DSpace at VNU: Generalized Random Spectral Measures

DSpace at VNU: Generalized Random Spectral Measures tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn...

J Theor Probab 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 measures are introduced and investigated It is shown that each generalized random spectral measure on (C, B(C)) admits a modification which is a random spectral measure Keywords Random operators · Random normal operator · Random Hermitian bounded operators · Random projection operator · Random spectral measure · Generalized random spectral measure · Integral with respect to generalized spectral measure Mathematics Subject Classification (2000) 60B11 · 45R05 Primary 60H25 · Secondary 60H05 · 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 mapping ω → (ω, x) is a Y -valued random variable Equivalently, can be viewed D H Thang (B)· N Thinh Department of Mathematics, Hanoi University of Sciences, Hanoi, Vietnam e-mail: hungthang.dang53@gmail.com N Thinh e-mail: thinhj@gmail.com T X Quy Department of Mathematics, College of Sciences, Thai Nguyen University, Thai Nguyen City, Vietnam e-mail: xuanquy@gmail.com 123 J Theor Probab as a mapping : X → L 0Y ( ), where L 0Y ( ) stands for the space of X -valued random variables and is equipped with the topology of convergence in probability If a random operator : X → L 0Y ( ) 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 a random generalization of usual deterministic 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 random fixed points of random operators, random operator equations, 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 spaces were published in [11,12] This study was closely connected with the investigation of the spectra of random matrices of order approaching infinity in the context of actual physical 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 important role in functional analysis and has many applications (see [1,4,10]) In Sect 2, the notion of random spectral measures, random normal operators, and random Hermittian operators are defined as a family of spectral measures, normal operators and Hermitian operators, respectively, indexed by the parameter set satisfying certain measurability It is shown that such random operators can be represented as the limit of the integral with respect to certain random spectral measure Next the notion of random spectral measure is generalized further To this, the notion of random projection operators is introduced in the Sect Then the notion of generalized random spectral measure, the integral of complex-valued bounded functions with respect to generalized random spectral measures are introduced and investigated in Sect The main result in this section is the claim that every generalized random spectral measures on (C, B) admits a modification which is a random spectral measure 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 (∅) = and E(S) = I (3) If M, N ∈ A then U (M ∩ N ) = U (M)U (N ) (4) If (Mi ) is a sequence of pairwise disjoint sets from A then for each x ∈ H U ∞ ∪i=1 Mi ∞ U (Mi )x, x= i=1 where the series is convergent in H 123 J Theor Probab • A linear operator T on H is called a normal operator if T is bounded and T T ∗ = T ∗ T • A linear operator T on H is called a Hermitian operator if if T is bounded and T = T ∗ The spectral theorem for normal operators and Hermitian operators states that Theorem 2.2 ([10]) (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 zU (dz) T = σ (T ) (2) Let T be a Hermitian operator on H and σ (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 = λU (λ) σ (T ) Our aim in this section is to obtain the random version of the spectral theorem To this end, the notion of random normal operators and the notion of random spectral measures 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 mapping ω → 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 variable The random spectral theorem for random normal operators and random Hermitian operators is expressed as follows: Theorem 2.4 (1) Let T (ω) be a random normal operator Then there is a random spectral measure U (ω) on (C, B, H ) such that for each ω ∈ , x ∈ H T (ω)x = lim zU (ω)(dz)x, n (1) Bn where Bn = {z ∈ C : |z| n} 123 J Theor Probab (2) Let T (ω) be a random Hermitian operator Then there is a random spectral measure U (ω) on (R, B, H ) such that for each x ∈ H n T (ω)x = lim n λU (ω)x(ω) (2) −n Proof (1) For each ω, by Theorem 2.2 there is a spectral measure U (ω) defined on the Borel subsets of the spectrum σ (T (ω)) of T (ω) such that T (ω)x = zU (ω)(dz)x σ (T (ω)) 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 Dn = {ω : T (ω) < n}, Bn = {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 ∈ Dn implies σ (T (ω)) ⊂ Bn Hence for each ω ∈ Dn T (ω)x = zU (ω)(dz)x = σ (T (ω)) Step Let P(z) = k zU (ω)(dz)x Bn ak z h be a polynomial If P(T (ω)) = P(z)U (ω)(dz) σ (T (ω)) then the mapping ω → P(T (ω))x from Dn into H is measurable Indeed, P(U (ω))x = P(z)U (ω)(dz)x = σ (T (ω)) ak T (ω)k x k 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 Dn into H is measurable Step If f (z) is a continuous function defined on Bn then the mapping ω → f (T (ω))x from Dn into H is measurable By the Weierstrass theorem there exists a sequence of polynomial Pk (z) converges uniformly to f (z) on Bn Hence |Pk (z)| is uniformly bounded Because σ (T (ω)) ⊂ Bn for ω ∈ Dn by Lemma in [4, p 899], we get 123 J Theor Probab lim Pk (T (ω))x = f (T (ω))x for each ω ∈ Dn Hence the conclusion follows from Step Step 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 Mn = {s : d(s, M) ≥ 1/n then Mn is closed and M ∩Mn = ∅ By Urysohn Lemma there exists a continuous function f n such that f n (z) = f (z) If z ∈ / M then fot z ∈ M and f n (z) = for z ∈ Mn and there is n such that d(z, M) ≥ 1/n ≥ 1/n for n ≥ n Hence f n (z) = for n ≥ n Consequently lim f n (z) = M (z) Since the mapping M → E(ω)(M)x is a H -valued measure by Theorem in [2, p 56], we get for each ω ∈ Dn f n (z)U (ω)(dz)x = lim n σ (T (ω) M (s)U (ω)(dz)x, σ (T (ω) i.e., limn f n (T (ω))x = U (ω)(M)x ∀ω ∈ Dn From Step ,it follows that the mapping ω → U (ω)(M)x from Dn into H is measurable But Dn is measurable and = ∪n Dn so the mapping ω → U (ω)(M)x from into H is measurable Last step Let M is the class of Borel sets M such that the map ω → U (ω)(M)x from into H is measurable Then M is a σ -algebra Indeed by the definition of spectral measure for each ω 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) (∩An ) x = lim U (ω)(An )x if (An ) ↓ U (ω) (∪An ) x = lim U (ω)(An )x if (An ) ↑ From this M is closed under the intersection operation, the union operation and it is a monotone class Hence M 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, fix ω ∈ Because Dn ↑ there exists n (ω) such that ω ∈ Dn for every n > n (ω) Thus if n > n (ω) then σ (T (ω)) ⊂ Bn which implies that T (ω)x = zU (ω)(dz)x = σ (T (ω) zU (ω)(dz)x, Bn i.e., zU (ω)(dz)x T (ω)x = lim n Bn 123 J Theor Probab (2) By Theorem 2.2 for each ω , there is a spectral measure U (ω) defined on the Borel subsets of the spectrum σ (T (ω)) ⊂ R of T (ω) By the same argument as the above proof, we conclude that (2) holds Random Projection Operators Let ( , F, P) be a complete probability space, and X be a separable Banach space A measurable mapping ξ 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 0X ( ) We not distinguish two X -random variables which are equal almost surely The space L 0X ( ) is equipped with the topology of convergence in probability If a sequence (ξn ) of L 0X ( ) converges to ξ in probability then we write p − lim ξn = ξ Definition 3.1 ([15,18]) Let X, Y be separable Banach spaces A linear continuous mapping A from X into L 0Y ( ) is said to be random linear operators or a random operator for short.(We omit the word “linear” because from now on only random linear operators are considered.) Definition 3.2 ([18]) A random operator A : X → L 0Y ( ) is said to be bounded if there exists a real-valued random variable k(ω) such that for each x ∈ X Ax(ω) k(ω) x a.s (3) Noting that the exceptional set in (3) may depend on x In general, a random operator need not be bounded For examples of random operators and random bounded operators, we refer to[18] Theorem 3.3 ([18]) A random operator A : X → L 0Y ( ) is bounded if and only if there is a mapping T A : → L(X, Y ) such that Ax(ω) = T A (ω)x a.s (4) (1) (2) It is easy to see that the mapping T A is unique in the sence: if T A , T A satisfy (4) (1) (2) then T A (ω) = T A (ω) a.s Theorem 3.4 Let A : X → L 0Y ( ) be a bounded random operator Define a mapping A˜ on L 0X ( ) by ˜ Au(ω) = T A (ω)(u(ω)) Then A˜ is a linear continuous mapping from L 0X ( ) into L 0Y ( ) 123 J Theor Probab ˜ Proof At first, we show that if u ∈ L 0X ( ) then Au(ω) ∈ L 0Y ( ) Indeed, if u is an X -valued simple random variable of the form n u(ω) = Ei (ω)xi i=1 then for almost all ω n ˜ Au(ω) = T A (ω) n Ei (ω)xi i=1 Ei (ω)T A (ω)xi = i=1 n = Ei (ω)Axi (ω) ∈ L 0Y ( ) i=1 Now assume that u ∈ L 0X ( ) Then there exists a sequence (u n ) of X -valued simple ˜ On the ˜ n = Au random variables converging to u in probability Hence p − limn Au other hand, there exists a subsequence (vn k ) such that limk→∞ 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 lim u n k (ω) = u(ω) k→∞ Since T A (ω) ∈ L(X, Y ) we get ˜ n k (ω) = lim T A (ω) u n k (ω) = T A (ω)(u(ω)) Au k→∞ ˜ = Au(ω) ∀ω ∈ D, ˜ ˜ ˜ n k (ω) = A(ω)(u(ω)) a.s Hence A(ω)(u(ω)) ∈ L 0Y ( ) The which implies lim Au k→∞ ˜ Put k(ω) = T A (ω) From (5), linearity of A˜ is clear We show the continuity of A it follows that there is a set D with P(D) = such that Axn (ω) = T A (ω)xn for all ω ∈ D, n = 1, 2, For each ω ∈ D k(ω) = T A (ω) = sup T A (ω)xn = sup Axn (ω) n n 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 > and u ∈ L 0X ( ), we have ˜ > t) = P( Au ˜ > t, u ˜ > t, u > r ) P( Au r ) + P( Au P(k(ω) > t/r ) + P( u > r ) 123 J Theor Probab Let (u n ) be a sequence in L 0X ( ) converging to u in probability We have P ˜ n − Au ˜ >t Au P{k(ω) > t/r } + P{ u n − u > r } Then lim sup P n ˜ n − Au ˜ >t Au P{k(ω) > t/r } Letting r → 0, we obtain lim sup P n ˜ n − Au ˜ > t = Au The above theorem shows that each bounded random operator A : X → L 0Y ( ) can be extended to a linear continuous mapping A˜ : L 0X ( ) → L 0Y ( ) by putting ˜ Au(ω) = T A (ω)(u(ω)) for each u ∈ L 0X ( ) Definition 3.5 If A, B : X → L 0X ( ) are two random bounded operators then the composition AB : X → L 0X ( ) is defined by ˜ (AB)(x) = A(Bx) From now on, for brevity the extension A˜ of A is also denoted by A Hence, we write (AB)(x) = A(Bx) Lemma If A, B : X → L 0X ( ) are two random bounded operators then AB is also a random bounded operator and T AB (ω) = T A (ω)TB (ω) a.s (5) Proof There exists a set D of probability one such that for each ω ∈ D, we have ABx(ω) = A(Bx) = T A (ω)(Bx(ω)) = T A (ω)(TB (ω)x) = (T A (ω)TB (ω)) x Consequently, AB 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 0H ( ) is said to be a random operator on H 123 J Theor Probab 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 Suppose that the random operator A is a random bounded operator Then A admits an adjoint A∗ Moreover A∗ is bounded and (1) T A∗ (ω) = T A (ω)∗ , a.s., (6) (2) For u, v ∈ L 0H ( ) 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 all ω ∈ 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 (1) If P is a random projection operator on H then TP (ω) is a projection operator on H for almost all ω (2) For each u ∈ L 0H ( ), we have Pu(ω) u(ω) a.s 123 J Theor Probab (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 all ω, the operator TP (ω) is a projection operator (2) From (1) it follows that Pu(ω) = TP (ω)(u(ω)) u(ω) a.s (3) From Lemma 2, we get Pu(ω), v(ω) = u(ω), P ∗ v(ω) = u(ω), Pv(ω) a.s 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) (2) (3) (4) E(M) is a random projection operator on H for each M ∈ A; E(∅) = and E(S) = I ; If M, N ∈ A then E(M ∩ N ) = E(M)E(N ); If (Mi ) is a sequence of pairwise disjoint sets from A then for each x ∈ H ∞ E ∞ Mi E(Mi )x(ω) a.s x(ω) = i=1 i=1 Noting that the exceptional set in (4) may depend on (Mi ) 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 TE(M) (ω) = U (ω)(M) From Lemma it is easy to check that the conditions (1)–(3) holds Since U (ω) is spectral measure hence for all ω we have ∞ U (ω) i=1 123 ∞ Mi U (ω)(Mi )x, x= i=1 J Theor Probab i.e., ∞ ∞ E E(Mi )x(ω) ∀ ω ∈ x(ω) = Mi i=1 i=1 Hence E is a generalized random spectral measure Lemma 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 (Mi ) is a sequence of pairwise disjoint sets from A then for each x ∈ H ∞ E ∞ Mi x(ω) i=1 = E(Mi )x(ω) a.s (7) i=1 Proof (1) Put A = E(M), B = E(N ), v = E(N )y = By Then by Lemma 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 ∞ ∞ Mi E i=1 E(Mi )x(ω) x(ω) = i=1 ∞ are pairwise orthogonal in H, we Taking account of the fact that (E(Mi )x(ω))i=1 get (7) Let B(S) be a Banach space of bounded complex-valued measurable functions defined on S with the supremum norm f = sups∈S | f (s)| Now we are going to define the integral f (s)E(ds) S of f ∈ B(S) with respect to the generalized random spectral measure E on (S, A, H ) 123 J Theor Probab n Lemma For each simple function f (s) = ci Mi (s) and x ∈ H put i=1 n I ( f )x(ω) = ci E(Mi )x(ω) i=1 Then the mapping I ( f ) : x → I ( f )x defines a random bounded operator on H and I ( f )x(ω) f x a.s (8) Proof The linearity of I ( f ) is clear Now we shall show (8) By Lemma for almost all ω, we have n I ( f )x(ω) = n ci E(Mi )x(ω), i=1 n n = = c j E(M j )x(ω) j=1 ci c j E(Mi )x(ω), E(M j )x(ω) i=1 j=1 n |ci |2 i=1 n E(Mi )x(ω) n E(Mi )x(ω) 2= f f i=1 E Mi x(ω) f x i=1 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 limn→∞ f n − f = By (8) for almost all ω, we have I ( f n )x(ω) − I ( f m )x(ω) = I ( f n − f m )x(ω) fn − fm x Hence, the sequence {I ( f n )x(ω)} converges a.s in H Put I ( f )x(ω) = lim I ( f n )x(ω) n→∞ 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 I ( f )x(ω) = f (s)E(ds)x S From the Lemma by passing to the limit, we conclude that the mapping I ( f ) : x → I ( f )x from H into L 0H ( ) is a random operator on H and I ( f )x(ω) 123 f x a.s (9) J Theor Probab Hence I ( f ) is a random bounded operator on H., I ( f ) is denoted by f (s)E(ds) I( f ) = S 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 I ( f + g) = I ( f ) + I (g); I (c f ) = cI ( f ) (2) I ( f )∗ = f (s)E(ds) S (3) For f, g ∈ B(S), we have I ( f )I (g) = I ( f g) = f (s)g(s)E(ds) S Proof (1) It is obvious (2) We shall show that for every x, y ∈ H f (s)E(ds)x(ω), y = x, S f (s)E(ds)y(ω) (10) S n At first assume that f (s) = ci Mi (s) is simple For each i, since E(Mi ) is a i=1 random projection operator, we have E(Mi )x(ω), y = x, E(Mi )y(ω) a.s Hence for almost all ω n n I ( f )x(ω), y = ci E(Mi )x(ω), y = i=1 n = ci E(Mi )x(ω), y i=1 n ci E(Mi )x(ω), y = i=1 n = ci x, E(Mi )y(ω) i=1 n x, ci E(Mi )y(ω) = x, i=1 ci E(Mi )y(ω) i=1 = x, I ( f )y(ω) 123 J Theor Probab 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 fact that I ( f )x(ω) = lim I ( f n )x(ω) a.s., n→∞ we have I ( f )x(ω), y = lim I ( f n )x(ω), y = lim x, I ( f n )y(ω) n→∞ n→∞ = x, I ( f )y(ω) (3) At first, we show that if f, g ∈ B(S) then for almost all ω I (g)x(ω), I ( f )y(ω) = I ( f g)(ω)x, y Indeed, suppose that f (s) = n ci Mi (s), g(s) = i=1 n (11) di Mi (s) are simple func- i=1 tions Then n I (g)x(ω), I ( f )y(ω) = n ci E(Mi )x(ω), i=1 n = d j E(M j )y(ω) j=1 n ci d j E(M j )x(ω), E(Mi )y(ω) = i, j=1 ci di E(Mi )x(ω), y i=1 n ci di E(Mi )x(ω), y = I ( f g)x(ω), y = i=1 Next, for brevity put A = I ( f ), An = I ( f n ), B = I (g), Bn = I (gn ) where ( f n ), (gn ) ∈ B(S) and limn→∞ f n = f, limn→∞ gn = g in B(S) We have Bn x(ω), A∗n y(ω) = I (gn )x(ω), I ( f n )y(ω) = I ( f n gn )x(ω), y Let n → ∞, we get for almost all ω Bx(ω), A∗ y(ω) = I ( f g)x(ω), y Now by Lemma 6, we have (AB)x(ω), y = A[Bx(ω)], y = Bx(ω), A∗ y(ω) 123 J Theor Probab Hence, (AB)x(ω), y = I ( f g)x(ω), y → (AB)x(ω) = I ( f g)x(ω), i.e., AB = f (s)g(s)E(ds) S The following theorem is the bounded convergence theorem for spectral measures and generalized random spectral measures Theorem 4.3 (1) Let U be a spectral measure on (S, A, H ) Suppose that ( f n ) is a M for all n If limn→∞ f n = f pointwise, sequence in B(S) such that f n then for each x ∈ H f n (s)U (ds)x = lim n→∞ S f (s)U (ds)x (12) S (2) Let E be a generalized random spectral measure on (S, A, H ) Suppose that ( f n ) M for all n If limn→∞ f n = f pointwise, is a sequence in B(S) such that f n then for each x ∈ H f n (s)E(ds)x(ω) = lim n→∞ S f (s)E(ds)x(ω) in probability (13) S Proof (1) By definition of spectral measures, for each x ∈ H , the mapping Fx : M → U (M)x is a H -valued vector measure denoted by U (ds)x The integral S f (s)U (ds)x defined for each bounded function f : S → C (see [10]) can be regarded as the integral of f with respect to the vector measure Fx Applying the Bartle’s bounded convergence theorem in the theory of vector measures (Theorem 1, [2, p 56]) we get (12) (2) Let L 2H ( ) be the space of H -valued random variables u such that E u = u(ω) d P(ω) < ∞ L 2H ( ) becomes a Hilbert space with the inner product given by [u, v] = u(ω), v(ω) d P(ω) We claim that for each fixed x ∈ H • The mapping G x : M → E(M)x 123 J Theor Probab defines a countably additive, vector measure G x taking values in the Hilbert space L 2H ( ) • For each f ∈ B(S) the integral f (s)G x (ds) is an element of L 2H ( ) and S f (s)E(ds)x(ω) = f (s)G x (ds)(ω) S (14) S Indeed, • By Lemma 3, E(M)x(ω) x a.s so E(M)x(ω) d P x dP = x which show that G x (M) ∈ L 2H ( ) Now we prove that G x is countably additive n ∞ Put Sn (ω) = i=1 E(Mi )x(ω) = i=1 G x (Mi ) By the condition (4) in the ∞ Definition 4.1, the sequence Sn converges a.s to S(ω) = E i=1 Mi x(ω) = ∞ Gx i=1 Mi We have n Sn (ω) = n E(Mi )x(ω) = E i=1 Mi x(ω) a.s i=1 By the dominated theorem, we conclude that the sequence Sn converges to S in L 2H ( ), i.e., ∞ Gx ∞ G x (Mi ), = Mi i=1 i=1 where the series is convergent in L 2H ( ) • Clearly (14) holds for each simple function f Suppose that f ∈ B(S) and (gn ) is the sequence of simple functions converging to f Since gn (s)E(ds)x(ω) = lim n→∞ S S gn (s)G x (ds)(ω) = lim n→∞ S and S f (s)E(ds)x(ω) a.s gn (s)E(ds)x(ω) = f (s)G x (ds)(ω) in L 2H ( ), S S gn (s)G x (ds)(ω), we infer that (14) holds Again using the Bartle’s bounded convergence theorem (Theorem 1, [2, p 56]), we obtain f n (s)E(ds)x(ω) = lim n→∞ S 123 f (s)E(ds)x(ω) in L 2H ( ) S J Theor Probab which implies (13) The following theorem shows that each generalized random spectral measure on (C, B, H ) have a modification which is a random spectral measure Theorem 4.4 Let E be a generalized random spectral measure on (C, B, H ) Then there exists a random spectral measures U (ω) on (C, B, H ) such that for every x ∈ H, M ∈ B E(M)x(ω) = U (ω)(M)x a.s Proof Lemma Given a positive integer n Let E be a generalized random spectral measure concentrated on Bn = {z ∈ C : |z| n}, i.e., E(M) = if M ∩ Bn = ∅ Then there is a family U (ω) of spectral measures indexed by the parameter set such that for each M ∈ B(Bn ) and x ∈ H E(M)x(ω) = U (ω)(M)x a.s Proof of Lemma Put Ax(ω) = z E(dz)x(ω) (15) Bn Then by Theorem 4.2, A A∗ = A∗ A and by (8) Ax(ω) n x a.s By Lemma and Lemma 2, there exists a family {T (ω)} of normal operators indexed by the parameter set such that for each x ∈ H , we have Ax(ω) = T (ω)x Hence T (ω)x n x ⇒ T (ω) n By the Theorem 1.14 (see [1, p 259]), for each ω there exists a spectral measure U (ω) for T (ω) such that T (ω) = zU (ω)(dz) (16) σ (T (ω)) 123 J Theor Probab By the spectral radius formula r (T ) T , where r (T ) is the spectral radius of a operator T , we infer that σ (T (ω)) ⊂ Bn so zU (ω)(dz) = T (ω)) zU (ω)(dz) a.s Bn From (15) and (16), we get z E(dz)x(ω) = Bn Step Let P(z) = k zU (ω)(dz)x (17) Bn ak z k be a polynomial We have ⎛ ⎞⎛ ⎜ z E(dz)x(ω) = ⎝ ⎟⎜ z E(dz)(ω)⎠ ⎝ Bn Bn ⎟ z E(dz)x(ω)⎠ Bn ⎛ ⎞⎛ ⎜ z U (ω)(dz)x = ⎝ ⎟⎜ zU (ω)(dz)⎠ ⎝ Bn ⎞ ⎞ ⎟ zU (ω)(dz)x ⎠ Bn Bn z E(dz)x(ω) = z U (ω)(dz)x Hence by (17), we get Bn Bn By the induction z k E(dz)x(ω) = Bn z k U (ω)(dz)x Bn Hence P(z)E(dz)x(ω) = Bn P(z)U (ω)(dz)x a.s Bn If f (z) is a continuous function defined on Bn then by the Weierstrass theorem there exists a sequence of polynomial Pk (z) that converges uniformly to f (z) on Bn Hence by Theorem 4.3, we get for each ω Pk (z)U (ω)(dz)x = lim k→∞ Bn 123 f (z)U (ω)(dz)x Bn J Theor Probab By Theorem 4.3 Pk (z)E(dz)x(ω) = lim k→∞ Bn f (z)E(dz)x(ω) in probability Bn Hence for each continuous function f on Bn , we have f (z)E(dz)x(ω) = Bn f (z)U (ω)(dz)x a.s Bn Step Suppose that M is a closed set of C For each k let Mk = {z : d(z, M) 1/k} then Mk is closed and M ∩ Mk = ∅ By Uryxon’s theorem there exists a continuous f k (z) function f k such that f k (z) = for z ∈ M and f k (z) = for z ∈ Mk and 1/n 1/n for n n Hence If z ∈ / M then there is n such that d(z, M) By f k (z) = for n n Consequently, limk→∞ f k (z) = M (z) and supk f k Theorem 4.3, we get for each ω f k (z)U (ω)(dz)x = lim k→∞ Bn M (z))U (ω)(dz)x = U (ω)(M)x (18) Bn Again by Theorem 4.3 f k (z)E(dz)x(ω) = lim k→∞ Bn M (z)E(dz)x(ω) = E(M)x(ω) in probability Bn Hence, for each closed set M, we get E(M)x(ω) = U (ω)(M)x a.s Step Let M ⊂ B is the class of sets M ∈ B such that E(M)x(ω) = U (ω)(M)x By the definition of spectral measures and generalized random spectral measures and Theorem 4.3, we get for each ω 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) n=1 ∞ U (ω) n=1 An x = lim U (ω)(An )x if (An ) ↓; An x = lim U (ω)(An )x if (An ) ↑; n→∞ n→∞ 123 J Theor Probab and E(M ∩ N )x(ω) = E(M)[E(N )x(ω)] a.s.; E(M ∪ N )x(ω) = E(M)x(ω) + E(N )x(ω) − E(M ∩ N )x(ω) a.s.; ∞ E An x(ω) = p- lim E(An )x(ω) if (An ) ↓; An x(ω) = p- lim U (ω)(An )x if (An ) ↑ n→∞ n=1 ∞ E n→∞ n=1 From this M is closed under the intersection operation, the union operation and it is a monotone class Hence M is a σ -algebra From this M is closed under the intersection operation and the union operation Hence M is a algebra By Step 2, M contains the closed sets So M coincides the the class B of all Borel sets of C The lemma is proved Let E n = E| Bn be the restriction of E on Bn By Lemma 6, there is a family Un (ω) of spectral measures indexed by the parameter set such that for each M ∈ B(Bn ) and x ∈ H E n (M)x(ω) = Un (ω)(M)x a.s Fix m > n We shall show that Um (ω)| Bn = Un (ω) a.s (19) Put Un (ω) = Um (ω)| Bn We have for each x ∈ H zUn (ω)(dz)x = Bn zUm | Bn (ω)(dz)x Bn = z E m (dz)x(ω) = Bn = z E n (dz)x(ω) Bn zUn (ω)(dz)x Bn Since H is separable there is a set D of probability one such that zUn (ω)(dz) = Bn zUn (ω)(dz) for each ω ∈ D Bn By the uniqueness of the spectral representation it follows that Um (ω) = Un (ω) 123 for each ω ∈ D, J Theor Probab i.e., (19) holds as claimed From this there is a set ω ∈ then for of probability one such that if Um (ω)| Bn = Un (ω) whenever m > n (20) Fix ω ∈ We shall construct the projection operator U (ω)(M) (M ∈ B(C)) as follows Put Mn = M ∩ Bn and Pn = Un (ω)(Mn ) Then Pn is a projection operator onto Hn = Pn (H )., Hn is a closed subspace of H and by (19) Hn ⊂ Hn+1 For fixed x ∈ H , we have Pn+1 x Pn x ⇔ Pn+1 x, Pn+1 x Pn x, x This means the sequence n Pn x, Pn x ⇔ x, Pn+1 x is non-decreasing and bounded by x x, Pn x Hence, lim | (Pn − Pm )x, x | = m,n→∞ Put h(x, y) = (Pn − Pm )x, y , n > m Using the Schwarz inequality for non-negative bilinear form h(x, y), we get (Pn − Pm )x = (Pn − Pm )x, (Pn − Pm )x = h(x, (Pn − Pm )x) h(x, x)h((Pn − Pm )x, (Pn − Pm )x) = (Pn − Pm )x, x (Pn − Pm )2 x, (Pn − Pm )x (Pn − Pm )x, x Pn − Pm x x (Pn − Pm )x, x → as m, n → ∞ which proves that for each x ∈ H, lim Pn x exists Put P x = lim Pn x We have n→∞ n→∞ = x − Pn x, Pn x −→ x − P x, P x = Hence, P is the projection operator Put U (ω)(M) = P we get lim Un (ω)(Mn )x = U (ω)(M)x ⇔ lim Un (ω)(M ∩ Bn )x = U (ω)(M)x n→∞ n→∞ We verify that U (ω) is a spectral measure We have Un (ω)(M ∩ N ∩ Bn )x = Un (ω)(M ∩ Bn )[Un (ω)(N ∩ Bn )x] = Pn yn , where yn = Un (ω)(N ∩ Bn )x We have lim yn = U (ω)(N )x = y Noting that since n→∞ 1, we have lim Pn yn = P y whenever lim yn = y let n → ∞ we get Pn n→∞ n→∞ U (ω)(M ∩ N )x = U (ω)(M)[U (ω)(N )x] 123 J Theor Probab Next, we show that if (Ai ) is a sequence of disjoint sets from A a nd x ∈ H then ∞ ∞ U (ω) U (ω)(Ai )x x= Ai i=1 (21) i=1 Indeed, we have ∞ U (ω) ∞ Ai x = lim Un (ω) n→∞ i=1 ∞ Ai ∩ Bn x = lim n→∞ i=1 U (ω)(Ai ∩ Bn )x, i=1 (22) and ∞ ∞ U (ω)(Ai )x = i=1 lim Un (ω)(Ai ∩ Bn )x i=1 n→∞ Put V0 = ∅, V j = B j \ B j−1 , j = U j (ω)(Ai ∩ V j ) = Un (ω)(Ai ∩ V j ) for j n then n Un (ω)(Ai ∩ Bn ) = j j=1 and j , akh = 0, (i, j) = (k, h) One has ∞ ∞ lim Un (ω)(Ai ∩ Bn )x = i=1 n→∞ n j , lim i=1 n→∞ j=1 and ∞ U (ω) ∞ Ai x = lim i=1 n→∞ ∞ n Un (ω)(Ai ∩ Bn )x = lim n→∞ i=1 j i=1 j=1 Therefore, we have to show that ∞ n→∞ ∞ n j = lim i=1 j=1 i=1 To prove (23), we need the following lemma 123 n j lim n→∞ j=1 (23) J Theor Probab Lemma Let j (k, h) If ∈ H satisfy the condition j , akh = 0, ∀(i, j) = (i, j)∈N2 ∞ ∞ j exists i=1 j=1 then ∞ ∞ ∞ ∞ ∞ j=1 i=1 then j i=1 j=1 ∞ ∞ Proof of the Lemma Since ∞ j = j exists and j=1 i=1 j exists and j , akh = 0, ∀(i, j) = (k, h) i=1 j=1 ∞ ∞ j < ∞ It is easy to see that i=1 j=1 ∞ ∞ ∞ j=1 i=1 From j ∞ 2 j ∞ = j j < ∞ i=1 j=1 < ∞, then i=1 ∞ j ∞ ∞ = j j=1 ∞ j=1 j < ∞, j < ∞ j=1 i=1 and similarily ∞ j ∞ ∞ = i=1 j ∞ i=1 i=1 j=1 By the triangle inequality and the fact that j , akh = 0, ∀(i, j) = (k, h) we get n ∞ ∞ ∞ j − j=1 i=1 n m + n ∞ i=1 j=1 m ∞ j=1 i=1 m i=1 j=1 m j=1 i=m+1 j ∞ ∞ ∞ j i=1 j=1 ∞ j + i=1 j=n+1 ∞ j − i=1 j=1 j + m j=1 i=1 ∞ j + j − j=1 i=1 n ∞ n j − j j i=m+1 j=1 123 J Theor Probab ⎛ =⎝ j ∞ +⎝ j 2⎠ ⎞1/2 ∞ j 2⎠ i=1 j=n+1 ⎞1/2 ∞ m +⎝ 2⎠ j=1 i=m+1 ⎛ ⎛ ⎞1/2 ∞ n i=m+1 j=1 From this ∞ 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Research in theory of random operators has been carried out in many directions such as random fixed points of random operators, random operator equations, and random lineart operators (e.g., [3,5,8,11]–[22]