1. Introduction

In this chapter, we shall assume that the reader is familiar with the fundamental results and the standard notation of the integral operators theory [1–3, 5, 6, 8–12]. Let X be an arbitrary set, μ be a σ� finite measure on X (μ is defined on a σ� algebra of subsets of X; we do not indicate this <sup>σ</sup>� algebra), and <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � � the Hilbert space of square integrable functions with respect to μ. Instead of writing "μ� measurable," "μ� almost everywhere," and "(dμð Þx )," we write "measurable," "a e," and "dx."

A linear operator <sup>A</sup>: D Að Þ ���! <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �, where the domain D Að Þ is a dense linear manifold in <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �, is said to be integral if there exist a measurable function <sup>K</sup> on <sup>X</sup> � <sup>X</sup>, a kernel, such that, for every f ∈ D Að Þ,

$$Af(\mathbf{x}) = \int\_{X} K(\mathbf{x}, y) f(y) dy \quad \text{a.e.} \tag{1}$$

A kernel <sup>K</sup> on <sup>X</sup> � <sup>X</sup> is said to be Carleman, if K xð Þ ; <sup>y</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � � for almost every fixed x, that is to say

© 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited. © 2018 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.

$$\int\_X |K(x, y)|^2 dy \quad < \text{es} \quad \text{a.e.} \tag{2}$$

Moreover, we have

2. Position operator

� �<sup>∞</sup>

n o<sup>∞</sup>

p¼0 :

where the coefficients anð Þ n∈ N are scalars.

We say that two formal elements f <sup>¼</sup> <sup>P</sup>

The sequence að Þ<sup>n</sup> <sup>n</sup> is said to generate the formal element f .

If φ is a scalar function defined for each an, we set

We start this section by defining some formal spaces.

Definition 1. (see [7]) We call formal element any expression of the form

φ X n

anψ<sup>n</sup> !

Let ψ ¼ ψ<sup>n</sup>

sequence ψpð Þx

2.1. Formal elements

n∈ N.

or in another form,

For example, let

φλð Þ¼ <sup>x</sup> <sup>X</sup><sup>∞</sup>

φak

8 ><

>:

p¼0

ð Þ¼ x ψkð Þx ,

Nλ being the defect space associated with λ (see [3, 4]):.

γp

ap � <sup>λ</sup>ψpð Þ<sup>x</sup> <sup>∈</sup> <sup>N</sup>λ, <sup>λ</sup> <sup>∈</sup> <sup>C</sup>, <sup>λ</sup> 6¼ ak, k <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …

<sup>n</sup>¼<sup>0</sup> be a fixed Carleman sequence in <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �. It is clear from the foregoing that <sup>ψ</sup>

<sup>L</sup><sup>ψ</sup> <sup>¼</sup> span <sup>ψ</sup>n; <sup>n</sup><sup>∈</sup> <sup>N</sup> � �: (11)

anψn, (12)

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

<sup>n</sup> <sup>∈</sup> <sup>N</sup> bnψ<sup>n</sup> are equal if an ¼ bn for all

φð Þ an ψn, (13)

is not a complete sequence in L<sup>2</sup> X; μ � �. We denote by L<sup>ψ</sup> the closure of the linear span of the

<sup>f</sup> <sup>¼</sup> <sup>X</sup> n ∈ N

Definition 2. We say that f is the zero formal element, and we note f ¼ 0 if an ¼ 0 for all n∈ N:

<sup>n</sup><sup>∈</sup> <sup>N</sup> anψ<sup>n</sup> and g <sup>¼</sup> <sup>P</sup>

<sup>¼</sup> <sup>X</sup> n

φða1; a2;…; an; …Þ ¼ ð Þ φð Þ a<sup>1</sup> ;φð Þ a<sup>2</sup> ; …;φð Þ an ;… : (14)

(10)

51

An integral operator A (1) with a kernel K is called Carleman operator, if K is a Carleman kernel (2). Every Carleman kernel <sup>K</sup> defines a Carleman function <sup>k</sup> from <sup>X</sup> to <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � � by k xð Þ¼ K xð Þ ; : for all <sup>x</sup> in <sup>X</sup> for which K xð Þ ; : <sup>∈</sup> <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �:.

Now we consider the Carleman integral operator (1) of second class [3, 8] generated by the following symmetric kernel:

$$\mathcal{K}(\mathbf{x}, \mathbf{y}) = \sum\_{p=0}^{\infty} a\_p \psi\_p(\mathbf{x}) \overline{\psi\_p(\mathbf{y})}.\tag{3}$$

where the overbar in (3) denotes the complex conjugation and ψpð Þx n o<sup>∞</sup> p¼0 is an orthonormal sequence in L<sup>2</sup> X; μ � � such that

$$\sum\_{p=0}^{\infty} \left| \psi\_p(\mathbf{x}) \right|^2 < \infty \text{ a.e.}\tag{4}$$

and ap � �<sup>∞</sup> <sup>p</sup>¼<sup>0</sup> is a real number sequence verifying

$$\sum\_{p=0}^{\infty} a\_p^2 \left| \psi\_p(\mathbf{x}) \right|^2 < \text{\textquotedbl{}a.e.}\tag{5}$$

We call ψpð Þx n o<sup>∞</sup> p¼0 a Carleman sequence.

Moreover, we assume that there exist a numeric sequence γ<sup>p</sup> n o<sup>∞</sup> p¼0 such that

$$\sum\_{p=0}^{\infty} \gamma\_p \psi\_p(\mathbf{x}) = 0 \text{ a.e.}\tag{6}$$

and

$$\sum\_{p=0}^{\infty} \left| \frac{\mathcal{V}\_p}{a\_p - \lambda} \right|^2 < \infty. \tag{7}$$

With the conditions (6) and (7), the symmetric operator <sup>A</sup> <sup>¼</sup> <sup>A</sup><sup>∗</sup> ð Þ<sup>∗</sup> admits the defect indices ð Þ 1; 1 (see [3]), and its adjoint operator is given by

$$A^\*f(\mathbf{x}) = \sum\_{p=0}^{\infty} a\_p \left( f, \psi\_p \right) \psi\_p(\mathbf{x}) . \tag{8}$$

$$D(A^\*) = \left\{ f \in L^2(X, \mu) : \sum\_{p=0}^n a\_p \binom{f}{p} \psi\_p(\mathbf{x}) \in L^2(X, \mu) \right\}.\tag{9}$$

Moreover, we have

ð X

K xð Þ ; : for all <sup>x</sup> in <sup>X</sup> for which K xð Þ ; : <sup>∈</sup> <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �:.

50 Perturbation Methods with Applications in Science and Engineering

following symmetric kernel:

sequence in L<sup>2</sup> X; μ � � such that

and ap � �<sup>∞</sup>

and

We call ψpð Þx

n o<sup>∞</sup>

p¼0

j j K xð Þ ; y 2

K xð Þ¼ ; <sup>y</sup> <sup>X</sup><sup>∞</sup>

where the overbar in (3) denotes the complex conjugation and ψpð Þx

<sup>p</sup>¼<sup>0</sup> is a real number sequence verifying

a Carleman sequence.

Moreover, we assume that there exist a numeric sequence γ<sup>p</sup>

ð Þ 1; 1 (see [3]), and its adjoint operator is given by

X∞ p¼0

X∞ p¼0 a2 <sup>p</sup> ψpð Þx � � �

X∞ p¼0

> X∞ p¼0

f xð Þ¼ <sup>X</sup><sup>∞</sup>

p¼0

X∞ p¼0

A∗

D A<sup>∗</sup> ð Þ¼ <sup>f</sup> <sup>∈</sup>L<sup>2</sup> <sup>X</sup>; <sup>μ</sup> � � :

8 < : � � � �

γp ap � λ

With the conditions (6) and (7), the symmetric operator <sup>A</sup> <sup>¼</sup> <sup>A</sup><sup>∗</sup> ð Þ<sup>∗</sup> admits the defect indices

ap f ; ψ<sup>p</sup> � �

> ap f ;ψ<sup>p</sup> � �

� � � � 2

An integral operator A (1) with a kernel K is called Carleman operator, if K is a Carleman kernel (2). Every Carleman kernel <sup>K</sup> defines a Carleman function <sup>k</sup> from <sup>X</sup> to <sup>L</sup><sup>2</sup> <sup>X</sup>; <sup>μ</sup> � � by k xð Þ¼

Now we consider the Carleman integral operator (1) of second class [3, 8] generated by the

p¼0

ψpð Þx � � �

� � � 2

> � � � 2

dy < ∞ a:e:: (2)

apψpð Þx ψpð Þy , (3)

p¼0

< ∞ a:e:, (4)

< ∞a:e:: (5)

such that

< ∞: (7)

<sup>ψ</sup>pð Þ<sup>x</sup> , (8)

9 =

;: (9)

n o<sup>∞</sup>

p¼0

<sup>ψ</sup>pð Þ<sup>x</sup> <sup>∈</sup>L<sup>2</sup> <sup>X</sup>; <sup>μ</sup> � �

γpψpð Þ¼ x 0 a:e:, (6)

is an orthonormal

n o<sup>∞</sup>

$$\begin{cases} \boldsymbol{\varrho}\_{\boldsymbol{\lambda}}(\mathbf{x}) = \sum\_{p=0}^{m} \frac{\mathcal{V}\_{p}}{a\_{p} - \boldsymbol{\lambda}} \boldsymbol{\psi}\_{p}(\mathbf{x}) \in \mathfrak{N}\_{\overline{\boldsymbol{\lambda}}'} \; \; \; \boldsymbol{\lambda} \in \mathbb{C}, \; \; \; \boldsymbol{\lambda} \neq a\_{k}, \; \; k = 1, 2, \dots \\\ \boldsymbol{\varrho}\_{\boldsymbol{a}\_{k}}(\mathbf{x}) = \boldsymbol{\psi}\_{k}(\mathbf{x}), \end{cases} \tag{10}$$

Nλ being the defect space associated with λ (see [3, 4]):.
