Open access peer-reviewed chapter

Analytical Applications on Some Hilbert Spaces

By Fethi Soltani

Submitted: June 5th 2019Reviewed: October 30th 2019Published: April 23rd 2020

DOI: 10.5772/intechopen.90322

Downloaded: 62

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 D.

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 square-integrable 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, 6, 7, 8, 9]. We expect that the results of this paper will be useful when discussing (in Section 2) uncertainty inequality for Hilbert space Hand minimizer function associated with a bounded linear operator Tfrom Hinto a Hilbert space K. As applications, we consider Hardy and Dirichlet spaces as follows.

Let Cbe the complex plane and D=zC:z<1the open unit disk. The Hardy space HDis the set of all analytic functions fin the unit disk Dwith the finite integral:

02πfe2dθ.E1

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

fgHD=12π02πfege¯dθ.E2

Over the years, the applications of Hardy space HDplay 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 HDthe following two operators:

fz=fz,Lfz=z2fz+zfz,E3

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

T1fz=1zfzf0.E4

In Section 4, we consider the Dirichlet space DD, which is the set of all analytic functions fin the unit disk Dwith the finite Dirichlet integral:

Dfz2dxdyπ,z=x+iy.E5

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

fgDD=f0g0¯+Dfzgz¯dxdyπ,z=x+iy.E6

This space is the objective of many applicable works [5, 13, 14, 15, 16, 17] and plays a background to our contribution. For example, we study on DDthe following two operators:

Λfz=fzf0,Xfz=z2fz,E7

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

T2fz=1zfzzf0f0.E8

2. Generalized results

Let Hbe a Hilbert space equipped with the inner product ..H. And let Aand Bbe the two operators defined on H. We define the commutator ABby

ABABBA.E9

The adjoint of Adenoted by Ais defined by

AfgH=fAgH,E10

for fDomAand gDomA.

Theorem 2.1. For fDomAADomAA, one has

AfH2=AfH2+AAffH.E11

Proof. Let fDomAADomAA. Then AAfand AAfbelong to H. Therefore AAfH. Hence one has

AfH2=AAffH=AAffH+AAffHE12
=AfH2+AAffH.E13

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

Theorem 2.2. Let Aand Bbe the self-adjoint operators on a Hilbert space H. Then

AafHBbfH12ABffH,E14

for all fDomABDomBA, and all a,bR.

Theorem 2.3. Let fDomAADomAA. For all a,bR, one has

A+AafHAA+ibfHAfH2AfH2,E15

where iis the imaginary unit.

Proof. Let us consider the following two operators on DomAADomAAby

P=A+A,Q=iAA.E16

It follows that, for fDomAADomAA, we have Pf,QfH. The operators Pand Qare self-adjoint and PQ=2iAA. Thus the inequality (15) follows from Theorems 2.1 and 2.2.

Theorem 2.4. Let fDomAADomAA. Then

ΔH+fΔHffH4AfH2AfH22,E17

where

ΔH±f=fH2A±AfH2A±AffH2.E18

Proof. Let fDomAADomAA. The operator Pgiven by (16) is self-adjoint; then for any real a, we have

PafH2=PfH2+a2fH22aPffH.E19

This shows that

minaRPafH2=PfH2PffH2fH2,E20

and the minimum is attained when a=PffHfH2. In other words, we have

minaRA+AafH2=A+AfH2A+AffH2fH2.E21

Similarly

minbRAA+ibfH2=AAfH2AAffH2fH2.E22

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

Let λ>0and let T:HKbe a bounded linear operator from Hinto 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 kKand for any λ>0, the problem

inffHλfH2+TfkK2E23

has a unique minimizer given by

fλ,k=λI+TT1Tk.E24

Proof. The problem (23) is solved elementarily by finding the roots of the first derivative DΦof the quadratic and strictly convex function Φf=λfH2+TfkK2. Note that for convex functions, the equation DΦf=0is a necessary and sufficient condition for the minimum at f. The calculation provides

DΦf=2λf+2TTfk,E25

and the assertion of the theorem follows at once.

Theorem 2.6. If T:HKis an isometric isomorphism; then for any kKand for any λ>0, the problem

inffHλfH2+TfkK2E26

has a unique minimizer given by

fλ,h=1λ+1T1k.E27

Proof. We have T=T1and TT=I. Thus, by (24), we deduce the result.

3. The Hardy space HD

Let Cbe the complex plane and D=zC:z<1the open unit disk. The Hardy space HDis the set of all analytic functions fin the unit disk Dwith the finite integral:

02πfe2dθ.E28

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

fgHD=12π02πfege¯dθ.E29

If f,gHDwith fz=n=0anznand gz=n=0bnzn, then

fgHD=n=0anbn¯.E30

The set znn=0forms an Hilbert’s basis for the space HD.

The Szegő kernel Szgiven for zD, by

Szw=n=0z¯nwn=11z¯w,wD,E31

is a reproducing kernel for the Hardy space HD, meaning that SzHD, and for all fHD, we have fSzHD=fz.

For zD, the function uz=Sz¯wis the unique analytic solution on Dof the initial problem:

uz=wzuz+uz,wD,u0=1.E32

In the next of this section, we define the operators , , and Lon HDby

fz=fz,ℜfz=zfz,Lfz=z2fz+zfz.E33

These operators satisfy the commutation rule:

L=LL=2+I,E34

where Iis the identity operator.

We define the Hilbert space UDas the space of all analytic functions fin the unit disk Dsuch that

fUD2=12π02πfe2dθ<.E35

If fUDwith fz=n=0anzn, then

fUD2=n=1n2an2.E36

Thus, the space UDis a subspace of the Hardy space HD.

Theorem 3.1.

  1. For fUD, then f, ℜfand Lfbelong to HD.

  2. =L.

  3. For fUD, one has

LfHD2=fHD2+fHD2+2ℜffHD.E37

Proof.

  1. Let fUDwith fz=n=0anzn. Then

    fz=n=0n+1an+1zn,ℜfz=n=1nanzn,E38

    and

    Lfz=n=1nan1zn.E39

    Therefore

    fHD2=n=0n+12an+12=fUD2,E40
    ℜfHD2=n=1n2an2=fUD2,E41

    and

    LfHD2=n=0n+12an2f02+4fUD2.E42

    Consequently f, ℜf, and Lfbelong to HD.

  2. For f,gUDwith fz=n=0anznand gz=n=0bnzn, one has

    fgHD=n=0n+1an+1bn¯=n=1nanbn1¯=fLgHD.E43

    Thus =L.

  3. Let fUD. By (ii) and (34), we deduce that

    LfHD2=LffHDE44
    =LffHD+LffHDE45
    =fHD2+fHD2+2ℜffHD.E46

Theorem 3.2. Let fUD. For all a,bR, one has

+LafHDL+ibfHDfHD2+2ℜffHD.E47

Theorem 3.3. Let T1be the difference operator defined on HDby

T1fz=1zfzf0.E48
  1. The operator T1maps continuously from HDto HD, and

    T1fHDfHD.E49

  2. For fHDand zD, we have

    T1fz=zfz,T1T1fz=fzf0.E50

  3. For any hHDand for any λ>0, the problem

    inffHDλfHD2+T1fhHD2E51

    has a unique minimizer given by

    fλ,hz=1λ+1zhz,zD.E52

Proof.

  1. If fHDwith fz=n=0anzn, then T1fz=n=0an+1znand

    T1fHD2=n=1an2fHD2.E53

  2. If f,gHDwith fz=n=0anznand gz=n=0bnzn, then

    T1fgHD=n=0an+1bn¯=n=1anbn1¯=fT1gHD,E54

    where T1gz=zgz, for zD. And therefore

    T1T1fz=zT1fz=fzf0.E55

  3. From Theorem 2.5 we have

    λI+T1T1fλ,hz=T1hz.E56

By (ii) we deduce that

λ+1fλ,hzfλ,h0=zhz.E57

And from this equation, fλ,h0=0. Hence

fλ,hz=1λ+1zhz.E58

4. The Dirichlet space DD

The Dirichlet space DDis the set of all analytic functions fin the unit disk Dwith the finite Dirichlet integral:

Dfz2dxdyπ,z=x+iy.E59

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

fgDD=f0g0¯+Dfzgz¯dxdyπ,z=x+iy.E60

If f,gDDwith fz=n=0anznand gz=n=0bnzn, then

fgDD=a0b0¯+n=1nanbn¯.E61

The set 1znnn=1forms an Hilbert’s basis for the space DD.

The function Kzgiven for zD, by

Kzw=1+log11z¯w,wD,E62

is a reproducing kernel for the Dirichlet space DD, meaning that KzDD, and for all fDD, we have fKzDD=fz.

For zD, the function uz=Kz¯wis the unique analytic solution on Dof the initial problem:

uzu0z=wuz,wD,u0=1.E63

In the next of this section, we define the operators Λ, , and Xon DDby

Λfz=fzf0,ℜfz=zfz,Xfz=z2fz.E64

These operators satisfy the following commutation relation:

ΛX=ΛXXΛ=2.E65

We define the Hilbert space VDas the space of all analytic functions fin the unit disk Dsuch that

fVD2=Dfz2z2dxdyπ<,z=x+iy.E66

If fVDwith fz=n=0anzn, then

fVD2=n=1n3an2.E67

Thus, the space VDis a subspace of the Dirichlet space DD.

Theorem 4.1.

  1. For fVD, then Λf, ℜf, and Xfbelong to DD.

  2. Λ=X.

  3. For fVD, one has

    XfDD2=ΛfDD2+2ℜffDD.E68

Proof.

  1. Let fVDwith fz=n=0anzn. Then

    Λfz=n=1n+1an+1zn,ℜfz=n=1nanzn,E69

    and

    Xfz=n=2n1an1zn.E70

    Therefore

    ΛfDD2=n=1nn+12an+12n=2n3an2fVD2,E71
    ℜfDD2=n=1n3an2=fVD2,E72

    and

    XfDD2=n=1n+1n2an22fVD2.E73

    Consequently Λf, ℜf, and Xfbelong to DD.

  2. For f,gVDwith fz=n=0anznand gz=n=0bnzn, one has

    ΛfgDD=n=1nn+1an+1bn¯=n=2nn1anbn1¯=fXgDD.E74

  3. Let fVD. By (ii) and (65), we deduce that

    XfDD2=ΛXffDDE75
    =XΛffDD+ΛXffDDE76
    =ΛfDD2+2ℜffDD.E77

Theorem 4.2. Let fVD. For all a,bR, one has

Λ+XafDDΛX+ibfDD2ℜffDD.E78

Theorem 4.3. Let T2be the difference operator defined on DDby

T2fz=1zfzzf0f0.E79
  1. The operator T2maps continuously from DDto DD, and

    T2fDDfDD.E80

  2. For fDDwith fz=n=0anzn, we have

    T2fz=n=2n1nan1zn,T2T2fz=n=2n1nanzn.E81

  3. For any dDDand for any λ>0, the problem

    inffDDλfDD2+T2fdDD2E82

    has a unique minimizer given by

    fλ,dz=dΨzDD,zD,E83
    Ψzw=n=1z¯n+1λn+1+nwn,wD.E84

Proof.

  1. If fDDwith fz=n=0anzn, then T2fz=n=1an+1znand

    T2fDD2=n=2n1an2n=2nan2fDD2.E85

  2. If f,gDDwith fz=n=0anznand gz=n=0bnzn, then

    T2fgDD=n=1nan+1bn¯=n=2n1anbn1¯=fT2gDD,E86

    where

    T2gz=n=2n1nbn1zn,zD.E87

    And therefore

    T2T2fz=n=2n1nanzn.E88

  3. We put dz=n=0dnznand

    fλ,dz=n=0cnzn.E89

From (ii) and the equation

λI+T2T2fλ,dz=T2dz,E90

we deduce that

c1=c0=0,cn=n1λn+n1dn1,n2.E91

Thus

fλ,dz=n=1ndnλn+1+nzn+1=dΨzDD,zD.E92

© 2020 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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Fethi Soltani (April 23rd 2020). Analytical Applications on Some Hilbert Spaces, Functional Calculus, Kamal Shah and Baver Okutmuştur, IntechOpen, DOI: 10.5772/intechopen.90322. Available from:

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