DSpace at VNU: Generalized Random Spectral Measures

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DOI 10.1007/s10959-012-0461-0

Generalized Random Spectral Measures

Dang Hung Thang · Nguyen Thinh · Tran Xuan Quy

Received: 30 January 2012 / Revised: 14 September 2012

© Springer Science+Business Media New York 2012

Abstract In an attempt to examine the random version of the spectral theorem,the notion of random spectral measures and generalized random spectral measuresare introduced and investigated It is shown that each generalized random spectralmeasure on(C, B(C)) admits a modification which is a random spectral measure.

Keywords Random operators· Random normal operator · Random Hermitianbounded operators · Random projection operator · Random spectral measure ·Generalized random spectral measure· Integral with respect to generalized spectralmeasure

Mathematics Subject Classification (2000) Primary 60H25· Secondary 60H05 ·60B11· 45R05

1 Introduction

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

 × X → Y is said to be a random operator from X into Y if for each x ∈ X, the

mappingω → (ω, x) is a Y -valued random variable Equivalently,  can be viewed

as a mapping : X → L Y

0(), where L Y

0() stands for the space of X-valued random

variables and is equipped with the topology of convergence in probability If a randomoperator : X → L Y

0() is linear and continuous then it is called a random linear

operator from X into Y

The interest in random operators has been arouse not only for its own right as arandom generalization of usual deterministic operators, but also for their widespreadapplications in other areas Research in theory of random operators has been carried out

in many directions such as random fixed points of random operators, random operatorequations, and random lineart operators (e.g., [3,5,8,11]–[22] and references therein).The spectral properties of certain classes of random linear operators on Hilbert spaceswere published in [11,12] This study was closely connected with the investigation ofthe spectra of random matrices of order approaching infinity in the context of actualphysical model (see [7,9]) The aim of this paper is to examine the random version

of the spectral theory of linear operators on Hilbert spaces which plays an importantrole in functional analysis and has many applications (see [1,4,10]) In Sect.2, thenotion of random spectral measures, random normal operators, and random Hermittianoperators are defined as a family of spectral measures, normal operators and Hermitianoperators, respectively, indexed by the parameter set satisfying certain measurability.

It is shown that such random operators can be represented as the limit of the integralwith respect to certain random spectral measure Next the notion of random spectralmeasure is generalized further To do this, the notion of random projection operators

is introduced in the Sect.3 Then the notion of generalized random spectral measure,the integral of complex-valued bounded functions with respect to generalized randomspectral measures are introduced and investigated in Sect.4 The main result in thissection is the claim that every generalized random spectral measures on(C, B) admits

a modification which is a random spectral measure

2 Random Spectral Measures

Recall that

Definition 2.1 ([6,10]) Let(S, A) be a measurable space and H be a complex Hilbert

separable space

• By a spectral measure U on (S, A, H) we mean a mapping U : A → L(H, H),

having the following properties

(1) U (M) is a projection for each M ∈ A

(2) U (∅) = 0 and E(S) = I

(3) If M , N ∈ A then U(M ∩ N) = U(M)U(N).

(4) If(M i ) is a sequence of pairwise disjoint sets from A then for each x ∈ H

• A linear operator T on H is called a normal operator if T is bounded and T T∗=

(1) Let T is a normal operator on H and σ(T ) ⊂ C is the spectrum of T Then σ (T )

is a compact set and there exists a spectral measure U defined on the Borel subsets

of the spectrum σ(T ) such that

T =



σ(T ) zU(dz)

(2) Let T be a Hermitian operator on H andσ (T ) is the spectrum of T Then σ (T ) ⊂ R

is a compact set and there exists a spectral measure U defined on the Borel subsets

of the spectrum σ (T ) such that

T =



σ(T ) λU(λ)

Our aim in this section is to obtain the random version of the spectral theorem Tothis end, the notion of random normal operators and the notion of random spectralmeasures can be defined as follows:

Definition 2.3 Let(S, A) be a measurable space and H be a complex Hilbert separable

space

(1) By a random normal operator T (ω), we mean a family {T (ω), ω ∈ } of normal

operators indexed by the parameter set such that for every x ∈ H the mapping

ω → T (ω)x is H-valued random variable.

(2) By a random Hermitian operator T (ω), we mean a family {T (ω), ω ∈ } of

Hermitian operators indexed by the parameter set such that for every x ∈ H the

mappingω → T (ω)x is H-valued random variable.

(3) By a random spectral measure U (ω) on (S, A, H), we mean a family {U(ω), ω ∈

} of spectral measures on (S, A, H) indexed by the parameter set  such that

for every x ∈ H, M ∈ A, the mapping ω → U(ω)(M)x is a H-valued random

(2) Let T (ω) be a random Hermitian operator Then there is a random spectral sure U (ω) on (R, B, H) such that for each x ∈ H

Proof (1) For each ω, by Theorem2.2there is a spectral measure U (ω) defined on

the Borel subsets of the spectrumσ(T (ω)) of T (ω) such that

T (ω)x =



σ(T (ω))

zU (ω)(dz)x.

At first, we shall show that for every x ∈ H, M ∈ B, the mapping ω → U(ω)(M)x

is measurable, i.e., U (ω) define a random spectral measure.

For each n and put D n = {ω : T (ω) < n}, B n = {z ∈ C : |z|  n} From the inequality r (T (ω))  T (ω), where r(T (ω)) is the spectral radius of a operator

T (ω) it follows that ω ∈ D nimpliesσ (T (ω)) ⊂ B n Hence for eachω ∈ D n

then the mappingω → P(T (ω))x from D n into H is measurable Indeed,

For each H -valued r.v u , ω → T (ω)u(ω) is a H-valued r.v From this by induction

for each k the mapping ω → T (ω) k x is meassurable Hence, the mapping ω → P(T (ω))x from D n into H is measurable.

Step 2 If f (z) is a continuous function defined on B n then the mappingω →

f (T (ω))x from D n into H is measurable.

By the Weierstrass theorem there exists a sequence of polynomial P k (z) converges

uniformly to f (z) on B n Hence|P k (z)| is uniformly bounded Because σ (T (ω)) ⊂

B nforω ∈ D nby Lemma 8 in [4, p 899], we get

lim P k (T (ω))x = f (T (ω))x

for eachω ∈ D n Hence the conclusion follows from Step 1

Step 3 For each the closed set M of C and x ∈ H, the mapping ω → U(ω)(M)x

from into H is measurable.

Indeed for each n let M n = {s : d(s, M) ≥ 1/n then M n is closed and M∩M n= ∅

By Urysohn Lemma there exists a continuous function f n such that f n (z) = 1

fot z ∈ M and f n (z) = 0 for z ∈ M n and 0  f (z)  1 If z /∈ M then there is n0 such that d (z, M) ≥ 1/n0 ≥ 1/n for n ≥ n0 Hence f n (z) = 0 for

n ≥ n0 Consequently lim f n (z) = 1 M (z) Since the mapping M → E(ω)(M)x

is a H -valued measure by Theorem 1 in [2, p 56], we get for eachω ∈ D n

from into H is measurable Then M is a σ-algebra Indeed by the definition of

spectral measure for eachω

U(ω)(M ∩ N)x = U(ω)(M)[U(ω)(N)x]

U(ω)(M ∪ N)x = U(ω)(M)x + U(ω)(N)x − U(M ∩ N)(ω)x

U (ω) (∩A n ) x = lim U(ω)(A n )x if (A n ) ↓

U (ω) (∪A n ) x = lim U(ω)(A n )x if (A n ) ↑.

From thisM is closed under the intersection operation, the union operation and

it is a monotone class HenceM is a σ-algebra.

By Step 3,M contains the closed sets So M coincides the the class of all Borel

sets Consequently, we have shown that for every x ∈ H, M ∈ B, the mapping

ω → U(ω)(M)x is measurable as claimed.

Next we show that (1) holds Indeed, fixω ∈  Because D n ↑  there exists n0 (ω)

such thatω ∈ D n for every n > n0(ω) Thus if n > n0(ω) then σ (T (ω)) ⊂ B n

which implies that

(2) By Theorem2.2for eachω , there is a spectral measure U(ω) defined on the Borel

subsets of the spectrumσ(T (ω)) ⊂ R of T (ω) By the same argument as the

3 Random Projection Operators

Let(, F, P) be a complete probability space, and X be a separable Banach space.

A measurable mappingξ from (, F) into (X, B(X)) is called a X-valued random

variable The set of all X -valued random variables is denoted by L X0() We do not

distinguish two X -random variables which are equal almost surely The space L X0()

is equipped with the topology of convergence in probability If a sequence(ξ n ) of

L0X () converges to ξ in probability then we write p − lim ξ n = ξ.

Definition 3.1 ([15,18]) Let X , Y be separable Banach spaces A linear continuous

mapping A from X into L Y0() is said to be random linear operators or a random

operator for short.(We omit the word “linear” because from now on only randomlinear operators are considered.)

Definition 3.2 ([18]) A random operator A : X → L Y

0() is said to be bounded if

there exists a real-valued random variable k (ω) such that for each x ∈ X

Noting that the exceptional set in (3) may depend on x.

In general, a random operator need not be bounded For examples of random operatorsand random bounded operators, we refer to[18]

Theorem 3.3 ([18]) A random operator A : X → L Y

0() is bounded if and only if there is a mapping T A :  → L(X, Y ) such that

0().

Proof At first, we show that if u ∈ L X

Now assume that u ∈ L X

0() Then there exists a sequence (u n ) of X-valued simple

random variables converging to u in probability Hence p− limn ˜Au n = ˜Au On the

other hand, there exists a subsequence(v n k ) such that lim k→∞u n k (ω) = u(ω) a.s.

Hence there exists a set D of probability one such that for each ω ∈ D, we have

which implies lim

k→∞ ˜Au n k (ω) = ˜A(ω)(u(ω)) a.s Hence ˜A(ω)(u(ω)) ∈ L Y

0() The

linearity of ˜A is clear We show the continuity of ˜ A Put k (ω) = T A (ω) From (5),

it follows that there is a set D with P (D) = 1 such that Ax n (ω) = T A (ω)x nfor all

ω ∈ D, n = 1, 2, For each ω ∈ D

k(ω) = T A (ω) = sup

n T A (ω)x n = sup

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

˜Au(ω)  T A (ω) u(ω) = k(ω)u(ω) a.s.

For each t , r > 0 and u ∈ L X

0(), we have P( ˜Au > t) = P( ˜Au > t, u  r) + P( ˜Au > t, u > r)

 P(k(ω) > t/r) + P(u > r).

From now on, for brevity the extension ˜A of A is also denoted by A Hence, we write

Consequently, A B is also a random bounded operator and (5) holds 

Let H be a complex Hilbert separable space with the inner product ·, · From now

on, for brevity a random operator A : H → L H

0() is said to be a random operator

on H

Definition 3.6 ([16]) Let A be a random operator on H A random operator B on H

is called an adjoint of A if for every x ∈ H, y ∈ H, we have

Ax(ω), y = x, By(ω) a.s.

In general, a random operator need not admit an adjoint ([16]) It is easy to see that

the adjoint of a random operaror A, if it exists, is unique The adjoint of A is denoted

by A∗.

Lemma 2 Suppose that the random operator A is a random bounded operator Then

A admits an adjoint A∗ Moreover A∗is bounded and

(1)

(2) For u , v ∈ L H

0()

Au(ω), v(ω) = u(ω), A∗v(ω) a.s.

Proof (1) Define a random operator B by Bx (ω) = T A (ω)∗x From the equality

Ax(ω), y = T A (ω)x, y = x, T A (ω)∗y  = x, By(ω) a.s.

it follows that B is an adjoint of A and

A∗y(ω) = By(ω) = T A (ω)∗y a.s.

showing that A∗is bounded and (6) holds.

(2) We have for almost allω ∈ 

Au(ω), v(ω) = T A (ω)(u(ω)), v(ω) = u(ω), T A (ω)∗v(ω)

= u(ω), A∗v(ω).



Definition 3.7 A random operator P on H is said to be a random projection operator

on H if P is bounded, self-adjoint (i.e., P = P∗) and P P = P.

Lemma 3 (1) If P is a random projection operator on H then T P (ω) is a projection operator on H for almost all ω.

(2) For each u ∈ L H

0(), we have

Pu(ω)  u(ω) a.s.

(3) For each pair (u, v) of H-valued random variables we have

Pu(ω), v(ω) = u(ω), Pv(ω) a.s.

Proof (1) It follows from the (3) and (4) that P is a random projection operator on H

if and only if for almost allω, the operator T P (ω) is a projection operator.

(2) From (1) it follows that

Pu(ω) = T P (ω)(u(ω))  u(ω) a.s.

(3) From Lemma2, we get

Pu(ω), v(ω) = u(ω), P∗v(ω) = u(ω), Pv(ω) a.s.



4 Generalized Random Spectral Measure

Definition 4.1 Let (S, A) be a measurable space and H be a Hilbert space By a

generalized random spectral measure E on (S, A, H), we mean a mapping E from A

into the space of random operators on H such that

(1) E (M) is a random projection operator on H for each M ∈ A;

(2) E (∅) = 0 and E(S) = I ;

(3) If M , N ∈ A then E(M ∩ N) = E(M)E(N);

(4) If(M i ) is a sequence of pairwise disjoint sets from A then for each x ∈ H

E

∞

Noting that the exceptional set in (4) may depend on(M i ) and x.

Example Let {U(ω), ω ∈ } be a random spectral measures on (S, A, H) Define a mapping E from A into the space of random operators on H by

E (M)x(ω) = U(ω)(M)x.

We have T E (M)(ω) = U(ω)(M) From Lemma2it is easy to check that the conditions

(1)–(3) holds Since U (ω) is spectral measure hence for all ω we have

U(ω)

∞

E

∞

Hence E is a generalized random spectral measure.

Lemma 4 Let E be a generalized random spectral measure on (S, A, H).

(1) If x, y ∈ H and M, N ∈ A then

E(M)x(ω), E(N)y(ω) = x, E(M ∩ N)y(ω) a.s.

In particular, if M ∩ N = ∅ then E(M)x(ω) and E(N)y(ω) are orthogonal in H

a.s.

(2) If (M i ) is a sequence of pairwise disjoint sets from A then for each x ∈ H

E

∞

Proof (1) Put A = E(M), B = E(N), v = E(N)y = By Then by Lemma2

E(M)x(ω), E(N)y(ω) = Ax(ω), v(ω) = x, Av(ω) = x, A(By)(ω)

= x, (AB)y(ω) = x, E(M ∩ N)y(ω) a.s.

(2) By the condition (4) in the Definition 3.1

E

∞

Taking account of the fact that(E(M i )x(ω))∞

i=1are pairwise orthogonal in H , we

get (7)



Let B (S) be a Banach space of bounded complex-valued measurable functions defined

on S with the supremum norm f = sup s ∈S | f (s)| Now we are going to define the

Lemma 5 For each simple function f (s) =n

i=1

M i



x(ω)2 f 2x2.

Let f ∈ B(S) Fix x ∈ H Because the set of simple functions is dense in B(S),

there is a sequence( f n ) ∈ B(S) such that lim n→∞ f n − f = 0 By (8) for almostallω, we have

I ( f n )x(ω) − I ( f m )x(ω) = I ( f n − f m )x(ω)  f n − f m x

Hence, the sequence{I ( f n )x(ω)} converges a.s in H Put

I ( f )x(ω) = lim

n→∞I ( f n )x(ω).

By a standard argument, it is easy to show that this limit does not depend on the choice

of the sequence( f n ) and is denoted by

Hence I ( f ) is a random bounded operator on H., I ( f ) is denoted by

I ( f ) =



S

f (s)E(ds)

and is called the integral of f w.r.t the generalized random spectral measure E.

Theorem 4.2 (1) For f , g ∈ B(S), c ∈ C, we have

At first assume that f (s) = n

i=1c i1M i (s) is simple For each i, since E(M i ) is a

random projection operator, we have

E(M i )x(ω), y = x, E(M i )y(ω) a.s.

Hence for almost allω

Next assume that f is bounded Then there is a sequence ( f n ) of simple functions

which uniformly converges to f Then f n converges uniformly to f Using the

Now by Lemma6, we have

(AB)x(ω), y = A[Bx(ω)], y = Bx(ω), A∗y(ω).