DSpace at VNU: Generalized random linear operators on a Hilbert space

Generalized random linear operators on a Hilbert spaceDang Hung Thang* and Nguyen Thinh Department of Mathematics, Hanoi University of Science, Hanoi, VietnamReceived 7 July 2011; final

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To cite this article: Dang Hung Thang & Nguyen Thinh (2013) Generalized random linear operators

on a Hilbert space, Stochastics An International Journal of Probability and Stochastic Processes:formerly Stochastics and Stochastics Reports, 85:6, 1040-1059, DOI: 10.1080/17442508.2012.736995

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Generalized random linear operators on a Hilbert space

Dang Hung Thang* and Nguyen Thinh

Department of Mathematics, Hanoi University of Science, Hanoi, Vietnam(Received 7 July 2011; final version received 12 September 2012)

In this paper, we are concerned with generalized random linear operators on a separableHilbert space Generalized random linear bounded operators, generalized randomlinear normal operators and generalized random linear self-adjoint operators aredefined and investigated The spectral theorems for generalized random linear normaloperators and generalized random linear self-adjoint operators are obtained

Keywords: random operator; generalized bounded random operator; generalizedrandom normal operators; generalized random self-adjoint operators; random spectralmeasure

2000 Mathematics Subject Classification: 60H025; 47B80; 47H40; 60B11

1 Introduction

Let (V,F, P) be a probability space and X, Y be Banach spaces A mapping f : V £ X ! Y

is said to be a random operator from X into Y if for each x [ X, the mappingv7! f (v, x) is

a Y-valued random variable (r.v.’s) Equivalently, a random operator from X into Y is a rule

f that assigns each x [ X into a r.v.’s f(x) with values in Y The interest in random operatorshas been aroused not only for its own right as a random generalization of usualdeterministic operators but also for their widespread applications in other areas Research

in theory of random operators has been carried out in many directions such as randomlinear operators, random fixed points of random operators and random operator equations(e.g [4,5,7,8 – 11,13 – 20] and references therein)

A random operator f from X into Y can be considered as an action which transformseach deterministic input x [ X into a random output f(x) with values in Y Taking intoaccount many circumstances in which the inputs are also subject to the influence of arandom environment, there arise the need to define the action of f on some random inputs,i.e on some class of X-valued r.v.’s

In this paper, we are concerned with generalized random linear operators on aseparable Hilbert space In Section 2, a definition and some examples of generalizedrandom linear operators are introduced In particular, the adjoint of a generalized randomlinear operator and the notion of generalized random linear bounded operators are definedand discussed Sections 3 and 4 are devoted to generalized random linear normal operatorsand generalized random linear self-adjoint operators, respectively The main results ofthese sections are random spectral theorems which state that every generalized randomlinear normal operators as well as every generalized random linear self-adjoint operators

*Corresponding author Email: thangdh@vnu.edu.vn

Vol 85, No 6, 1040–1059, http://dx.doi.org/10.1080/17442508.2012.736995

can be represented as the limit of the integrations with respect to certain random spectralmeasures.

2 Generalized random linear operators

Let (V,F, P) be a probability complete space and H be a complex Hilbert separable spacewith the inner , , A mapping u : V ! H is said to be a H-valued random variable (H-r.v.’s) if u is (F, B)-measurable, where B denotes the Borels-algebra of H The space of all(equivalence classes of) H-r.v.’s and the space of all (equivalence classes of) complex-valued r.v.’s are denoted by L0(H) and L0, respectively Convergence in L0(H) and L0

means the convergence in probability If a sequence (un) in L0(H) (or L0) converges to u inprobability, we write p-lim un¼ u If u, v [ L0(H) then the mappingsv7!, uðvÞ; vðvÞ and v7! ku(v)k are complex-valued r.v.’s and are denoted by , u, v and kuk,respectively It is easy to show that if lim un¼ u, lim vn¼ v in L0(H), then lim , un,

vn.¼, u; v in L0

A linear continuous mapping A : H ! L0(H) is said to be a random linear operator on

H Examples of random linear operators are numerous Some results on random operatorscan be found in [11,14,16,18,19]

Under the definition, random operators on H cannot be applied to H-valued r.v.’s.There are many situations, however, in which there arises the need to define theapplication of A to some H-valued r.v.’s For example, let A be a random operator on

L2[0, 1] defined by the Wiener stochastic integral

AxðtÞ ¼

ðt 0

uðs;vÞ dWðsÞ:

A definition of the application of Gaussian random operators on a Hilbert space H to someH-valued r.v’s was introduced in [2] in connection with the stochastic integral with non-adapted integrals in [6] The need to define the action of a random operator to somerandom inputs leads us to the following definition

As usual, the domain M of F is denoted byD(F)

(iii) A random linear mapping F :D(F) ! L0(H) with the dense domainD(F) is called ageneralized random linear operator or a generalized random operator for short (We omitthe word ‘linear’ because only random linear mapping is considered in this paper.)

1Eixi; xi[H; Ei[F :

Since H ,D(F) it follows that S , D(F) Since S is dense in L0(H) so isD(F)

2 Let A : H ! L0(H) be a linear mapping Then A can be extended to a generalizedrandom operator Indeed let M be the set of H-valued r.v.’s of the forms

u ¼Xn i¼1

where xi[ H;ji[ L0 Clearly, M is a random linear space containing H Define amapping F : M ! L0(H) by

Fu ¼Xn i¼1

jiAxi;

for u is of the form (1) It is easy to verify that this definition is well-defined and F

is a generalized random operator extending A

Example 1 Let A : H ! L0(H) be a random operator and e ¼ ðenÞ1n¼1stands for the basis ofH

Denote by M the set of all H-valued r.v.’s u for which the series

X1 n¼1

ðu; enÞAen ð2Þ

converges in L0(H) If u [ M then the sum (2) is denoted by Fu

It is easily shown that M is a random linear space and the mapping F : M ! L0(H) is arandom linear mapping

Since A is continuous and

x ¼X1 n¼1

ðx; enÞen;

we get

Ax ¼X1 n¼1

ðu; enÞAen;

which shows that H ,D(F) By Remark 1, F is a generalized random operator

Definition 2.2 Let F :D(F) ! L0(H) be a generalized random operator and V be thecollection of v [ L0(H) for which there exists g [ L0(H) such that for all u [D(F)

, Fu; v ¼, u; g :

Such g is uniquely determined Indeed suppose that g1; g2[ L0ðHÞ such that , Fu; v ¼, u; g1.¼, u; g2 ;u [DðFÞ Then , u; g12 g2.¼ 0 ;u [DðFÞ SinceD(F) is dense, there is a sequence (un) in D(F) such that lim un¼ g12 g2 Hence, g12 g2; g12 g2.¼ lim , un; g12 g2.¼ 0 which implies that g1¼ g2.

Putting g ¼ F*v we get a mapping F* : V ! L0(H) and the domain V of F* is denoted

byD(F*) It is easy to check that D(F*) is a random linear space and F* is a randomlinear mapping and it is called the adjoint of F Thus the adjoint F* is defined by therelation

, Fu; v ¼, u; F* v ; ð3Þfor all u [DðFÞ; v [ DðF* Þ

In general, the domain of F* need not be dense in L0(H) i.e F* need not be ageneralized random operator (see Example 3 below)

Example 2 Let (ji) be a sequence of standard Gaussian i.i.d r.v.’s Denote by M the set ofall H-valued r.v.’s u for which the series

X1 n¼1

jnðu; enÞen ð4Þ

converges in probability If u [ M then sum (4) is denoted by Fu It is easily shown that M

is a random linear space and the mapping F : M ! L0(H) is a random linear mapping.Since for each x [ H

X1 n¼1

kðx; enÞenk2¼ kxk2, 1;

the series

X1 n¼1

Take a [ H; a – 0 Define a mapping F : M ! L0ðHÞ by Fu ¼ aTu It is easy to see that

F is a random linear mapping Moreover, for each x [ H

X1 n¼1

Ejðx; enÞjnj2¼X1

n¼1

j , x; en j2¼ kxk2, 1;

which implies the seriesP1

n¼1ðx; enÞjnconverges a.s Hence H , M By Remark 1, F is ageneralized random operator

As usual, M is denoted byD(F) Now, we claim that(i) F :D(F) ! L0(H) is not continuous but the restriction of F on H is continuous.Indeed, put vk¼ ð1=kÞPk

i¼1jiei Then vk[D(F) Because

Ekvkk2 ¼ 1

k2

Xk i¼1

Ej2i ¼1

k;

we conclude that p-lim vk¼ 0 However, Fvk¼ aTvk¼ að1=kÞPk

i¼1j2k By the law

of large numbers p-lim Fvk¼ a – 0 Hence F : D(F) ! L0(H) is not continuous.Since for every x [ H, EkFxk2¼ kak2kxk2 we conclude that F : H ! L0(H) iscontinuous

(ii) F* is not a generalized random operator

Indeed v [D(F*) if and only if there is g [ L0(H) such that for all u [D(F)

Since a – 0,D(F*) is not dense in L0(H)

Proposition 2.3 Suppose that F* :D(F*) ! L0(H) is a generalized random operator and

H ,D(F*) Then the restriction of F* to H is a random operator

Proof We have to show that the restriction of F* to H is continuous Suppose that (yn) is asequence in H such that lim yn¼ y and p-lim F*yn¼ g Then for every u [ D(F)

, Fu; y ¼, u; F* y :

Definition 2.4 A generalized random operator F :D(F) ! L0(H) is said to be bounded

if there exists a non-negative random variable k(v) such that for each u [D(F)

kFukðvÞ # kðvÞkukðvÞ a:s:

FuðvÞ ¼ TðvÞðuðvÞÞ a:s: ð5Þ

3 Conversely, let T : V ! L(H, H) be a mapping such that for every x [ H, themapping v7! T(v)x is a H-valued r.v.’s Then the mapping F : L0ðHÞ ! L0ðHÞdefined by (5) is a generalized random bounded operator

4 If F : L0(H) ! L0(H) is a generalized random bounded operator thenF* : L0(H) ! L0(H) is also a generalized random bounded operator and

F* vðvÞ ¼ T* ðvÞðvðvÞÞ a:s:

Proof

(1) For each t, r 0 and u [D(F) we have

PðkFuk tÞ ¼ PðkFuk t; kuk # rÞ þ PðkFuk t; kuk rÞ

lim supP kFuf n2 Fumk tg # P{kðvÞ t=r}:

(2) The restriction of F to H is a random bounded operator By Theorem 3.1 [19] there

is a mapping T : V ! L(H, H) such that

FxðvÞ ¼ TðvÞx a:s: ð6Þ

If u is a H-valued simple random variable of the form

uðvÞ ¼Xn i¼1

k(v) ¼ u(v) a.s Hence there exists a set D of probability one such that foreachv[ D we have

FunkðvÞ ¼ TðvÞðun kðvÞÞ and lim

FuðvÞ ¼ TðvÞðuðvÞÞ a:s:

Now we show that T is unique Assume that T1, T2satisfy (5) Let (xn) be countabledense subset of H Then there exists a set D of probability one such that for each

T1ðvÞxn¼ T2ðvÞxn¼ FxnðvÞ; for all xn:Hence T1(v) ¼ T2(v) forv[ D, i.e T1(v) ¼ T2(v) a.s

(3) By the same argument as part (2), it can be shown that Fu defined by (5) is aH-valued r.v.’s and F : L(H) ! L(H) is a generalized random operator Now we

shall show that F is bounded Let (xn) be a sequence dense in B ¼ {x [

which implies that kT(v)k is a non-negative random variable Hence

kFuðvÞk # kTðvÞkuðvÞk a:s:

and we are done

(4) Let u; v [ L0ðHÞ Then there is a set D with P(D) ¼ 1 such that for eachv[ D

, FuðvÞ; vðvÞ ¼, TðvÞ½uðvÞ
; vðvÞ ¼, uðvÞ; T* ðvÞðvðvÞ ;which prove that F*v(v) ¼ T*(v)(v(v) a.s By part (3) F* : L0(H) ! L0(H) is ageneralized random bounded operator A

Remark The generalized random operator F presented in Example 3 is not boundedbecauseD(F*) ¼ L0(H1) – L0(H), where H1¼ [a ]’ The generalized random operator Fpresented in Example 2 is also not bounded Indeed, suppose that F is bounded From thedefinition, it follows that supnkFenk , 1 a.s However, supnkFenk ¼ supnjjnj ¼ 1 a.s.and we get a contradiction

3 Generalized random normal operator

Definition 3.1 A generalized random bounded operator F is said to be a generalizedrandom normal operator if FF* ¼ F*F

By Theorem 2.5 every generalized random bounded operator F is of the form

FuðvÞ ¼ TðvÞðuðvÞÞ a:s:; ð7Þwhere T : V ! L(H, H) is mapping such that for every x [ H, the mappingv7! T(v)x is aH-valued random variable

Proposition 3.2 F is normal if and only if T(v) is normal a.s

Proof Suppose that F is normal For each x [ H by Theorem 2.5 we have

F½F* xðvÞ
¼ F* ½FxðvÞ
a:s:

!TðvÞ½T* ðvÞx
¼ T* ðvÞ½TðvÞx
a:s:

!½TðvÞT* ðvÞ
x ¼ ½T* ðvÞTðvÞ
x a:s:

Since H is separable it follows that T(v)T*(v) ¼ T*(v)T(v) a.s

Conversely suppose that T(v) is normal a.s Let u [ L0(H) There is a set D withP(D) ¼ 1 such that

TðvÞ½T* ðvÞx
¼ T* ðvÞ½TðvÞx
; ;v[ D; ;x [ H

!TðvÞ½T* ðvÞðuðvÞÞ
¼ T* ðvÞ½TðvÞðuðvÞÞ
; ;v[ DF½F* uðvÞ
¼ F* ½FuðvÞ
¼ ;v[ D

FF* u ¼ F* Fu

Hence FF* ¼ F*F

The main result in this section is the spectral theorem for generalized random normal

Let us recall the definition of spectral measure

Definition 3.3 ([12]) Let (S,A) be a measurable space and H be a complex Hilbertseparable space By a spectral measure E on (S,A, H) we mean a mapping E : A ! L(H, H),having the following properties

1 E(M) is a projection for each M [A

2 E(B) ¼ 0 and E(S) ¼ I

3 If M,N [A then E(M > N) ¼ E(M)E(N) In particular, if M, N are disjoint thenE(M) and E(N) are orthogonal

4 If (Mi) is a sequence of pairwise disjoint sets fromA then for each x [ H

E <1

i¼1Mi

x ¼X1 i¼1

EðMiÞx;

where the series is convergent in H

In other words, for each x [ H the mapping M 7! E(M)x is a H-valued measure.The spectral theorem states the following

Theorem 3.4 ([12])

1 If E is a spectral measure for (S, A, H) and f : S !C is a measurable, boundedfunction then the mapping T : H ! H defined by

Tx ¼ð

S

f ðsÞEðdsÞx

is a normal operator

2 Conversely, suppose T : H ! H is a normal operator ands(T) ,C is the spectrum

of T Thens(T) is a compact set and there exists a spectral measure E defined on theBorel subsets of the spectrums(T) such that

Tx ¼

ðzEðdzÞx ;x [ H;

Our aim is to obtain the random version of the spectral theorem To this end, the notion

of random spectral measure can be defined as follows

Definition 3.5 Let (S,A) be a measurable space and H be a Hilbert space By a randomspectral measure E(v) on (S,A, H) we mean a family {EðvÞ;v[ V} of spectral measures

on (S, A, H) indexed by the parameter set V such that for every x [ H; M [ A, themappingv7! E(v)(M)x is a H-valued random variable

The random spectral theorem states the following

Then F is a generalized random normal operator

2 Conversely, let F be a generalized random normal operator Then there is a randomspectral measure E(v) on (C, B, H) such that for eachv[ V; u [ L0ðHÞ

For each x [ H the mapping x 7! Tf(v)x is a H-valued r.v

At first, we prove that the claims hold for each simple function g Indeed, if g is

Since {E(v)} is a random spectral measure it follows that the mapping x 7! Tg(v)x

is a H-valued random variable

Next let f : S !C be a measurable bounded function Then there exists asequence (gn) of simple functions converging uniformly to f(s) From this

limTgnðvÞx ¼ TfðvÞx ;v: