Analytical Applications on Some Hilbert Spaces

*Fethi Soltani*

#### **Abstract**

In this paper, we establish an uncertainty inequality for a Hilbert space *H*. The minimizer function associated with a bounded linear operator from *H* into a Hilbert space *K* is provided. We come up with some results regarding Hardy and Dirichlet spaces on the unit disk .

**Keywords:** Hilbert space, Hardy space, Dirichlet space, uncertainty inequality, minimizer function

#### **1. Introduction**

Hilbert spaces are the most important tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, and ergodicity. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of squareintegrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Saitoh et al. applied the theory of Hilbert spaces to the Tikhonov regularization problems [1, 2]. Matsuura et al. obtained the approximate solutions for bounded linear operator equations with the viewpoint of numerical solutions by computers [3, 4]. During the last years, the theory of Hilbert spaces has gained considerable interest in various fields of mathematical sciences [5–9]. We expect that the results of this paper will be useful when discussing (in Section 2) uncertainty inequality for Hilbert space *H* and minimizer function associated with a bounded linear operator *T* from *H* into a Hilbert space *K*. As applications, we consider Hardy and Dirichlet spaces as follows.

Let be the complex plane and ¼ f g *z*∈ :j*z*j<1 the open unit disk. The Hardy space *H*ð Þ is the set of all analytic functions *f* in the unit disk with the finite integral:

$$\int\_{0}^{2\pi} \left| f\left(\dot{e}^{i\theta}\right) \right|^{2} d\theta. \tag{1}$$

It is a Hilbert space when equipped with the inner product:

$$\langle f, \mathbf{g} \rangle\_{H(\mathbb{D})} = \frac{1}{2\pi} \int\_0^{2\pi} f\left(e^{i\theta}\right) \overline{\mathbf{g}(e^{i\theta})} \,\mathrm{d}\,\theta. \tag{2}$$

Over the years, the applications of Hardy space *H*ð Þ play an important role in various fields of mathematics [5, 10] and in certain parts of quantum mechanics

[11, 12]. And this space is the background of some applications. For example, in Section 3, we study on *H*ð Þ the following two operators:

$$
\nabla f(\mathbf{z}) = f'(\mathbf{z}), \quad \mathbf{L}f(\mathbf{z}) = \mathbf{z}^2 f'(\mathbf{z}) + \mathbf{z}f(\mathbf{z}), \tag{3}
$$

and we deduce uncertainty inequality for this space. Next, we establish the minimizer function associated with the difference operator:

$$T\_{\hat{\mathbf{y}}}f(\mathbf{z}) = \frac{1}{\mathbf{z}}(|f(\mathbf{z}) - f(\mathbf{0})|). \tag{4}$$

∥*A*<sup>∗</sup> *f* ∥<sup>2</sup>

*Analytical Applications on Some Hilbert Spaces DOI: http://dx.doi.org/10.5772/intechopen.90322*

where *i* is the imaginary unit.

(15) follows from Theorems 2.1 and 2.2. □

Δ<sup>þ</sup> *<sup>H</sup>*ð Þ*f* Δ�

*<sup>H</sup>*ð Þ¼ *<sup>f</sup>* <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

<sup>∥</sup>ð Þ *<sup>P</sup>* � *<sup>a</sup> <sup>f</sup>* <sup>∥</sup><sup>2</sup>

min

*<sup>a</sup>*<sup>∈</sup> <sup>∥</sup> *<sup>A</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> ð Þ � *<sup>a</sup> <sup>f</sup>* <sup>∥</sup><sup>2</sup>

<sup>∥</sup> *<sup>A</sup>* � *<sup>A</sup>*<sup>∗</sup> ð Þ <sup>þ</sup> *ib <sup>f</sup>* <sup>∥</sup><sup>2</sup>

and the minimum is attained when *<sup>a</sup>* <sup>¼</sup> h i *Pf*, *<sup>f</sup> <sup>H</sup>*

*<sup>a</sup>* <sup>∈</sup> <sup>∥</sup>ð Þ *<sup>P</sup>* � *<sup>a</sup> <sup>f</sup>* <sup>∥</sup><sup>2</sup>

Δ�

self-adjoint; then for any real *a*, we have

Then

Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* by

where

This shows that

min

Similarly

min *b* ∈

**81**

The following result is proved in [18, 19].

<sup>¼</sup> <sup>∥</sup>*Af* <sup>∥</sup><sup>2</sup>

∥ð Þ *A* � *a f* ∥*H*∥ð Þ *B* � *b f* ∥*<sup>H</sup>* ≥

<sup>∥</sup> *<sup>A</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> ð Þ � *<sup>a</sup> <sup>f</sup>* <sup>∥</sup>*H*<sup>∥</sup> *<sup>A</sup>* � *<sup>A</sup>*<sup>∗</sup> ð Þ <sup>þ</sup> *ib <sup>f</sup>* <sup>∥</sup>*<sup>H</sup>* ≥ ∣∥*Af* <sup>∥</sup><sup>2</sup>

**Theorem 2.4**. Let *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* . Then

*<sup>H</sup>*ð Þ*<sup>f</sup>* ≥ ∥*<sup>f</sup>* <sup>∥</sup><sup>4</sup>

*<sup>H</sup>*<sup>∥</sup> *<sup>A</sup>* � *<sup>A</sup>*<sup>∗</sup> ð Þ*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>H</sup>* <sup>¼</sup> <sup>∥</sup>*Pf* <sup>∥</sup><sup>2</sup>

**Proof.** Let *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* . The operator *<sup>P</sup>* given by (16) is

*<sup>H</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

*<sup>H</sup>* <sup>¼</sup> <sup>∥</sup>*Pf* <sup>∥</sup><sup>2</sup>

*<sup>H</sup>* <sup>¼</sup> <sup>∥</sup> *<sup>A</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> ð Þ*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>H</sup>* <sup>¼</sup> <sup>∥</sup> *<sup>A</sup>* � *<sup>A</sup>*<sup>∗</sup> ð Þ*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

∥*f* ∥<sup>2</sup> *H*

∥*f* ∥<sup>2</sup>

*<sup>H</sup>* � h i *Pf*, *<sup>f</sup> <sup>H</sup>* 2

∥*f* ∥<sup>2</sup> *H*

*<sup>H</sup>* � <sup>h</sup> *<sup>A</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> j j ð Þ*f*, *<sup>f</sup>*i*<sup>H</sup>*

*<sup>H</sup>* � <sup>h</sup> *<sup>A</sup>* � *<sup>A</sup>*<sup>∗</sup> j j ð Þ*f*, *<sup>f</sup>*i*<sup>H</sup>*

∥*f* ∥<sup>2</sup> *H*

∥*f* ∥<sup>2</sup> *H*

. In other words, we have

for all *f* ∈ Domð Þ *AB* ∩ Domð Þ *BA* , and all *a*, *b*∈ .

**Theorem 2.2.** Let *A* and *B* be the self-adjoint operators on a Hilbert space *H*.

**Theorem 2.3.** Let *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* . For all *<sup>a</sup>*, *<sup>b</sup>*<sup>∈</sup> , one has

**Proof.** Let us consider the following two operators on Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup>

It follows that, for *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* , we have *Pf*, *Qf* <sup>∈</sup> *<sup>H</sup>*. The operators *<sup>P</sup>* and *<sup>Q</sup>* are self-adjoint and ½ �¼� *<sup>P</sup>*, *<sup>Q</sup>* <sup>2</sup>*i A*, *<sup>A</sup>*<sup>∗</sup> ½ �. Thus the inequality

*<sup>H</sup>* ∥*Af* ∥<sup>2</sup>

*<sup>H</sup>* <sup>¼</sup> *AA*<sup>∗</sup> h i *<sup>f</sup>*, *<sup>f</sup> <sup>H</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>∗</sup> h i *Af*, *<sup>f</sup> <sup>H</sup>* <sup>þ</sup> *<sup>A</sup>*, *<sup>A</sup>*<sup>∗</sup> h i ½ �*f*, *<sup>f</sup> <sup>H</sup>* (12)

1 2

*<sup>H</sup>* <sup>þ</sup> *<sup>A</sup>*, *<sup>A</sup>*<sup>∗</sup> h i ½ �*f*, *<sup>f</sup> <sup>H</sup>:* □ (13)

∣h i ½ � *A*, *B f*, *f <sup>H</sup>*∣, (14)

*<sup>H</sup>*∣, (15)

, (17)

, (20)

*:* (21)

*:* (22)

2

2

*:* (18)

2

*<sup>H</sup>* � 2*a Pf* h i , *f <sup>H</sup>:* (19)

*<sup>H</sup>* � <sup>∥</sup>*A*<sup>∗</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>P</sup>* <sup>¼</sup> *<sup>A</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> , *<sup>Q</sup>* <sup>¼</sup> *i A* � *<sup>A</sup>*<sup>∗</sup> ð Þ*:* (16)

*<sup>H</sup>* � <sup>∥</sup>*A*<sup>∗</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>H</sup>* � h *<sup>A</sup>* � *<sup>A</sup>*<sup>∗</sup> j j ð Þ*f*, *<sup>f</sup>*i*<sup>H</sup>*

<sup>2</sup>

*H*

In Section 4, we consider the Dirichlet space Dð Þ , which is the set of all analytic functions *f* in the unit disk with the finite Dirichlet integral:

$$\int\_{\mathcal{D}} \left| \, f'(z) \right|^2 \frac{\mathbf{d}x \, \mathrm{d}y}{\pi}, \quad z = x + i\mathrm{y}. \tag{5}$$

It is also a Hilbert space when equipped with the inner product:

$$\langle f, \mathbf{g} \rangle\_{\mathcal{D}(\mathbb{D})} = f(\mathbf{0}) \overline{\mathbf{g}(\mathbf{0})} + \int\_{\mathcal{D}} f'(\mathbf{z}) \overline{\mathbf{g'}(\mathbf{z})} \frac{\mathbf{d} \mathbf{x} \, \mathrm{d}y}{\pi}, \quad \mathbf{z} = \mathbf{x} + i \mathbf{y}. \tag{6}$$

This space is the objective of many applicable works [5, 13–17] and plays a background to our contribution. For example, we study on Dð Þ the following two operators:

$$
\Lambda f(z) = f'(z) - f'(0), \quad Xf(z) = z^2 f'(z), \tag{7}
$$

and we deduce the uncertainty inequality for this space Dð Þ . And we establish the minimizer function associated with the difference operator:

$$T\_2 f(z) = \frac{1}{z} \left( f(z) - zf'(0) - f(0) \right). \tag{8}$$

### **2. Generalized results**

Let *H* be a Hilbert space equipped with the inner product h i *:*, *: <sup>H</sup>*. And let *A* and *B* be the two operators defined on *H*. We define the commutator ½ � *A*, *B* by

$$[A,B] \coloneqq AB - BA.\tag{9}$$

The adjoint of *A* denoted by *A*<sup>∗</sup> is defined by

$$
\langle \mathsf{A}f, \mathsf{g} \rangle\_H = \langle f, \mathsf{A}^\* \mathsf{g} \rangle\_H,\tag{10}
$$

for *<sup>f</sup>* <sup>∈</sup> Domð Þ *<sup>A</sup>* and *<sup>g</sup>* <sup>∈</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ. **Theorem 2.1.** For *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* , one has

$$\|\|A^\*f\|\|\_{H}^2 = \|\|Af\|\|\_{H}^2 + \langle [A, A^\*]f, f\rangle\_{H}.\tag{11}$$

**Proof.** Let *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* . Then *AA*<sup>∗</sup> *<sup>f</sup>* and *<sup>A</sup>*<sup>∗</sup> *Af* belong to *<sup>H</sup>*. Therefore *<sup>A</sup>*, *<sup>A</sup>*<sup>∗</sup> ½ � *<sup>f</sup>* <sup>∈</sup> *<sup>H</sup>*. Hence one has

*Analytical Applications on Some Hilbert Spaces DOI: http://dx.doi.org/10.5772/intechopen.90322*

$$\|A^\*f\|\_{H}^2 = \langle AA^\*f, f\rangle\_{H} = \langle A^\*Af, f\rangle\_{H} + \langle [A, A^\*]f, f\rangle\_{H} \tag{12}$$

$$= \|Af\|\_{H}^{2} + \langle [A, A^\*]f, f\rangle\_{H}. \quad \Box \tag{13}$$

The following result is proved in [18, 19].

**Theorem 2.2.** Let *A* and *B* be the self-adjoint operators on a Hilbert space *H*. Then

$$\|\left(\left(A-a\right)f\right)\|\_{H}\|\left(B-b\right)f\right\|\_{H} \geq \frac{1}{2} |\langle \left[A,B\right]f, f\rangle\_{H}|,\tag{14}$$

for all *f* ∈ Domð Þ *AB* ∩ Domð Þ *BA* , and all *a*, *b*∈ .

$$\text{Theorem 2.3.}\text{ Let}\\
f \in \text{Dom}\left(A A^\*\right) \cap \text{Dom}\left(A^\* A\right).\\
\text{For all } a, b \in \mathbb{R}, \text{ one has}$$

$$\|(\mathbf{A} + \mathbf{A}^\* - a)f\|\_{H} \|(\mathbf{A} - \mathbf{A}^\* + ib)f\|\_{H} \ge |\|\mathbf{A}f\|\_{H}^2 - \|\mathbf{A}^\*f\|\_{H}^2|,\tag{15}$$

where *i* is the imaginary unit.

**Proof.** Let us consider the following two operators on Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* by

$$P = A + A^\* , \quad Q = i(A - A^\*) . \tag{16}$$

It follows that, for *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* , we have *Pf*, *Qf* <sup>∈</sup> *<sup>H</sup>*. The operators *<sup>P</sup>* and *<sup>Q</sup>* are self-adjoint and ½ �¼� *<sup>P</sup>*, *<sup>Q</sup>* <sup>2</sup>*i A*, *<sup>A</sup>*<sup>∗</sup> ½ �. Thus the inequality (15) follows from Theorems 2.1 and 2.2. □

**Theorem 2.4**. Let *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* . Then

$$
\Delta\_H^+(f)\Delta\_H^-(f) \ge \|f\|\_H^4 \left(\|Af\|\_H^2 - \|A^\*f\|\_H^2\right)^2,\tag{17}
$$

where

[11, 12]. And this space is the background of some applications. For example, in

ð Þ*<sup>z</sup>* , *Lf z*ð Þ¼ *<sup>z</sup>*<sup>2</sup> *<sup>f</sup>*

and we deduce uncertainty inequality for this space. Next, we establish the

*z*

In Section 4, we consider the Dirichlet space Dð Þ , which is the set of all analytic

0

ð Þþ *z zf z*ð Þ, (3)

ð Þ *f z*ð Þ� *f*ð Þ 0 *:* (4)

*<sup>π</sup>* , *<sup>z</sup>* <sup>¼</sup> *<sup>x</sup>* <sup>þ</sup> *iy:* (5)

0

ð Þ� <sup>0</sup> *<sup>f</sup>*ð Þ <sup>0</sup> � �*:* (8)

½ � *A*, *B* ≔ *AB* � *BA:* (9)

h i *Af*, *<sup>g</sup> <sup>H</sup>* <sup>¼</sup> *<sup>f</sup>*, *<sup>A</sup>*<sup>∗</sup> h i*<sup>g</sup> <sup>H</sup>*, (10)

*<sup>H</sup>* <sup>þ</sup> *<sup>A</sup>*, *<sup>A</sup>*<sup>∗</sup> h i ½ � *<sup>f</sup>*, *<sup>f</sup> <sup>H</sup>:* (11)

*<sup>π</sup>* , *<sup>z</sup>* <sup>¼</sup> *<sup>x</sup>* <sup>þ</sup> *iy:* (6)

ð Þ*z* , (7)

d*x*d*y*

ð Þ <sup>0</sup> , *Xf z*ð Þ¼ *<sup>z</sup>*<sup>2</sup> *<sup>f</sup>*

Section 3, we study on *H*ð Þ the following two operators:

0

*<sup>T</sup>*1*f z*ð Þ¼ <sup>1</sup>

It is also a Hilbert space when equipped with the inner product:

ð *f* 0 ð Þ*z g*<sup>0</sup> ð Þ*z*

This space is the objective of many applicable works [5, 13–17] and plays a background to our contribution. For example, we study on Dð Þ the following two

and we deduce the uncertainty inequality for this space Dð Þ . And we establish

Let *H* be a Hilbert space equipped with the inner product h i *:*, *: <sup>H</sup>*. And let *A* and *B*

be the two operators defined on *H*. We define the commutator ½ � *A*, *B* by

*<sup>z</sup> f z*ð Þ� *z f*<sup>0</sup>

minimizer function associated with the difference operator:

functions *f* in the unit disk with the finite Dirichlet integral:

ð *f* 0 ð Þ*<sup>z</sup>* � � � � <sup>2</sup> d*x*d*y*

h i *<sup>f</sup>*, *<sup>g</sup>* Dð Þ <sup>¼</sup> *<sup>f</sup>*ð Þ <sup>0</sup> *<sup>g</sup>*ð Þþ <sup>0</sup>

Λ*f z*ð Þ¼ *f*

The adjoint of *A* denoted by *A*<sup>∗</sup> is defined by

**Theorem 2.1.** For *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* , one has

*<sup>H</sup>* <sup>¼</sup> <sup>∥</sup>*Af* <sup>∥</sup><sup>2</sup>

**Proof.** Let *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* . Then *AA*<sup>∗</sup> *<sup>f</sup>* and *<sup>A</sup>*<sup>∗</sup> *Af* belong to *<sup>H</sup>*.

∥*A*<sup>∗</sup> *f* ∥<sup>2</sup>

for *<sup>f</sup>* <sup>∈</sup> Domð Þ *<sup>A</sup>* and *<sup>g</sup>* <sup>∈</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ.

Therefore *<sup>A</sup>*, *<sup>A</sup>*<sup>∗</sup> ½ � *<sup>f</sup>* <sup>∈</sup> *<sup>H</sup>*. Hence one has

**80**

0 ð Þ� *z f* 0

the minimizer function associated with the difference operator:

*<sup>T</sup>*2*f z*ð Þ¼ <sup>1</sup>

operators:

*Functional Calculus*

**2. Generalized results**

∇*f z*ð Þ¼ *f*

$$\Delta\_H^{\pm}(f) = \|f\|\_H^2 \|(A \pm A^\*)f\|\_H^2 - |\langle (A \pm A^\*)f, f\rangle\_H|^2. \tag{18}$$

**Proof.** Let *<sup>f</sup>* <sup>∈</sup> Dom *AA*<sup>∗</sup> ð Þ <sup>∩</sup> Dom *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>A</sup>* . The operator *<sup>P</sup>* given by (16) is self-adjoint; then for any real *a*, we have

$$\left\|(P-a)f\right\|\_{H}^{2} = \left\|Pf\right\|\_{H}^{2} + a^{2}\left\|f\right\|\_{H}^{2} - 2a\langle Pf, f \rangle\_{H}.\tag{19}$$

This shows that

$$\min\_{a \in \mathbb{R}} \left\|(P - a)f\right\|\_{H}^{2} = \left\|Pf\right\|\_{H}^{2} - \frac{\left|\langle Pf, f\rangle\_{H}\right|^{2}}{\left\|f\right\|\_{H}^{2}},\tag{20}$$

and the minimum is attained when *<sup>a</sup>* <sup>¼</sup> h i *Pf*, *<sup>f</sup> <sup>H</sup>* ∥*f* ∥<sup>2</sup> *H* . In other words, we have

$$\min\_{a \in \mathbb{R}} \|(A + A^\* - a)f\|\_{H}^2 = \|(A + A^\*)f\|\_{H}^2 - \frac{|\langle (A + A^\*)f, f\rangle\_H|^2}{\|f\|\_{H}^2}.\tag{21}$$

Similarly

$$\min\_{b \in \mathbb{R}} \|(A - A^\* + ib)f\|\_{H}^2 = \|(A - A^\*)f\|\_{H}^2 - \frac{|\langle (A - A^\*)f, f\rangle\_{H}|^2}{\|f\|\_{H}^2}.\tag{22}$$

Then by (15), (21), and (22), we deduce the inequality (17). □

Let *λ*>0 and let *T* : *H* ! *K* be a bounded linear operator from *H* into a Hilbert space *K*. Building on the ideas of Saitoh [2], we examine the minimizer function associated with the operator *T*.

**Theorem 2.5.** For any *k*∈*K* and for any *λ*>0, the problem

$$\inf\_{f \in H} \left\{ \lambda \| f \|\_{H}^{2} + \| Tf - k \|\_{K}^{2} \right\} \tag{23}$$

The set *<sup>z</sup><sup>n</sup>* f g<sup>∞</sup>

initial problem:

The Szegő kernel *Sz* given for *z*∈ , by

*Analytical Applications on Some Hilbert Spaces DOI: http://dx.doi.org/10.5772/intechopen.90322*

for all *<sup>f</sup>* <sup>∈</sup> *<sup>H</sup>*ð Þ , we have h i *<sup>f</sup>*, *Sz <sup>H</sup>*ð Þ <sup>¼</sup> *f z*ð Þ.

*u*0

0

where *I* is the identity operator.

If *<sup>f</sup>* <sup>∈</sup> *<sup>U</sup>*ð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup>

iii. For *f* ∈ *U*ð Þ , one has

∥*Lf* ∥<sup>2</sup>

i. Let *<sup>f</sup>* <sup>∈</sup> *<sup>U</sup>*ð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup>

<sup>∇</sup>*f z*ð Þ¼ <sup>X</sup><sup>∞</sup>

*n*¼0

∇*f z*ð Þ¼ *f*

unit disk such that

**Theorem 3.1.**

ii. <sup>∇</sup><sup>∗</sup> <sup>¼</sup> *<sup>L</sup>*.

**Proof.**

and

**83**

*Sz*ð Þ¼ *<sup>w</sup>* <sup>X</sup><sup>∞</sup>

ð Þ*z* , ℜ*f z*ð Þ¼ *z f*<sup>0</sup>

These operators satisfy the commutation rule:

∥*f* ∥<sup>2</sup>

*<sup>U</sup>*ð Þ <sup>¼</sup> <sup>1</sup> 2*π* ð<sup>2</sup>*<sup>π</sup>* 0 *f* <sup>0</sup> *e <sup>i</sup><sup>θ</sup>* � � � � � � 2

∥*f* ∥<sup>2</sup>

Thus, the space *U*ð Þ is a subspace of the Hardy space *H*ð Þ .

i. For *f* ∈ *U*ð Þ , then ∇*f*, ℜ*f* and *Lf* belong to *H*ð Þ .

*<sup>H</sup>*ð Þ <sup>¼</sup> ∥∇*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>n</sup>*¼<sup>0</sup>*anzn*, then

*<sup>U</sup>*ð Þ <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*<sup>H</sup>*ð Þ <sup>þ</sup> <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>n</sup>*¼<sup>0</sup>*anz<sup>n</sup>*. Then

*n*¼1

*Lf z*ð Þ¼ <sup>X</sup><sup>∞</sup>

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *an*þ<sup>1</sup>*z<sup>n</sup>*, <sup>ℜ</sup>*f z*ð Þ¼ <sup>X</sup><sup>∞</sup>

*n*¼1 *n*2 j j *an* <sup>2</sup>

*n*¼0

*<sup>n</sup>*¼<sup>0</sup> forms an Hilbert's basis for the space *<sup>H</sup>*ð Þ .

*<sup>z</sup>nwn* <sup>¼</sup> <sup>1</sup>

is a reproducing kernel for the Hardy space *H*ð Þ , meaning that *Sz* ∈ *H*ð Þ , and

For *z*∈ , the function *u z*ð Þ¼ *Sz*ð Þ *w* is the unique analytic solution on of the

In the next of this section, we define the operators ∇, ℜ, and *L* on *H*ð Þ by

We define the Hilbert space *U*ð Þ as the space of all analytic functions *f* in the

ð Þ¼ *z w zu*<sup>0</sup> ð Þ ð Þþ *z u z*ð Þ , *w* ∈ , *u*ð Þ¼ 0 1*:* (32)

0

½ �¼ ∇, *L* ∇*L* � *L*∇ ¼ 2ℜ þ *I*, (34)

ð Þ*<sup>z</sup>* , *Lf z*ð Þ¼ *<sup>z</sup>*<sup>2</sup> *<sup>f</sup>*

<sup>1</sup> � *zw* , *<sup>w</sup>* <sup>∈</sup> , (31)

ð Þþ *z zf z*ð Þ*:* (33)

d*θ* < ∞*:* (35)

*:* (36)

*<sup>H</sup>*ð Þ <sup>þ</sup> <sup>2</sup>h i <sup>ℜ</sup>*f*, *<sup>f</sup> <sup>H</sup>*ð Þ *:* (37)

*nan*�<sup>1</sup>*z<sup>n</sup>:* (39)

*nanzn*, (38)

*n*¼1

has a unique minimizer given by

$$f\_{\lambda,k}^{\*} = \left(\lambda I + T^\*T\right)^{-1}T^\*k.\tag{24}$$

**Proof.** The problem (23) is solved elementarily by finding the roots of the first derivative *<sup>D</sup>*<sup>Φ</sup> of the quadratic and strictly convex function <sup>Φ</sup>ð Þ¼ *<sup>f</sup> <sup>λ</sup>*∥*<sup>f</sup>* <sup>∥</sup><sup>2</sup> *<sup>H</sup>* þ ∥*Tf* � *k*∥<sup>2</sup> *<sup>K</sup>*. Note that for convex functions, the equation *D*Φð Þ¼ *f* 0 is a necessary and sufficient condition for the minimum at *f*. The calculation provides

$$D\Phi(f) = 2\mathcal{Y} + 2T^\*(Tf - k),\tag{25}$$

and the assertion of the theorem follows at once. □

**Theorem 2.6.** If *T* : *H* ! *K* is an isometric isomorphism; then for any *k*∈*K* and for any *λ*>0, the problem

$$\inf\_{f \in H} \left\{ \lambda \| f \|\_{H}^{2} + \| Tf - k \|\_{K}^{2} \right\} \tag{26}$$

has a unique minimizer given by

$$f\_{\lambda,h}^{\*} = \frac{1}{\lambda+1} T^{-1} k. \tag{27}$$

**Proof.** We have *<sup>T</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> and *<sup>T</sup>* <sup>∗</sup> *<sup>T</sup>* <sup>¼</sup> *<sup>I</sup>*. Thus, by (24), we deduce the result. □
