**4. The Dirichlet space** Dð Þ

The Dirichlet space Dð Þ is the set of all analytic functions *f* in the unit disk with the finite Dirichlet integral:

$$\int\_{\mathcal{D}} \left| \, f'(z) \right|^2 \frac{\mathbf{d} \boldsymbol{x} \, \mathrm{d}y}{\pi}, \quad z = \varkappa + i\boldsymbol{\eta}. \tag{59}$$

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

$$f(f, \mathbf{g})\_{\mathcal{D}(\mathbb{D})} = f(\mathbf{0})\overline{\mathbf{g}(\mathbf{0})} + \int\_{\mathbb{D}} f'(z)\overline{\mathbf{g'}(z)}\frac{\mathbf{d}\mathbf{x}\,\mathrm{d}y}{\pi}, \quad z = \infty + i\mathbf{y}.\tag{60}$$

If *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup> Dð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anz<sup>n</sup>* and *g z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*bnz<sup>n</sup>*, then

$$
\langle f, \mathfrak{g} \rangle\_{\mathcal{D}(\mathbb{D})} = a\_0 \overline{b\_0} + \sum\_{n=1}^{\infty} n a\_n \overline{b\_n}. \tag{61}
$$

The set 1, *<sup>z</sup><sup>n</sup>* ffiffi *n* p n o<sup>∞</sup> *n*¼1 forms an Hilbert's basis for the space Dð Þ . The function *Kz* given for *z*∈ , by

$$K\_x(w) = 1 + \log\left(\frac{1}{1 - \overline{z}w}\right), \quad w \in \mathbb{D}, \tag{62}$$

is a reproducing kernel for the Dirichlet space Dð Þ , meaning that *Kz* ∈ Dð Þ , and for all *<sup>f</sup>* <sup>∈</sup> Dð Þ , we have h i *<sup>f</sup>*,*Kz* Dð Þ <sup>¼</sup> *f z*ð Þ.

For *z*∈ , the function *u z*ð Þ¼ *Kz*ð Þ *w* is the unique analytic solution on of the initial problem:

$$\frac{u'(z) - u'(0)}{z} = \nu u'(z), \quad w \in \mathbb{D}, \quad u(0) = 1. \tag{63}$$

In the next of this section, we define the operators Λ, ℜ, and *X* on Dð Þ by

$$
\Lambda f(z) = f'(z) - f'(0), \quad \Re f(z) = zf'(z), \quad Xf(z) = z^2 f'(z). \tag{64}
$$

These operators satisfy the following commutation relation:

$$[\Lambda, X] = \Lambda X - X\Lambda = 2\Re.\tag{65}$$

∥ℜ*f* ∥<sup>2</sup>

∥*Xf* ∥<sup>2</sup>

Consequently Λ*f*, ℜ*f*, and *Xf* belong to Dð Þ .

iii. Let *f* ∈*V*ð Þ . By (ii) and (65), we deduce that

ii. For *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup>*V*ð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup>

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

*n*¼1

ii. For *<sup>f</sup>* <sup>∈</sup> Dð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup>

<sup>2</sup> *f z*ð Þ¼ <sup>X</sup><sup>∞</sup>

has a unique minimizer given by

*n*¼2

*T* ∗

**87**

h i <sup>Λ</sup>*f*, *<sup>g</sup>* Dð Þ <sup>¼</sup> <sup>X</sup><sup>∞</sup>

and

Dð Þ <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*n*¼1

Dð Þ <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*n n*ð Þ <sup>þ</sup> <sup>1</sup> *an*þ<sup>1</sup>*bn* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

∥*Xf* ∥<sup>2</sup>

<sup>¼</sup> <sup>∥</sup>Λ*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

**Theorem 4.3.** Let *T*<sup>2</sup> be the difference operator defined on Dð Þ by

i. The operator *T*<sup>2</sup> maps continuously from Dð Þ to Dð Þ , and

*λ*∥*f* ∥<sup>2</sup>

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

*<sup>n</sup>*¼<sup>0</sup>*anzn*, we have

Dð Þ <sup>þ</sup> <sup>∥</sup>*T*2*<sup>f</sup>* � *<sup>d</sup>*∥<sup>2</sup>

*z<sup>n</sup>*þ<sup>1</sup> *λ*ð Þþ *n* þ 1 *n*

*an*�<sup>1</sup>*z<sup>n</sup>*, *<sup>T</sup>*<sup>∗</sup>

**Theorem 4.2.** Let *f* ∈*V*ð Þ . For all *a*, *b*∈ , one has

*<sup>T</sup>*2*f z*ð Þ¼ <sup>1</sup>

*n* � 1 *n*

iii. For any *d*∈ Dð Þ and for any *λ*>0, the problem

inf *f* ∈ Dð Þ

*f* ∗

<sup>Ψ</sup>*z*ð Þ¼ *<sup>w</sup>* <sup>X</sup><sup>∞</sup>

*n*¼1

*n*¼1 *n*3

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*<sup>2</sup>

*n*¼2

j j *an* <sup>2</sup> <sup>¼</sup> <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>n</sup>*¼<sup>0</sup>*anz<sup>n</sup>* and *g z*ð Þ¼ <sup>P</sup><sup>∞</sup>

<sup>∥</sup>ð Þ <sup>Λ</sup> <sup>þ</sup> *<sup>X</sup>* � *<sup>a</sup> <sup>f</sup>* <sup>∥</sup>Dð Þ <sup>∥</sup>ð Þ <sup>Λ</sup> � *<sup>X</sup>* <sup>þ</sup> *ib <sup>f</sup>* <sup>∥</sup>Dð Þ <sup>≥</sup>2h i <sup>ℜ</sup>*f*, *<sup>f</sup>* Dð Þ *:* (78)

j j *an* <sup>2</sup> <sup>≤</sup> <sup>2</sup>∥*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>V</sup>*ð Þ , (72)

*<sup>V</sup>*ð Þ *:* (73)

*<sup>n</sup>*¼<sup>0</sup>*bnzn*, one has

*n n*ð Þ � <sup>1</sup> *anbn*�<sup>1</sup> <sup>¼</sup> h i *<sup>f</sup>*, *Xg* Dð Þ *:* (74)

Dð Þ <sup>¼</sup> h i <sup>Λ</sup>*Xf*, *<sup>f</sup>* Dð Þ (75)

Dð Þ <sup>þ</sup> <sup>2</sup>h i <sup>ℜ</sup>*f*, *<sup>f</sup>* Dð Þ *:* □ (77)

ð Þ� <sup>0</sup> *<sup>f</sup>*ð Þ <sup>0</sup> � �*:* (79)

∥*T*2*f* ∥Dð Þ ≤ ∥*f* ∥Dð Þ *:* (80)

*n*¼2

Dð Þ n o (82)

*<sup>λ</sup>*,*<sup>d</sup>*ð Þ¼ *<sup>z</sup>* h i *<sup>d</sup>*, <sup>Ψ</sup>*<sup>z</sup>* Dð Þ , *<sup>z</sup>*<sup>∈</sup> , (83)

*n* � 1 *n*

*w<sup>n</sup>*, *w* ∈ *:* (84)

*anz<sup>n</sup>:* (81)

<sup>2</sup> *<sup>T</sup>*2*f z*ð Þ¼ <sup>X</sup><sup>∞</sup>

<sup>¼</sup> h i *<sup>X</sup>*Λ*f*, *<sup>f</sup>* Dð Þ <sup>þ</sup> h i ½ � <sup>Λ</sup>,*<sup>X</sup> <sup>f</sup>*, *<sup>f</sup>* Dð Þ (76)

We define the Hilbert space *V*ð Þ as the space of all analytic functions *f* in the unit disk such that

$$\|\|f\|\|\_{V(\mathbb{D})}^2 = \int\_{\mathbb{D}} \left| \left. f'(z) \right|^2 \left| z \right|^2 \frac{\mathbf{d} \times \mathbf{d} \mathbf{y}}{\pi} < \infty, \quad z = \infty + i\mathbf{y}. \tag{66}$$

If *<sup>f</sup>* <sup>∈</sup>*V*ð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anz<sup>n</sup>*, then

$$\left\|f\right\|\_{V(\mathbb{D})}^2 = \sum\_{n=1}^{\infty} n^3 \left|a\_n\right|^2. \tag{67}$$

Thus, the space *V*ð Þ is a subspace of the Dirichlet space Dð Þ . **Theorem 4.1.**

i. For *f* ∈*V*ð Þ , then Λ*f*, ℜ*f*, and *Xf* belong to Dð Þ .

ii. <sup>Λ</sup><sup>∗</sup> <sup>¼</sup> *<sup>X</sup>*.

iii. For *f* ∈*V*ð Þ , one has

$$\|\mathbf{X}\mathbf{f}\|\_{\mathcal{D}(\mathbb{D})}^2 = \|\Lambda\mathbf{f}\|\_{\mathcal{D}(\mathbb{D})}^2 + 2\langle\mathfrak{R}\mathbf{f},\mathbf{f}\rangle\_{\mathcal{D}(\mathbb{D})}.\tag{68}$$

#### **Proof.**

i. Let *<sup>f</sup>* <sup>∈</sup>*V*ð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anzn*. Then

$$\Lambda f(z) = \sum\_{n=1}^{\infty} (n+1)a\_{n+1}z^n, \quad \Re f(z) = \sum\_{n=1}^{\infty} n a\_n z^n,\tag{69}$$

and

$$X\!f^{\!\!f}(z) = \sum\_{n=2}^{\infty} (n-1)a\_{n-1}z^n. \tag{70}$$

Therefore

$$\left\|\Lambda f\right\|\_{\mathcal{D}(\mathbb{D})}^2 = \sum\_{n=1}^{\infty} n(n+1)^2 |a\_{n+1}|^2 \le \sum\_{n=2}^{\infty} n^3 |a\_n|^2 \le \left\|f\right\|\_{V(\mathbb{D})}^2,\tag{71}$$

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

$$\|\Re f\|\_{\mathcal{D}(\mathbb{D})}^2 = \sum\_{n=1}^{\infty} n^3 |a\_n|^2 = \|f\|\_{V(\mathbb{D})}^2,\tag{72}$$

and

is a reproducing kernel for the Dirichlet space Dð Þ , meaning that *Kz* ∈ Dð Þ ,

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

In the next of this section, we define the operators Λ, ℜ, and *X* on Dð Þ by

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

<sup>2</sup> d*x*d*y*

ð Þ 0 , ℜ*f z*ð Þ¼ *z f*<sup>0</sup>

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

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

*n*¼1 *n*3 j j *an* <sup>2</sup>

∥*f* ∥<sup>2</sup>

Thus, the space *V*ð Þ is a subspace of the Dirichlet space Dð Þ .

Dð Þ <sup>¼</sup> <sup>∥</sup>Λ*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*Xf z*ð Þ¼ <sup>X</sup><sup>∞</sup>

*n n*ð Þ <sup>þ</sup> <sup>1</sup> <sup>2</sup>

*n*¼2

j j *an*þ<sup>1</sup>

<sup>2</sup> <sup>≤</sup> <sup>X</sup><sup>∞</sup> *n*¼2 *n*3

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

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

i. For *f* ∈*V*ð Þ , then Λ*f*, ℜ*f*, and *Xf* belong to Dð Þ .

∥*Xf* ∥<sup>2</sup>

These operators satisfy the following commutation relation:

ð *f* 0 ð Þ*<sup>z</sup>* � � � � 2 j j *z*

ð Þ*z* , *w* ∈ , *u*ð Þ¼ 0 1*:* (63)

0

*<sup>π</sup>* <sup>&</sup>lt; <sup>∞</sup>, *<sup>z</sup>* <sup>¼</sup> *<sup>x</sup>* <sup>þ</sup> *iy:* (66)

Dð Þ <sup>þ</sup> <sup>2</sup>h i <sup>ℜ</sup>*f*, *<sup>f</sup>* Dð Þ *:* (68)

*nanz<sup>n</sup>*, (69)

*<sup>V</sup>*ð Þ , (71)

*n*¼1

ð Þ *<sup>n</sup>* � <sup>1</sup> *an*�<sup>1</sup>*z<sup>n</sup>:* (70)

j j *an* <sup>2</sup> ≤ ∥*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*:* (67)

ð Þ*z :* (64)

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

½ �¼ Λ, *X* Λ*X* � *X*Λ ¼ 2ℜ*:* (65)

and for all *<sup>f</sup>* <sup>∈</sup> Dð Þ , we have h i *<sup>f</sup>*,*Kz* Dð Þ <sup>¼</sup> *f z*ð Þ.

ð Þ 0 *<sup>z</sup>* <sup>¼</sup> *wu*<sup>0</sup>

*u*0 ð Þ� *z u*<sup>0</sup>

0 ð Þ� *z f* 0

∥*f* ∥<sup>2</sup> *<sup>V</sup>*ð Þ ¼

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

Λ*f z*ð Þ¼ *f*

unit disk such that

**Theorem 4.1.**

ii. <sup>Λ</sup><sup>∗</sup> <sup>¼</sup> *<sup>X</sup>*.

**Proof.**

and

**86**

Therefore

iii. For *f* ∈*V*ð Þ , one has

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

∥Λ*f* ∥<sup>2</sup>

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

Dð Þ <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*n*¼1

*n*¼1

initial problem:

*Functional Calculus*

$$\|\mathbf{X}\mathbf{f}^{\prime}\|\_{\mathcal{D}(\mathbb{D})}^2 = \sum\_{n=1}^{\infty} (n+1)n^2|a\_n|^2 \le 2\|\mathbf{f}^{\prime}\|\_{V(\mathbb{D})}^2. \tag{73}$$

Consequently Λ*f*, ℜ*f*, and *Xf* belong to Dð Þ .

$$\text{iii.}\text{ For}\,f,\text{g}\in V(\mathbb{D})\text{ with}\,f(\mathbf{z})=\sum\_{n=0}^{\infty}a\_{n}\mathbf{z}^{n}\text{ and}\,\mathbf{g}(\mathbf{z})=\sum\_{n=0}^{\infty}b\_{n}\mathbf{z}^{n},\text{ one has}$$

$$\langle \langle \Lambda f, \mathbf{g} \rangle\_{\mathcal{D}(\mathbb{D})} = \sum\_{n=1}^{\infty} n(n+1)a\_{n+1}\overline{b\_n} = \sum\_{n=2}^{\infty} n(n-1)a\_n\overline{b\_{n-1}} = \langle f, \mathbf{Xg} \rangle\_{\mathcal{D}(\mathbb{D})}.\tag{74}$$

iii. Let *f* ∈*V*ð Þ . By (ii) and (65), we deduce that

$$\|\mathbf{X}\mathbf{f}^{\mathbf{f}}\|\_{\mathcal{D}(\mathbb{D})}^2 = \langle \Lambda \mathbf{X}\mathbf{f}, \mathbf{f} \rangle\_{\mathcal{D}(\mathbb{D})} \tag{75}$$

$$=\langle \mathbf{X}\Lambda\mathbf{f}, \mathbf{f} \rangle\_{\mathcal{D}(\mathbb{D})} + \langle [\Lambda, \mathbf{X}]\mathbf{f}, \mathbf{f} \rangle\_{\mathcal{D}(\mathbb{D})} \tag{76}$$

$$= \|\Lambda f\|\_{\mathcal{D}(\mathbb{D})}^2 + 2\langle \Re f, f \rangle\_{\mathcal{D}(\mathbb{D})}. \quad \square \tag{77}$$

**Theorem 4.2.** Let *f* ∈*V*ð Þ . For all *a*, *b*∈ , one has

$$\|(\Lambda + X - a)f\|\_{\mathcal{D}(\mathbb{D})} \|(\Lambda - X + ib)f\|\_{\mathcal{D}(\mathbb{D})} \ge 2\langle \Re f, f \rangle\_{\mathcal{D}(\mathbb{D})}.\tag{78}$$

**Theorem 4.3.** Let *T*<sup>2</sup> be the difference operator defined on Dð Þ by

$$T\_{\mathfrak{Y}}f(\mathbf{z}) = \frac{1}{\mathbf{z}} \left( f(\mathbf{z}) - zf'(\mathbf{0}) - f(\mathbf{0}) \right). \tag{79}$$

i. The operator *T*<sup>2</sup> maps continuously from Dð Þ to Dð Þ , and

$$\|\|T\_{\mathfrak{Y}}f\|\_{\mathcal{D}(\mathbb{D})} \le \|f\|\_{\mathcal{D}(\mathbb{D})}.\tag{80}$$

ii. For *<sup>f</sup>* <sup>∈</sup> Dð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anzn*, we have

$$T\_2^\* f(z) = \sum\_{n=2}^{\infty} \frac{n-1}{n} a\_{n-1} z^n, \quad T\_2^\* \, T\_3 f(z) = \sum\_{n=2}^{\infty} \frac{n-1}{n} a\_n z^n. \tag{81}$$

iii. For any *d*∈ Dð Þ and for any *λ*>0, the problem

$$\inf\_{f \in \mathcal{D}(\mathbb{D})} \left\{ \lambda \| f \|\_{\mathcal{D}(\mathbb{D})}^2 + \| T\_{\mathcal{Q}} f - d \|\_{\mathcal{D}(\mathbb{D})}^2 \right\} \tag{82}$$

has a unique minimizer given by

$$\langle f\_{\lambda,d}^\*(z) = \langle d, \Psi\_z \rangle\_{\mathcal{D}(\mathbb{D})}, \quad z \in \mathbb{D}, \tag{83}$$

$$\Psi\_x(w) = \sum\_{n=1}^{\infty} \frac{\overline{z}^{n+1}}{\lambda(n+1) + n} w^n, \quad w \in \mathbb{D}. \tag{84}$$

#### **Proof.**

$$\text{i. } \mathbf{If} \, f \in \mathcal{D}(\mathbb{D}) \text{ with} \\ f(\mathbf{z}) = \sum\_{n=0}^{\infty} a\_n \mathbf{z}^n \text{, then } \mathbf{T}\_2 f(\mathbf{z}) = \sum\_{n=1}^{\infty} a\_{n+1} \mathbf{z}^n \text{ and}$$

$$\|\mathbf{T}\_2 f\|\_{\mathcal{D}(\mathbb{D})}^2 = \sum\_{n=2}^{\infty} (n-1) |a\_n|^2 \le \sum\_{n=2}^{\infty} n |a\_n|^2 \le \|f\|\_{\mathcal{D}(\mathbb{D})}^2. \tag{85}$$

ii. If *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup> Dð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*anzn* and *g z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*bnz<sup>n</sup>*, then

$$\langle T\_{\mathfrak{J}}f, \mathfrak{g} \rangle\_{\mathcal{D}(\mathbb{D})} = \sum\_{n=1}^{\infty} n a\_{n+1} \overline{b\_n} = \sum\_{n=2}^{\infty} (n-1) a\_n \overline{b\_{n-1}} = \langle f, T\_2^\* \mathfrak{g} \rangle\_{\mathcal{D}(\mathbb{D})},\tag{86}$$

where

$$T\_2^\*\mathbf{g}(\mathbf{z}) = \sum\_{n=2}^{\infty} \frac{n-1}{n} b\_{n-1} \mathbf{z}^n, \quad \mathbf{z} \in \mathbb{D}.\tag{87}$$

And therefore

$$T\_2^\* \, T\_2 f(z) = \sum\_{n=2}^{\infty} \frac{n-1}{n} a\_n z^n. \tag{88}$$

iii. We put *d z*ð Þ¼ <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*dnz<sup>n</sup>* and

$$f\_{\lambda,d}^\*(\mathbf{z}) = \sum\_{n=0}^{\infty} c\_n \mathbf{z}^n. \tag{89}$$

From (ii) and the equation

$$(\lambda I + T\_2^\* T\_2) f\_{\lambda d}^\*(z) = T\_2^\* d(z),\tag{90}$$

**Author details**

Fethi Soltani1,2

**89**

1 Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire

2 Université de Carthage, Ecole Nationale d'Ingénieurs de Carthage, Tunis, Tunisia

© 2020 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,

d'Analyse Mathématique et Applications LR11ES11, Tunis, Tunisia

\*Address all correspondence to: fethi.soltani@fst.utm.tn

provided the original work is properly cited.

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

we deduce that

$$c\_1 = c\_0 = 0, \quad c\_n = \frac{n-1}{\lambda n + n - 1} d\_{n-1}, \quad n \ge 2. \tag{91}$$

Thus

$$\,\_1f\_{\lambda,d}^\*(z) = \sum\_{n=1}^\infty \frac{nd\_n}{\lambda(n+1) + n} z^{n+1} = \langle d, \Psi\_z \rangle\_{\mathcal{D}(\mathbb{D})}, \quad z \in \mathbb{D}. \quad \square \tag{92}$$

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

**Proof.**

*Functional Calculus*

where

And therefore

iii. We put *d z*ð Þ¼ <sup>P</sup><sup>∞</sup>

From (ii) and the equation

we deduce that

*f* ∗

*<sup>λ</sup>*,*<sup>d</sup>*ð Þ¼ *<sup>z</sup>* <sup>X</sup><sup>∞</sup>

*n*¼1

Thus

**88**

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

∥*T*2*f* ∥<sup>2</sup>

ii. If *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup> Dð Þ with *f z*ð Þ¼ <sup>P</sup><sup>∞</sup>

*n*¼1

*T*∗

h i *<sup>T</sup>*2*f*, *<sup>g</sup>* Dð Þ <sup>¼</sup> <sup>X</sup><sup>∞</sup>

Dð Þ <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*n*¼2

*nan*þ<sup>1</sup>*bn* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

<sup>2</sup> *g z*ð Þ¼ <sup>X</sup><sup>∞</sup>

*T*∗

*<sup>n</sup>*¼<sup>0</sup>*dnz<sup>n</sup>* and

*f* ∗

*<sup>λ</sup><sup>I</sup>* <sup>þ</sup> *<sup>T</sup>*<sup>∗</sup> <sup>2</sup> *T*<sup>2</sup> � �*f* <sup>∗</sup>

*<sup>c</sup>*<sup>1</sup> <sup>¼</sup> *<sup>c</sup>*<sup>0</sup> <sup>¼</sup> 0, *cn* <sup>¼</sup> *<sup>n</sup>* � <sup>1</sup>

*ndn λ*ð Þþ *n* þ 1 *n*

*n*¼2

*n* � 1 *n*

*n*¼2

*n*¼0

*<sup>λ</sup>*,*<sup>d</sup>*ð Þ¼ *<sup>z</sup> <sup>T</sup>*<sup>∗</sup>

*λn* þ *n* � 1

*<sup>λ</sup>*,*<sup>d</sup>*ð Þ¼ *<sup>z</sup>* <sup>X</sup><sup>∞</sup>

*n* � 1 *n*

*n*¼2

<sup>2</sup> *<sup>T</sup>*2*f z*ð Þ¼ <sup>X</sup><sup>∞</sup>

*<sup>n</sup>*¼<sup>0</sup>*anzn*, then *<sup>T</sup>*2*f z*ð Þ¼ <sup>P</sup><sup>∞</sup>

*<sup>n</sup>*¼<sup>0</sup>*anzn* and *g z*ð Þ¼ <sup>P</sup><sup>∞</sup>

*n*¼2

ð Þ *<sup>n</sup>* � <sup>1</sup> *anbn*�<sup>1</sup> <sup>¼</sup> *<sup>f</sup>*, *<sup>T</sup>*<sup>∗</sup>

*n a*j j *<sup>n</sup>* <sup>2</sup> ≤ ∥*<sup>f</sup>* <sup>∥</sup><sup>2</sup>

ð Þ *<sup>n</sup>* � <sup>1</sup> j j *an* <sup>2</sup> <sup>≤</sup> <sup>X</sup><sup>∞</sup>

*<sup>n</sup>*¼<sup>1</sup>*an*þ<sup>1</sup>*zn* and

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

*bn*�<sup>1</sup>*zn*, *<sup>z</sup>*<sup>∈</sup> *:* (87)

*anzn:* (88)

*cnz<sup>n</sup>:* (89)

<sup>2</sup> *d z*ð Þ, (90)

*dn*�1, *n*≥2*:* (91)

*zn*þ<sup>1</sup> <sup>¼</sup> h i *<sup>d</sup>*, <sup>Ψ</sup>*<sup>z</sup>* Dð Þ , *<sup>z</sup>*<sup>∈</sup> *:* □ (92)

<sup>2</sup> *<sup>g</sup>* � �

Dð Þ *:* (85)

Dð Þ , (86)
