Open access peer-reviewed chapter

# A Formal Perturbation Theory of Carleman Operators

Written By

Sidi Mohamed Bahri

Submitted: 16 November 2017 Reviewed: 22 May 2018 Published: 17 October 2018

DOI: 10.5772/intechopen.79022

From the Edited Volume

## Perturbation Methods with Applications in Science and Engineering

Edited by İlkay Bakırtaş

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## Abstract

In this chapter, we introduce a multiplication operation that allows us to give to the Carleman integral operator of second class the form of a multiplication operator. Also we establish the formal theory of perturbation of such operators.

### Keywords

• Carleman kernel
• defect indices
• integral operator
• formal series

## 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, 2, 3, 5, 6, 8, 9, 10, 11, 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 σ algebra), and L2Xμ the Hilbert space of square integrable functions with respect to μ. Instead of writing “μ measurable,” “μ almost everywhere,” and “(x),” we write “measurable,” “a e,” and “dx.”

A linear operator A: DA L2Xμ, where the domain DA is a dense linear manifold in L2Xμ, is said to be integral if there exist a measurable function K on X×X, a kernel, such that, for every fDA,

Afx=XKxyfydya.e..E1

A kernel K on X×X is said to be Carleman, if KxyL2Xμ for almost every fixed x, that is to say

XKxy2dy<a.e..E2

An integral operator A (1) with a kernel K is called Carleman operator, if K is a Carleman kernel (2). Every Carleman kernel K defines a Carleman function k from X to L2Xμ by kx = Kx.¯ for all x in X for which Kx. L2Xμ..

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

Kxy=p=0apψpxψpy¯,E3

where the overbar in (3) denotes the complex conjugation and ψpxp=0 is an orthonormal sequence in L2Xμ such that

p=0ψpx2<a.e.,E4

and app=0 is a real number sequence verifying

p=0ap2ψpx2<a.e..E5

We call ψpxp=0 a Carleman sequence.

Moreover, we assume that there exist a numeric sequence γpp=0 such that

p=0γpψpx=0a.e.,E6

and

p=0γpapλ2<.E7

With the conditions (6) and (7), the symmetric operator A=A admits the defect indices 11 (see [3]), and its adjoint operator is given by

Afx=p=0apfψpψpx,E8
DA=fL2Xμ:p=0apfψpψpxL2Xμ.E9

Moreover, we have

φλx=p=0γpapλψpxNλ¯,λC,λak,k=1,2,φakx=ψkx,E10

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

## 2. Position operator

Let ψ=ψnn=0 be a fixed Carleman sequence in L2Xμ. It is clear from the foregoing that ψ is not a complete sequence in L2Xμ. We denote by Lψ the closure of the linear span of the sequence ψpxp=0:

Lψ=spanψnnN¯.E11

We start this section by defining some formal spaces.

### 2.1. Formal elements

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

f=nNanψn,E12

where the coefficients annN are scalars.

The sequence ann is said to generate the formal element f.

Definition 2. We say that f is the zero formal element, and we note f=0 if an=0 for all nN.

We say that two formal elements f=nNanψn and g=nNbnψn are equal if an=bn for all nN.

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

φnanψn=nφanψn,E13

or in another form,

φa1a2an=φa1φa2φan.E14

For example, let

φx=1x,x0.E15

If an0 for all nN, then the formal element

φnanψn=n1anψnE16

is called inverse of the formal element f=nanψn.

Furthermore, we define the conjugate of a formal element f by

f¯=nan¯ψn.E17

Denote by Fψ the set of all formal elements (12).

On Fψ, we define the following algebraic operations:

the sum

+:Fψ×FψFψnanψn+nbnψn=nan+bnψnE18

and the product

:C×FψFψλnanψn=nλ.anψn.E19

Hence, we obtain a complex vector space structure for Fψ.

### 2.2. Bounded formal elements

Definition 3. A formal element f=nNanψn is bounded if its sequence ann is bounded.

We denote by Bψ the set of all bounded formal elements.

It is clear that Bψ is a subspace of Fψ.

We claim that:

1. Lψ is a subspace of Bψ.

2. Furthermore we have the strict inclusions:

LψBψFψ.E20

We define a linear form .. on Fψ by setting:

nanψnnbnψn=nanbn¯E21

with the series converging on the right side of (21).

Proposition 4. The form (21) verifies the properties of scalar product.

Proof. Indeed, let

f=nanψn,g=nbnψn,f1=nan1ψnandf2=nan2ψn

in Fψ.

We have then:

1. fg=nanbn¯=nanbn¯¯=fg¯.

2. λfg=λnanψnnanψn=nλanψnnbnψn=nλanbn¯=λnanψnnbnψn=λfg.

3. f1+f2g=nan1+an2ψnnbnψn=nan1+an2bn¯=nan1bn¯+nan2bn¯=f1g+f2g.

4. ff=nan20andff>0iff0.

Remark 5. On Lψ, the scalar product .. coincides with the scalar product .. of L2Xμ.

### 2.3. The multiplication operation

Here, we introduce the crucial tool of our work.

Definition 6. We call multiplication with respect to the Carleman sequence ψnn, the operation denoted “” and defined by:

fg=nfψngψn=nanbnψn,fgFψ2.E22

Definition 7. We call position operator in Lψ any unitary self-adjoint (see [1]) operator satisfying

Ufg=UfUg,forallf,gLψ.E23

The term “position operator” comes from the fact (as it will be shown in the following theorem) that for the elements of the sequence ψ=ψnn, the operator U acts as operator of change of position of these elements.

### 2.4. Main results

Theorem 8. A linear operator defined on Lψ is a position operator if and only if there exist an involution j (i.e., j2 =Id) of the set N such that for all nN

Uψn=ψjn.E24

Proof.

1. It is easy to see that if (24) holds, then U is a position operator.

2. Let U be a position operator. According to 1, we can write

Uψn=kαn,kψkwithkαn,k2=1E25

since UψnLψ..

On the other hand, we have

kαn,kψk=kαn,k2ψkE26

as

Uψn=Uψnψn=UψnUψn.

The equalities (26) lead to the resolution of the system:

nαn,k2=1,αn,k2=αn,k,kN.E27

We get then

nN!knN:αn,kn=1,αn,k=0kkn.

Let us now consider the following application:

j:NN,njn=kn.

It’s clear that j is injective.

Now let mN. Since U2=I, then

UUψm=Uψjm=ψjjm=ψm.

Hence,

jjm=m.

Finally j is well defined as involution.

Notation In the sequel , jn will be noted by nv. We write

Uψn=ψjn=ψnvE28

and

Uf=Unanψn=nanψnv=fv.E29

Remark 9. The position operator U can be extended over Fψ as follows:

If f= nanψnFψ, then

Uf=fv=nanψnv.E30

## 3. Carleman operator in Fψ

### 3.1. Case of defect indices 11

Let α=pαpψpFψ; we introduce the operator Aα defined in Lψ by

Aαf=αf=nαψnfψnψn.E31

It is clear that Aα is a Carleman operator induced by the kernel

kxy=αnψnxψny¯,E32

with domain

DAα=fLψ:nαnfψn2<.E33

Moreover, if α=α¯, Aα is self-adjoint.

Now let Θ=pγpψpFψ and ΘLψi.e.pγp2=. We introduce the following set

HΘ=f+μΘ:fLψμCE34

which verifies the following properties.

Proposition 10. 1. HΘ is a subset of Fψ.

2. Hθ=LψCθ, i.e., direct sum of Lψ with Cθ=μθ:μC.

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

Let g1=f1+μ1θ and g2=f2+μ2θ, two formal elements in Hθ. Then

g1=g2f1f2=μ2μ1θ.

This last equality is verified only if μ2=μ1. Therefore, f1=f2. ■

Denote by Q the projector of HΘ on Lψ, that is to say: if gHΘ,

g=f+μΘwithfLψandμC

then

Qg=f.

We define the operator Bα by:

Bαf=Qαf,fLψ.E35

It is clear that

DBα=fLψ:αfHΘ.E36

Theorem 11 Bα is a densely defined and closed operator.

Proof.

1. Since

spanψnnNDBα

and that ψnn is complete in Lψ, then

DBα¯=Lψ.

2. Let fnn be a sequence of elements in DBα. Checking:

fnfBαfngconvergence in theL2sense.

We have then

Bαfn=Qαfn,

with

αfn=gn+μΘ,gnLψ.

Then

gn=αfnμnΘLψ,

This implies that

gnψm=αmfnψmμnγmψmmN.

Or, when n tends to , we have

gngandfnf.

Therefore, there exist μC such that

limnμn=μ.

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

αfHΘandg=Qαf.

Finally fDBα and g=Bαf.

It follows from this theorem that the adjoint operator Bα exists and Bα=Bα.

Let us denote by Aα the operator adjoint of Bα,

Aα=Bα.E37

In the case α=α¯, the operator Aα is symmetric and we have the following results:

Theorem 12. Aα admits defect indices 11 if and only if

φλ=αλ1ΘLψ.E38

In this case φλNλ¯(defect space associated with λ, [3]).

Proof. We know (see [3]) that Aα has the defect indices 11 if and only if its defect subspaces Nλ¯ and Nλ are unidimensional.

We have

Nλ¯=kerAαλI=kerBαλI.

So it suffices to solve the system:

Bαφλ=λφλφλLψ

that is,

Qαφλ=λφλφλLψαφλ=λφλ+μΘ,μCφλLψ
αλφλ=ΘφλLψ
φλ=αλ1ΘφλLψ.

### 3.2. Case of defect indices mm

In this section, we give the generalization for the case of defect indices mm,m>1.

Let Θ1,Θ2,,Θm,m be formal elements not belonging to Lψ, and let

HΘ=f+k=1mμkΘkfLψμkCk=1m.E39

We consider the operator Bα defined by

Bαf=QαffDBα,DBα=fLψ:αfHΘE40

We assume that α=α¯ and we set

Aα=Bα.E41

By analogy to the case of defect indices 11, we also have the following:

Theorem 13. The operator Bα is densely defined and closed.

Theorem 14. The operator Aα admits defect indices mm if and only if

φλk=αλΘkLψ,k=1,,m.E42

In this case, the functions φλkk=1m are linearly independent and generate the defect space Nλ¯.

## 4. Conclusion

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 Carleman operators:

Hxy=Kxy+j=1mbjψjxφjy,φjL2XμψjL2Xμj=1,m¯,E43

with Kxy a Carleman kernel.

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

## References

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Written By

Sidi Mohamed Bahri

Submitted: 16 November 2017 Reviewed: 22 May 2018 Published: 17 October 2018