Open access peer-reviewed chapter

A Formal Perturbation Theory of Carleman Operators

By Sidi Mohamed Bahri

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

DOI: 10.5772/intechopen.79022

Downloaded: 217

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 Xbe an arbitrary set, μbe a σfinite measure on X(μis defined on a σalgebra of subsets of X; we do not indicate this σalgebra), and L2Xμthe Hilbert space of square integrable functions with respect to μ. Instead of writing “μmeasurable,” “μalmost everywhere,” and “(x),” we write “measurable,” “a e,” and “dx.”

A linear operator A: DAL2Xμ, where the domain DAis a dense linear manifold in L2Xμ, is said to be integral if there exist a measurable function Kon X×X, a kernel, such that, for every fDA,

Afx=XKxyfydya.e..E1

A kernel Kon X×Xis said to be Carleman, if KxyL2Xμfor almost every fixed x,that is to say

XKxy2dy<a.e..E2

An integral operator A(1) with a kernel Kis called Carleman operator, if Kis a Carleman kernel (2). Every Carleman kernel Kdefines a Carleman function kfrom Xto L2Xμby kx=Kx.¯for all xin Xfor 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=0is an orthonormal sequence in L2Xμsuch that

p=0ψpx2<a.e.,E4

and app=0is a real number sequence verifying

p=0ap2ψpx2<a.e..E5

We call ψpxp=0a Carleman sequence.

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

p=0γpψpx=0a.e.,E6

and

p=0γpapλ2<.E7

With the conditions (6) and (7), the symmetric operator A=Aadmits 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=0be a fixed Carleman sequence in L2Xμ. It is clear from the foregoing that ψis not a complete sequence in L2Xμ. We denote by Lψthe 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 annNare scalars.

The sequence annis said to generate the formal element f.

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

We say that two formal elements f=nNanψnand g=nNbnψnare equal if an=bnfor all nN.

If φis 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 an0for 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 fby

f¯=nan¯ψn.E17

Denote by Fψthe 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=nNanψnis bounded if its sequence annis bounded.

We denote by Bψthe set of all bounded formal elements.

It is clear that Bψis a subspace of Fψ.

We claim that:

  1. Lψis a subspace of Bψ.

  2. Furthermore we have the strict inclusions:

LψBψFψ.E20

We define a linear form ..on Fψby 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 Lψany 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 Uacts as operator of change of position of these elements.

2.4. Main results

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

Uψn=ψjn.E24

Proof.

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

  2. Let Ube 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 jis injective.

Now let mN.Since U2=I,then

UUψm=Uψjm=ψjjm=ψm.

Hence,

jjm=m.

Finally jis well defined as involution.

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

Uψn=ψjn=ψnvE28

and

Uf=Unanψn=nanψnv=fv.E29

Remark 9. The position operator Ucan be extended over Fψas 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 Aαdefined in Lψby

Aαf=αf=nαψnfψnψn.E31

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

kxy=αnψnxψny¯,E32

with domain

DAα=fLψ:nαnfψn2<.E33

Moreover, if α=α¯,Aαis self-adjoint.

Now letΘ=pγpψpFψand ΘLψi.e.pγp2=. We introduce the following set

HΘ=f+μΘ:fLψμCE34

which verifies the following properties.

Proposition 10. 1. HΘis a subset of Fψ.

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

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

Let g1=f1+μ1θand 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 Qthe projector of HΘon Lψ, that is to say: if gHΘ,

g=f+μΘwithfLψandμC

then

Qg=f.

We define the operator Bαby:

Bαf=Qαf,fLψ.E35

It is clear that

DBα=fLψ:αfHΘ.E36

Theorem 11 Bαis a densely defined and closed operator.

Proof.

1. Since

spanψnnNDBα

and that ψnnis complete in Lψ,then

DBα¯=Lψ.

2. Let fnnbe 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 ntends to , we have

gngandfnf.

Therefore, there exist μCsuch that

limnμn=μ.

And as Qis a closed operator, then we can write

αfHΘandg=Qαf.

Finally fDBαand g=Bαf.

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

Let us denote by Aαthe operator adjoint of Bα,

Aα=Bα.E37

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

Theorem 12. Aαadmits defect indices 11if and only if

φλ=αλ1ΘLψ.E38

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

Proof. We know (see [3]) that Aαhas the defect indices 11if and only if its defect subspaces Nλ¯and Nλare 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.

LetΘ1,Θ2,,Θm,mbe formal elements not belonging to Lψ, and let

HΘ=f+k=1mμkΘkfLψμkCk=1m.E39

We consider the operator Bαdefined 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 Bαis densely defined and closed.

Theorem 14. The operator Aαadmits defect indices mmif and only if

φλk=αλΘkLψ,k=1,,m.E42

In this case, the functions φλkk=1mare 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 Kxya Carleman kernel.

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

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Sidi Mohamed Bahri (October 17th 2018). A Formal Perturbation Theory of Carleman Operators, Perturbation Methods with Applications in Science and Engineering, İlkay Bakırtaş, IntechOpen, DOI: 10.5772/intechopen.79022. Available from:

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