3.1. Case of defect indices 1ð Þ ; 1

Let <sup>α</sup> <sup>¼</sup> <sup>P</sup> <sup>p</sup> αpψ<sup>p</sup> ∈ Fψ; we introduce the operator A<sup>∘</sup> <sup>α</sup> defined in L<sup>ψ</sup> by

$$
\mathring{A}\_{\mathfrak{a}}f = \alpha \bullet f = \sum\_{n} \langle \alpha, \psi\_{n} \rangle \langle f, \psi\_{n} \rangle \psi\_{n}.\tag{31}
$$

It is clear that A ∘ <sup>α</sup> is a Carleman operator induced by the kernel

$$k(\mathbf{x}, \mathbf{y}) = \sum \alpha\_n \psi\_n(\mathbf{x}) \overline{\psi\_n(\mathbf{y})} \tag{32}$$

with domain

$$D\left(\stackrel{\bullet}{A}\_{\alpha}\right) = \left\{ f \in \mathcal{L}\_{\psi} : \sum\_{n} \left| \alpha\_{n}(f, \psi\_{n}) \right|^{2} < \infty \right\}.\tag{33}$$

Moreover, if <sup>α</sup> <sup>¼</sup> <sup>α</sup>, A<sup>∘</sup> <sup>α</sup> is self-adjoint.

$$\text{Now let } \begin{aligned} \Theta = \sum\_{p} \gamma\_{p} \psi\_{p} \in \mathcal{F}\_{\psi} \text{ and } \Theta \notin \mathcal{L}\_{\psi} \left( \text{i.e., } \sum\_{p} \left| \boldsymbol{\mathcal{V}}\_{p} \right|^{2} = \bullet \right). \text{ We introduce the following set} \\\\ \mathcal{H}\_{\Theta} = \left\{ f + \mu \Theta : f \in \mathcal{L}\_{\psi}, \mu \in \mathbb{C} \right\} \end{aligned} \tag{34}$$

which verifies the following properties.

Proposition 10. 1. H<sup>Θ</sup> is a subset of Fψ.

2. <sup>H</sup><sup>θ</sup> <sup>¼</sup> <sup>L</sup><sup>ψ</sup> <sup>⊕</sup> <sup>C</sup>θ, i.e., direct sum of <sup>L</sup><sup>ψ</sup> with <sup>C</sup><sup>θ</sup> <sup>¼</sup> μθ : <sup>μ</sup><sup>∈</sup> <sup>C</sup> � �:

Proof. The first property is easy to establish. We show the uniqueness for the second.

Let g<sup>1</sup> ¼ f <sup>1</sup> þ μ1θ and g<sup>2</sup> ¼ f <sup>2</sup> þ μ2θ , two formal elements in Hθ: Then

$$g\_1 = g\_2 \Leftrightarrow f\_1 - f\_2 = (\mu\_2 - \mu\_1)\theta.$$

This last equality is verified only if μ<sup>2</sup> ¼ μ1: Therefore, f <sup>1</sup> ¼ f <sup>2</sup>. ■ Denote by Q the projector of H<sup>Θ</sup> on Lψ, that is to say: if g∈ HΘ,

$$\mathcal{g} = f + \mu \Theta \text{ with } f \in \mathcal{L}\_{\psi} \text{ and } \mu \in \mathbb{C}$$

then

Qg ¼ f :

We define the operator B<sup>α</sup> by:

$$B\_{\mathfrak{a}}f = \mathbb{Q}(\mathfrak{a} \circ f), f \in \mathcal{L}\_{\psi}. \tag{35}$$

2. Let f <sup>n</sup> 

We have then

with

Then

■

This implies that

Or, when n tends to ∞, we have

Finally f ∈ D Bð Þ<sup>α</sup> and g ¼ B<sup>α</sup> f .

Therefore, there exist μ∈ C such that

And as Q is a closed operator, then we can write

It follows from this theorem that the adjoint operator B<sup>∗</sup>

Theorem 12. A<sup>α</sup> admits defect indices ð Þ 1; 1 if and only if

Let us denote by A<sup>α</sup> the operator adjoint of Bα,

<sup>n</sup> be a sequence of elements in D Bð Þ<sup>α</sup> : Checking:

<sup>B</sup><sup>α</sup> <sup>f</sup> <sup>n</sup> ! <sup>g</sup> convergence in the <sup>L</sup><sup>2</sup> sense :

B<sup>α</sup> f <sup>n</sup> ¼ Q α ∘f <sup>n</sup>

α ∘f <sup>n</sup> ¼ gn þ μΘ, gn ∈Lψ:

gn ¼ α ∘f <sup>n</sup> � μnΘ ∈Lψ,

gn ! g and f <sup>n</sup> ! f :

lim<sup>n</sup>!<sup>∞</sup> <sup>μ</sup><sup>n</sup> <sup>¼</sup> <sup>μ</sup>:

ð Þ α ∘f ∈ H<sup>Θ</sup> and g ¼ Qð Þ α ∘f :

<sup>A</sup><sup>α</sup> <sup>¼</sup> <sup>B</sup><sup>∗</sup>

In the case α ¼ α, the operator A<sup>α</sup> is symmetric and we have the following results:

<sup>α</sup> exists and B∗∗

φλ <sup>¼</sup> ð Þ <sup>α</sup> � <sup>λ</sup> �<sup>1</sup> <sup>∘</sup> <sup>Θ</sup> <sup>∈</sup>Lψ: (38)

<sup>α</sup> ¼ Bα:

A Formal Perturbation Theory of Carleman Operators http://dx.doi.org/10.5772/intechopen.79022 57

<sup>α</sup>: (37)

� <sup>μ</sup>nγmψ<sup>m</sup> <sup>∀</sup><sup>m</sup> <sup>∈</sup> <sup>N</sup>:

,

f <sup>n</sup> ! f

gn;ψ<sup>m</sup>

<sup>¼</sup> <sup>α</sup><sup>m</sup> <sup>f</sup> <sup>n</sup>;ψ<sup>m</sup>

It is clear that

$$D(\mathcal{B}\_{\alpha}) = \left\{ f \in \mathcal{L}\_{\psi} : (\alpha \circ f) \in \mathcal{H}\_{\Theta} \right\}. \tag{36}$$

Theorem 11 B<sup>α</sup> is a densely defined and closed operator.

#### Proof.

1. Since

$$\operatorname{span}\left\{\psi\_n, n \in \mathbb{N}\right\} \subset D(B\_a)$$

and that ψ<sup>n</sup> � � <sup>n</sup> is complete in Lψ, then

D Bð Þ¼ <sup>α</sup> Lψ:

2. Let f <sup>n</sup> <sup>n</sup> be a sequence of elements in D Bð Þ<sup>α</sup> : Checking:

$$\begin{array}{rcl} \begin{cases} f\_n & \rightarrow & f \\ B\_\alpha f\_n & \rightarrow & \emptyset \end{cases} \text{(convergence in the } L^2 \text{ sense)}. \end{array}$$

We have then

$$B\_{\alpha}f\_n = Q(\alpha \bullet f\_n)\_{\prime}$$

with

Moreover, if <sup>α</sup> <sup>¼</sup> <sup>α</sup>, A<sup>∘</sup>

which verifies the following properties. Proposition 10. 1. H<sup>Θ</sup> is a subset of Fψ.

Now let <sup>Θ</sup> <sup>¼</sup> <sup>P</sup>

then

It is clear that

Proof. 1. Since

and that ψ<sup>n</sup>

� �

We define the operator B<sup>α</sup> by:

Theorem 11 B<sup>α</sup> is a densely defined and closed operator.

<sup>n</sup> is complete in Lψ, then

<sup>α</sup> is self-adjoint.

56 Perturbation Methods with Applications in Science and Engineering

<sup>p</sup> <sup>γ</sup>pψ<sup>p</sup> <sup>∈</sup> <sup>F</sup><sup>ψ</sup> and <sup>Θ</sup> <sup>∉</sup>L<sup>ψ</sup> <sup>i</sup>:e:; <sup>P</sup>

2. <sup>H</sup><sup>θ</sup> <sup>¼</sup> <sup>L</sup><sup>ψ</sup> <sup>⊕</sup> <sup>C</sup>θ, i.e., direct sum of <sup>L</sup><sup>ψ</sup> with <sup>C</sup><sup>θ</sup> <sup>¼</sup> μθ : <sup>μ</sup><sup>∈</sup> <sup>C</sup> � �:

This last equality is verified only if μ<sup>2</sup> ¼ μ1: Therefore, f <sup>1</sup> ¼ f <sup>2</sup>. ■ Denote by Q the projector of H<sup>Θ</sup> on Lψ, that is to say: if g∈ HΘ,

Let g<sup>1</sup> ¼ f <sup>1</sup> þ μ1θ and g<sup>2</sup> ¼ f <sup>2</sup> þ μ2θ , two formal elements in Hθ: Then

Proof. The first property is easy to establish. We show the uniqueness for the second.

g<sup>1</sup> ¼ g<sup>2</sup> ⇔ f <sup>1</sup> � f <sup>2</sup> ¼ μ<sup>2</sup> � μ<sup>1</sup>

g ¼ f þ μΘ with f ∈L<sup>ψ</sup> and μ∈ C

Qg ¼ f :

D Bð Þ¼ <sup>α</sup> f ∈L<sup>ψ</sup> : ð Þ α ∘f ∈ H<sup>Θ</sup>

span <sup>ψ</sup>n; <sup>n</sup><sup>∈</sup> <sup>N</sup> � �<sup>⊂</sup> D Bð Þ<sup>α</sup>

D Bð Þ¼ <sup>α</sup> Lψ:

� �θ:

Bαf ¼ Qð Þ α ∘f , f ∈Lψ: (35)

� �: (36)

<sup>p</sup> γ<sup>p</sup> � � � � � � 2 ¼ ∞

<sup>H</sup><sup>Θ</sup> <sup>¼</sup> <sup>f</sup> <sup>þ</sup> <sup>μ</sup><sup>Θ</sup> : <sup>f</sup> <sup>∈</sup>Lψ; <sup>μ</sup><sup>∈</sup> <sup>C</sup> � � (34)

. We introduce the following set

� �

$$
\alpha \circ f\_n = \mathcal{g}\_n + \mu \Theta \prime \mathcal{g}\_n \in \mathcal{L}\_{\psi}.
$$

Then

$$\mathcal{g}\_n = \alpha \circ f\_n - \mu\_n \Theta \in \mathcal{L}\_{\psi \wedge}$$

This implies that

$$
\langle \mathcal{g}\_n, \psi\_m \rangle = \alpha\_m \langle f\_n, \psi\_m \rangle - \mu\_n \gamma\_m \psi\_m \quad \forall m \in \mathbb{N}.
$$

Or, when n tends to ∞, we have

$$
\mathfrak{g}\_n \to \mathfrak{g} \text{ and } f\_n \to f.
$$

Therefore, there exist μ∈ C such that

lim<sup>n</sup>!<sup>∞</sup> <sup>μ</sup><sup>n</sup> <sup>¼</sup> <sup>μ</sup>:

And as Q is a closed operator, then we can write

$$(\alpha \circ f) \in \mathcal{H}\_{\Theta} \text{ and } \mathfrak{g} = \mathcal{Q}(\alpha \circ f).$$

Finally f ∈ D Bð Þ<sup>α</sup> and g ¼ B<sup>α</sup> f .

■

It follows from this theorem that the adjoint operator B<sup>∗</sup> <sup>α</sup> exists and B∗∗ <sup>α</sup> ¼ Bα:

Let us denote by A<sup>α</sup> the operator adjoint of Bα,

$$A\_{\alpha} = B\_{\alpha}^\*. \tag{37}$$

In the case α ¼ α, the operator A<sup>α</sup> is symmetric and we have the following results:

Theorem 12. A<sup>α</sup> admits defect indices ð Þ 1; 1 if and only if

$$
\varphi\_{\lambda} = (\alpha - \lambda)^{-1} \circ \Theta \in \mathcal{L}\_{\psi}.\tag{38}
$$

In this case φλ ∈ N λ (defect space associated with λ, [3]).

Proof. We know (see [3]) that A<sup>α</sup> has the defect indices 1ð Þ ; 1 if and only if its defect subspaces N λ and N <sup>λ</sup> are unidimensional.

Theorem 13. The operator B<sup>α</sup> is densely defined and closed.

In this case, the functions φð Þ<sup>k</sup>

space N λ.

4. Conclusion

Carleman operators:

Author details

Sidi Mohamed Bahri

Mostaganem, Algeria

References

1993

with K xð Þ ; y a Carleman kernel.

Theorem 14. The operator A<sup>α</sup> admits defect indices mð Þ ; m if and only if

H xð Þ¼ ; <sup>y</sup> K xð Þþ ; <sup>y</sup> <sup>X</sup><sup>m</sup>

<sup>λ</sup> ¼ ð Þ α � λ ∘ Θ<sup>k</sup> ∈Lψ, k ¼ 1, …, m: (42)

A Formal Perturbation Theory of Carleman Operators http://dx.doi.org/10.5772/intechopen.79022

<sup>λ</sup> ð Þ k ¼ 1;…; m are linearly independent and generate the defect

(43)

59

We have seen the interest of multiplication operators in reducing Carleman integral operators and how they simplify the spectral study of these operators with some perturbation. In the same way, we can easily generalize this perturbation theory to the case of the non-densely defined

> j¼1 bjψ<sup>j</sup> ð Þx φ<sup>j</sup> ð Þy ,

<sup>∉</sup>L<sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �; <sup>j</sup> <sup>¼</sup> <sup>1</sup>, m � �,

<sup>φ</sup><sup>j</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �; <sup>ψ</sup><sup>j</sup>

It should be noted that this study allows the estimation of random variables.

Laboratory of Pure and Applied Mathematics, Abdelhamid Ibn Badis University,

Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1970 N 7, 3–12

[1] Akhiezer NI, Glazman IM. Theory of Linear Operators in Hilbert Space. New York: Dover;

[2] Aleksandrov EL, On the resolvents of symmetric operators which are not densely defined,

Address all correspondence to: sidimohamed.bahri@univ-mosta.dz

φð Þ<sup>k</sup>

We have

$$\mathcal{N}\_{\overline{\lambda}} = \ker(A\_{\alpha}^\* - \lambda I) = \ker(B\_{\alpha} - \lambda I).$$

So it suffices to solve the system:

$$\begin{cases} \mathcal{B}\_a \varphi\_\lambda = \lambda \varphi\_\lambda \\ \varphi\_\lambda \in \mathcal{L}\_\psi \end{cases}$$

that is,

$$\begin{cases} Q(\boldsymbol{\alpha} \circ \boldsymbol{\rho}\_{\boldsymbol{\lambda}}) = \lambda \boldsymbol{\rho}\_{\boldsymbol{\lambda}} \Leftrightarrow \begin{cases} (\boldsymbol{\alpha} \circ \boldsymbol{\rho}\_{\boldsymbol{\lambda}}) = \lambda \boldsymbol{\rho}\_{\boldsymbol{\lambda}} + \mu \boldsymbol{\Theta}, \boldsymbol{\mu} \in \mathbb{C} \\ \boldsymbol{\varphi}\_{\boldsymbol{\lambda}} \in \mathcal{L}\_{\psi} \end{cases} \\ \boldsymbol{\Leftrightarrow} \begin{cases} (\boldsymbol{\alpha} - \boldsymbol{\lambda}) \bullet \boldsymbol{\rho}\_{\boldsymbol{\lambda}} = \boldsymbol{\Theta} \\ \boldsymbol{\varphi}\_{\boldsymbol{\lambda}} \in \mathcal{L}\_{\psi} \end{cases} \end{cases}$$
 
$$\boldsymbol{\Leftrightarrow} \begin{cases} \boldsymbol{\varphi}\_{\boldsymbol{\lambda}} = (\boldsymbol{\alpha} - \boldsymbol{\lambda})^{-1} \bullet \boldsymbol{\Theta} \\ \boldsymbol{\varphi}\_{\boldsymbol{\lambda}} \in \mathcal{L}\_{\psi} \end{cases}.$$

■
