear Algebra and its Applications. 1992;**162–164**: 615-626

[6] Zhang X, Tang X, Cao C. Preserver Problems on Spaces of Matrices. Beijing: Science Press; 2006

[7] Brešar M, Šemrl P. Linear transformations preserving potent matrices. Proceedings of the American Mathematical Society. 1993;**119**:81-86

[8] Zhang X, Cao CG. Linear k-power/kpotent preservers between matrix spaces. Linear Algebra and its Applications. 2006;**412**:373-379

[9] Kadison RV. Isometries of operator algebras. Annals of Mathematics. 1951; **54**:325-338

[10] Brešar M, Martindale WS, Miers CR. Maps preserving nth powers. Communications in Algebra. 1998;**26**: 117-138

[11] Cao C, Zhang X. Power preserving additive maps (in Chinese). Advances in Mathematics. 2004;**33**:103-109

[12] Cao C, Zhang X. Power preserving additive operators on triangular matrix algebra (in Chinese). Journal of Mathematics. 2005;**25**:111-114

[13] Uhlhorn U. Representation of symmetry transformations in quantum mechanics. Arkiv för Matematik. 1963; **23**:307-340

[14] Molnár L. Some characterizations of the automorphisms of ℬℋ and CX. Proceedings of the American Mathematical Society. 2002;**130**(1):111-120

[15] Li CK, Plevnik L, Šemrl P. Preservers of matrix pairs with a fixed inner product value. Operators and Matrices. 2012;**6**:433-464

[16] Huang H, Liu CN, Tsai MC, Szokol P, Zhang J. Trace and determinant preserving maps of matrices. Linear Algebra and its Applications. 2016;**507**: 373-388

[17] Leung CW, Ng CK, Wong NC. Transition probabilities of normal states determine the Jordan structure of a quantum system. Journal of Mathematical Physics. 2016;**57**:015212

[18] Huang H, Tsai MC. Maps Preserving Trace of Products of Matrices. Available from: https://arxiv.org/abs/2103.12552 [Preprint]

[19] Howard R. Linear maps that preserve matrices annihilated by a polynomial. Linear Algebra and its Applications. 1980;**30**:167-176

[20] Botta P, Pierce S, Watkins W. Linear transformations that preserve the

*Linear K-Power Preservers and Trace of Power-Product Preservers DOI: http://dx.doi.org/10.5772/intechopen.103713*

nilpotent matrices. Pacific Journal of Mathematics. 1983;**104**(1):39-46

[21] Li C-K, Pierce S. Linear operators preserving similarity classes and related results. Canadian Mathematical Bulletin. 1994;**37**(3):374-383

[22] Šemrl P. Linear mappings that preserve operators annihilated by a polynomial. Journal of Operator Theory. 1996;**36**(1):45-58

[23] Bai Z, Hou J. Linear maps and additive maps that preserve operators annihilated by a polynomial. Journal of Mathematical Analysis and Applications. 2002;**271**(1):139-154

[24] Hou J, Hou S. Linear maps on operator algebras that preserve elements annihilated by a polynomial. Proceedings of the American Mathematical Society. 2002;**130**(8):2383-2395

[25] Guterman AE, Kuzma B. Preserving zeros of a polynomial. Communications in Algebra. 2009;**37**(11):4038-4064

## **Chapter 2**
