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

This article was downloaded by: [University of Otago] On: 11 September 2014, At: 00:15 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochastics and Stochastics Reports Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gssr20 Generalized random linear operators on a Hilbert space a Dang Hung Thang & Nguyen Thinh a a Department of Mathematics , Hanoi University of Science , Hanoi , Vietnam Published online: 04 Dec 2012 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 To link to this article: http://dx.doi.org/10.1080/17442508.2012.736995 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Downloaded by [University of Otago] at 00:15 11 September 2014 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Stochastics: An International Journal of Probability and Stochastic Processes, 2013 Vol 85, No 6, 1040–1059, http://dx.doi.org/10.1080/17442508.2012.736995 Generalized random linear operators on a Hilbert space Dang Hung Thang* and Nguyen Thinh Department of Mathematics, Hanoi University of Science, Hanoi, Vietnam Downloaded by [University of Otago] at 00:15 11 September 2014 (Received July 2011; final version received 12 September 2012) In this paper, we are concerned with generalized random linear operators on a separable Hilbert space Generalized random linear bounded operators, generalized random linear normal operators and generalized random linear self-adjoint operators are defined and investigated The spectral theorems for generalized random linear normal operators and generalized random linear self-adjoint operators are obtained Keywords: random operator; generalized bounded random operator; generalized random normal operators; generalized random self-adjoint operators; random spectral measure 2000 Mathematics Subject Classification: 60H025; 47B80; 47H40; 60B11 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 mapping v 7! 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 operators has been aroused 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 linear 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 transforms each deterministic input x [ X into a random output f(x) with values in Y Taking into account many circumstances in which the inputs are also subject to the influence of a random 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 a separable Hilbert space In Section 2, a definition and some examples of generalized random linear operators are introduced In particular, the adjoint of a generalized random linear operator and the notion of generalized random linear bounded operators are defined and discussed Sections and are devoted to generalized random linear normal operators and generalized random linear self-adjoint operators, respectively The main results of these sections are random spectral theorems which state that every generalized random linear normal operators as well as every generalized random linear self-adjoint operators *Corresponding author Email: thangdh@vnu.edu.vn q 2013 Taylor & Francis Stochastics: An International Journal of Probability and Stochastic Processes 1041 can be represented as the limit of the integrations with respect to certain random spectral measures Downloaded by [University of Otago] at 00:15 11 September 2014 Generalized random linear operators Let (V, F, P) be a probability complete space and H be a complex Hilbert separable space with the inner , , A mapping u : V ! H is said to be a H-valued random variable (Hr.v.’s) if u is (F, B)-measurable, where B denotes the Borel s-algebra of H The space of all (equivalence classes of) H-r.v.’s and the space of all (equivalence classes of) complexvalued 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 in probability, we write p-lim un ¼ u If u, v [ L0(H) then the mappings v !, uðvÞ; vðvÞ and v 7! 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 un ¼ u, lim ¼ v in L0(H), then lim , un, 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 operators can 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 the application 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 Axtị ẳ t xsị dWsị: The domain of A can be extended to the class of random adapted functions u(t, v) with sample in L2[0, 1] by using the Ito stochastic integral Fut; vị ẳ ðt uðs; vÞ dWðsÞ: A definition of the application of Gaussian random operators on a Hilbert space H to some H-valued r.v’s was introduced in [2] in connection with the stochastic integral with nonadapted integrals in [6] The need to define the action of a random operator to some random inputs leads us to the following definition Definition 2.1 (i) A subset M of L0(H) is said to be a random linear space if for every u1 ; u2 [ M; j1 ; j2 [ L0 then j1 u1 þ j2 u2 [ M (ii) Let M be a random linear space A mapping F : M ! L0 ðHÞ is said to be a random linear mapping if for u1 ; u2 [ M; j1 ; j2 [ L0 then Fj1 u1 ỵ j2 u2 ị ẳ j1 Fu1 ị ỵ j2 Fu2 ị: As usual, the domain M of F is denoted by D(F) (iii) A random linear mapping F : D(F) ! L0(H) with the dense domain D(F) is called a generalized random linear operator or a generalized random operator for short (We omit the word ‘linear’ because only random linear mapping is considered in this paper.) 1042 D.H Thang and N Thinh Remark If F is a random linear mapping with the domain D(F) containing H as a subset, then F is a generalized random operator Indeed, let S be the linear subspace of L0(H) consisting of simple r.v.’s u of the form u¼ n X 1Ei xi ; xi [H; Ei [F : i¼1 Since H , D(F) it follows that S , D(F) Since S is dense in L0(H) so is D(F) Let A : H ! L0(H) be a linear mapping Then A can be extended to a generalized random operator Indeed let M be the set of H-valued r.v.’s of the forms Downloaded by [University of Otago] at 00:15 11 September 2014 uẳ n X ji xi ; 1ị i¼1 where xi [ H; ji [ L0 Clearly, M is a random linear space containing H Define a mapping F : M ! L0(H) by Fu ¼ n X ji Axi ; i¼1 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 Let A : H ! L0(H) be a random operator and e ẳ en ị1 nẳ1 stands for the basis of H Denote by M the set of all H-valued r.v.’s u for which the series X u; en ịAen 2ị nẳ1 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 a random linear mapping Since A is continuous and xẳ X x; en ịen ; nẳ1 we get Ax ẳ X u; en ịAen ; nẳ1 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 the collection of v [ L0(H) for which there exists g [ L0(H) such that for all u [ D(F) , Fu; v ¼, u; g : Stochastics: An International Journal of Probability and Stochastic Processes 1043 Such g is uniquely determined Indeed suppose that g1 ; g2 [ L0 Hị such that , Fu; v ẳ , u; g1 ẳ, u; g2 ;u [ DFị Then , u; g1 g2 ẳ ;u [ DFị Since D(F) is dense, there is a sequence (un) in D(F) such that lim un ¼ g1 g2 Hence , g1 g2 ; g1 g2 ¼ lim , un ; g1 g2 ¼ which implies that g1 ¼ g2 Putting g ¼ F*v we get a mapping F* : V ! L0(H) and the domain V of F* is denoted by D(F*) It is easy to check that D(F*) is a random linear space and F* is a random linear mapping and it is called the adjoint of F Thus the adjoint F* is defined by the relation ð3Þ Downloaded by [University of Otago] at 00:15 11 September 2014 , Fu; v ¼, u; F* v ; for 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 a generalized random operator (see Example below) Example Let (ji) be a sequence of standard Gaussian i.i.d r.v.’s Denote by M the set of all H-valued r.v.’s u for which the series X jn u; en ịen 4ị nẳ1 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 X 2 kðx; en Þen k ¼ kxk , 1; n¼1 the series X jn x; en ịen nẳ1 converges a.s Hence H , D(F) By Remark 1, F is a generalized random operator Next for every u [ D(F) and y [ H , Fu; y ¼ X jn ðu; en Þðy; en Þ ¼, u; g ; n¼1 P where g ẳ nẳ1 jn y; en ịen Hence H , DðF* Þ By Remark 1, F* is also a generalized random operator Example Let (jn) be a sequence of i.i.d real-valued P r.v.’s such that Ejn ¼ 0; Ejjn j ¼ Let M be the set of u [ L0(H) such that the series nẳ1 u; en ịjn converges in probability Clearly M is a random linear space For u [ M put Tu ¼ X n¼1 ðu; en Þjn : 1044 D.H Thang and N Thinh Take a [ H; a – Define a mapping F : M ! L0 ðHÞ by Fu ¼ aTu It is easy to see that F is a random linear mapping Moreover, for each x [ H X Ejx; en ịjn j ẳ n¼1 X 2 j , x; en j ¼ kxk , 1; n¼1 P which implies the series nẳ1 x; en ịjn converges a.s Hence H , M By Remark 1, F is a generalized random operator As usual, M is denoted by D(F) Now, we claim that Downloaded by [University of Otago] at 00:15 11 September 2014 (i) F : D(F) ! L0(H) is not continuous but the restriction of F on H is continuous P Indeed, put vk ẳ 1=kị kiẳ1 ji ei Then vk [ D(F) Because Ekvk k ¼ k X E j2 ¼ ; k i¼1 i k P we conclude that p-lim vk ¼ However, Fvk ẳ aTvk ẳ a1=kị kiẳ1 j2k By the law of large numbers p-lim Fvk ¼ a – Hence F : D(F) ! L0(H) is not continuous Since for every x [ H, EkFxk2 ¼ kak2kxk2 we conclude that F : H ! L0(H) is continuous (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) , Fu; v ¼, u; g ,, a; v Tu ¼, u; g ! , a; v Ten ¼, en ; g ; ;n 2 ! , a; v jn ¼, en ; g !j , a; v j jjn j ¼ j , en ; g j P1 2P 2 !j , a; v j n¼1 jjn j ¼ n¼1 j , en ; g j ¼ kgk It is easy to prove that P1 n¼1 jjn j ẳ ỵ1 a:s Hence j , a; v j ¼ ! , a; v ¼ 0: Hence DF* ị ẳ {v [ L0 Hị :, a; v ¼ 0}; i.e D(F*) ¼ L0(H1) where H1 ¼ [a ]’ 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 a sequence in H such that lim yn ¼ y and p-lim F*yn ¼ g Then for every u [ D(F) , Fu; yn ¼, u; F* yn : Stochastics: An International Journal of Probability and Stochastic Processes 1045 Since p lim , Fu; yn ¼, Fu; y n p lim , u; F* yn ¼, u; g ; n we obtain , Fu; y ¼, u; g which implies that g [ D(F*) and g ¼ F*y Hence the claims follow from the closed graph theorem A 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) Downloaded by [University of Otago] at 00:15 11 September 2014 kFukðvÞ # kðvÞkukðvÞ a:s: Theorem 2.5 Suppose that F : D(F) ! L0(H) is a generalized random bounded operator Then F can be uniquely extended to a generalized random continuous bounded operator ˜ : L0(H) ! L0(H) Hence, without the loss of generality it can be assumed that the F domain of a generalized random bounded operator is the whole space L0(H) If F : L0(H) ! L0(H) is a generalized random bounded operator then there exists uniquely a.s the mapping T : V ! L(H, H) such that Fuvị ẳ Tvịuvịị a:s: 5ị Conversely, let T : V ! L(H, H) be a mapping such that for every x [ H, the mapping v 7! T(v)x is a H-valued r.v.’s Then the mapping F : L0 ðHÞ ! L0 ðHÞ defined by (5) is a generalized random bounded operator If F : L 0(H) ! L0(H) is a generalized random bounded operator then F* : L0(H) ! L0(H) is also a generalized random bounded operator and F* vvị ẳ T* vịvvịị a:s: Proof (1) For each t, r and u [ D(F) we have PkFuk tị ẳ PkFuk t; kuk # rị ỵ PkFuk t; kuk rị # Pkvị t=rị ỵ Pkuk rị: Let u [ L0(H) Since D(F) is dense in L0(H) there exist a sequence (un) [ D(F) such that lim un ¼ u Hence PfkFun Fum k tg # P{kvị t=r} ỵ P{kun um k r}: Then lim supPfkFun Fum k tg # P{kðvÞ t=r}: n;m 1046 D.H Thang and N Thinh Letting r ! we obtain lim supPfkFun Fum k tg ¼ 0: n;m Hence there exists limnFun By a standard argument, it is easy to show that this ˜ u limit does not depend on the choice of the sequence (un) and is denoted by F ~ : L0 ðHÞ ! L0 ðHÞ is continuous Moreover, F (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 Fxvị ẳ Tvịx a:s: ð6Þ Downloaded by [University of Otago] at 00:15 11 September 2014 If u is a H-valued simple random variable of the form uvị ẳ n X 1Ei vịxi ; iẳ1 then for almost all v Fuvị ẳ n X 1Ei vịFxi vị ẳ iẳ1 n X 1Ei vịTvịxi ẳ Tvị iẳ1 n X ! 1Ei vịxi iẳ1 ẳ Tvịuvịị: Now assume that u [ L0(H) Then there exists a sequence (un) in L0(H) converging to u in probability By the part (1) F is continuous which implies p-limnFun ¼ Fu On the other hand, there exists a subsequence (unk) such that limkunk(v) ¼ u(v) a.s Hence there exists a set D of probability one such that for each v [ D we have Funk vị ẳ Tvịunk vịị and lim unk vị ẳ uvị: k Since T(v) [ L(H, H) we get lim Funk vị ẳ Tvịuvịị; k ;v [ D; i.e limk Funk vị ẳ Tvịuvịị a:s: which implies p-limn Funk ẳ Tvịuvịị Consequently Fuvị ẳ Tvịuvịị a:s: Now we show that T is unique Assume that T1, T2 satisfy (5) Let (xn) be countable dense subset of H Then there exists a set D of probability one such that for each v[D T vịxn ẳ T vịxn ẳ Fxn vị; for all xn : Hence T1(v) ¼ T2(v) for v [ 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 a H-valued r.v.’s and F : L0(H) ! L0(H) is a generalized random operator Now we Stochastics: An International Journal of Probability and Stochastic Processes 1047 shall show that F is bounded Let (xn) be a sequence dense in B ¼ {x [ H : kxk # 1} and kðvÞ ¼ supkFxn kðvÞ: n Then k(v) is a non-negative random variable From (6) it follows that there is a set D with P(D) ¼ such that Fxn(v) ¼ T(v)xn for all v [ D; n ¼ 1; 2; For each v [ D kTvịk ẳ supkTvịxn k ẳ supkFxn vịk ẳ kvị; n n which implies that kT(v)k is a non-negative random variable Hence Downloaded by [University of Otago] at 00:15 11 September 2014 kFuðvÞk # kTðvÞkuðvÞk a:s: and we are done (4) Let u; v [ L0 ðHÞ Then there is a set D with P(D) ¼ such that for each v [ D , Fuvị; vvị ẳ, Tvịẵuvị; vvị ẳ, uvị; T* vịvvị ; which prove that F*v(v) ẳ T*(v)(v(v) a.s By part (3) F* : L0(H) ! L0(H) is a generalized random bounded operator A Remark The generalized random operator F presented in Example is not bounded because D(F*) ¼ L0(H1) – L0(H), where H1 ¼ [a ]’ The generalized random operator F presented in Example is also not bounded Indeed, suppose that F is bounded From the definition, it follows that supnkFenk , a.s However, supnkFenk ¼ supnjjnj ¼ a.s and we get a contradiction Generalized random normal operator Definition 3.1 A generalized random bounded operator F is said to be a generalized random normal operator if FF* ¼ F*F By Theorem 2.5 every generalized random bounded operator F is of the form Fuvị ẳ TðvÞðuðvÞÞ a:s:; ð7Þ where T : V ! L(H, H) is mapping such that for every x [ H, the mapping v 7! T(v)x is a H-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 FẵF* xvị ẳ F* ẵFxvị a:s: !TvịẵT* vịx ẳ T* vịẵTvịx a:s: !ẵTvịT* vịx ẳ ẵT* vịTvịx a:s: Since H is separable it follows that T(v)T*(v) ¼ T*(v)T(v) a.s 1048 D.H Thang and N Thinh Conversely suppose that T(v) is normal a.s Let u [ L0(H) There is a set D with P(D) ẳ such that TvịẵT* vịx ẳ T* vịẵTvịx; ;v [ D; ;x [ H !TvịẵT* vịuvịị ẳ T* vịẵTvịuvịị; ;v [ D FẵF* uvị ẳ F* ẵFuvị ẳ ;v [ D FF* u ¼ F* Fu Downloaded by [University of Otago] at 00:15 11 September 2014 Hence FF* ¼ F*F The main result in this section is the spectral theorem for generalized random normal operators A Let us recall the definition of spectral measure Definition 3.3 ([12]) Let (S, A) be a measurable space and H be a complex Hilbert separable space By a spectral measure E on (S, A, H) we mean a mapping E : A ! L(H, H), having the following properties E(M) is a projection for each M [ A E(B) ¼ and E(S) ¼ I If M,N [ A then E(M > N) ¼ E(M)E(N) In particular, if M, N are disjoint then E(M) and E(N) are orthogonal If (Mi) is a sequence of pairwise disjoint sets from A then for each x [ H   E < Mi x ¼ i¼1 X EðM i Þx; i¼1 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]) If E is a spectral measure for (S, A, H) and f : S ! C is a measurable, bounded function then the mapping T : H ! H defined by Tx ¼ ð f ðsÞEðdsÞx S is a normal operator Conversely, suppose T : H ! H is a normal operator and s(T) , C is the spectrum of T Then s(T) is a compact set and there exists a spectral measure E defined on the Borel subsets of the spectrum s(T) such that Tx ẳ zEdzịx s Tị ;x [ H; Stochastics: An International Journal of Probability and Stochastic Processes 1049 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 random spectral 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, the mapping v 7! E(v)(M)x is a H-valued random variable The random spectral theorem states the following Downloaded by [University of Otago] at 00:15 11 September 2014 Theorem 3.6 Let EðvÞ be a random spectral measure on (S, A, H) and f : S ! C be a measurable bounded function Define a mapping F : L0(H) ! L0(H) by Fuvị ẳ f sịEvịdsịuvị: S Then F is a generalized random normal operator Conversely, let F be a generalized random normal operator Then there is a random spectral measure E(v) on (C, B, H) such that for each v [ V; u [ L0 Hị Fuvị ẳ lim n zEvịdzịuvị a:s:; 8ị Bn where Bn ¼ {z [ C:jzj # n} Proof For each v define the mapping Tf(v):H ! H defined by T f vịx ẳ f sịEvịdsịx: S We claim that: For each x [ H the mapping x 7! Tf(v)x is a H-valued r.v At first, we prove that the P claims hold for each simple function g Indeed, if g is a simple function, gsị ẳ i ci 1M i ðsÞ then we have T g ðvÞx ¼ ð gðsÞEðvÞðdsÞx; S X ci EðvÞðM i ÞxðvÞ ¼ X ci EðvÞðM i Þx: 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 a sequence (gn) of simple functions converging uniformly to f(s) From this lim T gn vịx ẳ T f vịx n ;v: 1050 D.H Thang and N Thinh Hence the claims follow from the fact that the mapping x 7! Tgn(v)x is a H-valued r.v for every n Now we have Fuvị ẳ T f vịẵuvị: By Theorem 3.4 and Proposition 3.2, we conclude that F is a generalized random normal operator By Theorem 3.2 there is a mapping T : V ! L(H, H) such that Downloaded by [University of Otago] at 00:15 11 September 2014 Fuvị ẳ TðvÞðuðvÞÞ ð9Þ a:s: and T(v) is a normal operator a.s Without the loss of generality assume that T(v) is a normal operator for every v By Theorem 3.4, there is a spectral measure U(v) defined on the Borel subsets of the spectrum s(T(v)) of T(v) Define spectral measure E(v) defined on a Borel subset M of C by EvịMị ẳ UvịM > s Tvịịị: Then Tvị ẳ zUvịdz ẳ s ðTðvÞÞ ð zEðvÞdz: s ðTðvÞÞ Next we shall show that a family {E(v), v [ V} defines a random spectral measure For each n put Dn ¼ {v : kTvịk , n}; Bn ẳ {z [ C : jzj # n} From the inequality r(T(v)) # kT(v)k, where r(T(v)) is the spectral radius of a operator T(v) it follows that v [ Dn implies s(T(v)) , Bn Hence for each v [ Dn Tvịx ẳ zEvịdzịx ẳ zEvịdzịx: s Tvịị Step Let Pzị ẳ P k ak z Bn h be a polynomial If ð PðzÞEðvÞðdzÞ; PTvịị ẳ s Tvịị then the mapping v 7! P(T(v))x from Dn into H is measurable Indeed, ð X PðzÞEðvÞðdzÞx ¼ ak TðvÞkx: PðEðvÞÞx ¼ s ðTðvÞÞ k For each H-valued r.v.’s u, v ! T(v)u(v) is a H-valued r.v.’s From this by induction for each k the mapping v 7! T(v)kx is measurable Hence the mapping v 7! P(T(v))x from Dn into H is measurable Step If f(z) is a continuous function defined on Bn then the mapping v 7! f(T(v))x from Dn into H is measurable By the Weierstrass theorem there exists a sequence of polynomial Pk(z) converging uniformly to f(z) on Bn Hence jPk(z)j is uniformly bounded Because Stochastics: An International Journal of Probability and Stochastic Processes 1051 s(T(v)) , Bn for v [ Dn by Lemma p 899 in [3] we get lim Pk Tvịịx ẳ f Tvịịx; Downloaded by [University of Otago] at 00:15 11 September 2014 for each v [ Dn Hence the conclusion follows from Step Step For each of the closed set M of C and x [ H, the mapping v 7! E(v)(M)x from V into H is measurable Indeed for each n let M n ¼ {s : dðs; MÞ $ 1=n then Mn is closed and M > Mn ¼ B By Urysohn’s lemma there exists a continuous function fn such that fn(z) ¼ for z [ M and fn(z) ¼ for z [ Mn and # f(z) # If z Ó M then there is n0 such that dðz; MÞ $ 1=n0 $ 1=n for n $ n0 Hence fn(z) ¼ for n $ n0 Consequently lim fn(z) ¼ 1M(z) Since the mapping M ! E(v)(M)x is a H-valued measure by Theorem p 56 in [1] we get for each v [ Dn f n zịEvịdzịx ẳ lim n s ðTðvÞ ð 1M ðsÞEðvÞðdzÞx; s ðTðvÞ i.e limn f n Tvịịx ẳ EvịMịx; ;v [ Dn From Step 2, it follows that the mapping v 7! E(v)(M)x from Dn into H is measurable But Dn is measurable and V ¼ Nịx ẳ EvịMịẵEvịNịx; EvịM < Nịx ẳ EvịMịx ỵ EvịNịx EM > Nịvịx; Evị >An ịx ẳ lim EvịAn ịx if An ị #; Evị G then jjvịkjukvị ẳ kukvị=kvkvị so kukvị $ ku0 kvị If v [ E > G c then kukðvÞ $ ¼ ku0 kðvÞk Hence kuk $ ku0 k a:s: A The proof of Theorem 4.3 Since M is a closed random linear space then u M is a closed convex random set By Lemma 4.4, there is u0 [ u M such that for each w [ u M we have ku0 k # kwk a:s Since u0 [ u M it follows that u0 ¼ u v0 ; v0 [ M! u ¼ u0 þ v0 Now we shall show that u0 ’ M Let v [ M Put G ¼ {v:v(v) – 0} and define l [ L0 by < ,u0 ;v.vị ; if v [ G; kvkvị lvị ẳ : 0; otherwise: 2 Since u0 [ u M ! u0 lv [ u M ! ku0 k # ku0 lvk a:s: Hence there is a set E of P(E) ¼ such that for v [ E > G 2 ku0 k ðvÞ # ku0 lvk vị ẳ, u0 lv; u0 lv vị ẳ 2 ku0 k l , u0 ; v ðvÞ l , v; u0 vị ỵ jlj kvk vị 2 ;v.vịj ;v.vịj ;v.j vị ẳ ku0 k vị j,ukvk j,ukvk ỵ j,ukvk 2 ðv Þ ð vÞ ðv Þ 2 ;v.ðvÞj ;v.ðvÞj ! j,ukvk # ! j,ukvk ¼0 2 ðv Þ ðv Þ ! , u0 ; v vị ẳ 0: For v [ E > G c clearly we have , u0 ; v vị ẳ Hence , u0,v ẳ a.s Hence u0 ’ M as required We show that the decomposition is unique Suppose that u ẳ u1 ỵ v1 ; v1 [ M; u1 ’ M Then u0 þ v0 ¼ u1 þ v1 ! u0 u1 ¼ v1 v0 [ M ! ðu0 ; u0 u1 ị ẳ u1 ; u0 2 u1 Þ ¼ ! ðu0 u1 ; u0 u1 ị ẳ a:s: ! ku1 u0 k ¼ a:s: Hence u1 ¼ u0 a.s which implies v1 ¼ v0 a.s A 1054 D.H Thang and N Thinh Now we come back to the proof of Theorem 4.2 At first we need the two following lemmas Lemma 4.5 Suppose that F is self-adjoint Then F is closed Proof Let (vn) [ D(F), p-lim ¼ v and p-limFvn ¼ g Then for each u [ D(F) , u; g ¼ p – lim , u; Fvn ¼ lim , Fu; ¼, Fu; v : Downloaded by [University of Otago] at 00:15 11 September 2014 This shows that v [ D(F*) and F*v ¼ g Since F ¼ F* we conclude that v [ D(F) and Fv ¼ g A Lemma 4.6 Let C : D(C) ! L0(H) be a random linear mapping If C is closed and bounded then D(C) is closed Proof Since C is bounded there is a real-valued random variable k(v) such that for each u [ D(C) kCukðvÞ # kðvÞkukðvÞ a:s: For each t, r and u [ D(C) we have PkCuk tị ẳ PkCuk t; kuk # rị ỵ PkCuk t; kuk rị # Pkvị t=rị ỵ Pkuk rị: Let (un) [ D(C) and p-lim un ¼ u Then PðkCun Cum k tị ẳ PkCun um ịk tị # Pkvị t=rị ỵ Pkun um k rị: Let n, m ! and r ! we get limn;m PkCun Cum k tị ẳ Hence there exists p-lim Cun ¼ g Since C is closed we conclude that u [ D(C) A The proof of Theorem 4.2 We have kFa uk ¼, au Fu; au Fu ẳ, ibu ỵ au Fuị; ibu ỵ au Auị 2 ẳ jbj kuk ỵ kau Fuk ỵ i , bu; au Fu 2i , au Fu; bu : Noting that , u; Fu ¼, Fu; u ¼ , u; Fu ! , u; Fu [ R a.s Hence , bu; au Fu ¼ bakuk b , u; Fu [ R a.s From this 2 2 kFa uk ¼ jbj kuk ỵ kau Fuk $ jbj kuk a:s: Since b – 0, Fau ¼ implies that u ¼ Hence Fa is injective Next we show that Fa is surjective, i.e R(Fa) ¼ L0(H) By Lemma 4.5 F is 21 closed so Fa is also closed Let F21 a : RðFa Þ ! L0 ðHÞ It is easy to see that Fa is Stochastics: An International Journal of Probability and Stochastic Processes 1055 a random linear mapping Since Fa is closed, F21 a is also closed Moreover for 2 2 2 v [ RFa ị ! v ẳ Fa u we have kvk ¼ kFa uk $ kbk kuk ¼ kbk kF21 a vk ! 21 21 kF21 a vk # kbk kvk a:s: hence Fa is bounded By Lemma 4.6 R(Fa) is closed To prove R(Fa) ¼ L0(H) by Theorem 4.2 it suffices to show that if v ’ R(Fa) then v ¼ Indeed, for every u [ L0(H) ! Fau [ R(Fa) we have ¼, Fa u; v ¼, au Fu; v ¼ a , u; v , Fu; v ¼ a , u; v , u; Fv ¼ a , u; v Fv ẳ, u; aI Fịv ẳ Downloaded by [University of Otago] at 00:15 11 September 2014 !ðaI Fịv ẳ ! v ẳ 0; since aI F ¼ Fa is injective as shown before We have shown above that F21 a : L0 ðHÞ ! L0 ðHÞ is a generalized random bounded operator By a similar argument as the case of linear deterministic mappings we have À 21 Á À Á21 À 21 Á À 21 Á 21 Fa * ¼ F*a ¼ ðaI Fị21 ẳ F21 Fa ẳ Fa Fa ị21 : a ; ðFa Þ Hence À Á À 21 Á À 21 Á À Á 21 21 ðFa Þ21 F21 Fa ¼ Fa ðFa Þ21 ¼ F21 a * ¼ ðFa Þ a * ðFa Þ ; which proves that (Fa)21 is a generalized random normal operator A Theorem 4.7 Let F : D(F) ! L0(H) be a generalized random self-adjoint operator Then there exists a random spectral measure E(v) on (R, B, H) such that & DFị ẳ u [ L0 Hị : lim n & ẳ u [ L0 Hị : ' lEvịdlịuvị exists a:s: n 2n ỵ1 ' l hEðvÞðdlÞuðvÞ; uðvÞi , a:s: : 21 Moreover, for each u [ D(F) we have ðn FuðvÞ ¼ lim lEðvÞðdlÞuðvÞ a:s: n 2n Proof Consider the random operator C ¼ (iI F)21 By Theorem 4.1, C is a generalized random normal operator Hence by Theorem 3.6, there exists a family {T(v), v [ V} of normal operators and a random spectral measure G(v) such that CðvÞ ¼ TðvÞðuðvÞÞ a:s: and for each v TðvÞuðvÞ ¼ ð z dGðvÞðdzÞuðvÞ: s ðTðvÞÞ 1056 D.H Thang and N Thinh Fix x [ H Put T(v)({0})x ẳ u(v) Then uvị ¼ Ð s ðTðvÞÞ 1{0} ðzÞGðvÞðdzÞx and for each v hÐ i zdGð v ÞðdzÞ ðzÞGð v ÞðdzÞx {0} s ðTðvÞÞ s ðTðvÞÞ hÐ i Ð zdGð v ÞðdzÞ ðzÞGð v ÞðdzÞ x {0} s ðTðvÞÞ s ðTðvÞÞ é ẳ s Tvịị z1{0} zịGvịdzịx ẳ Gvị{0}ịx ẳ uvị Tvịuvị ẳ é !Cu ẳ u ! u ẳ 0; Downloaded by [University of Otago] at 00:15 11 September 2014 since C is bijective Hence there is set Dx with P(Dx) ¼ such that G(v)({0})x ¼ for all v [ Dx Let (xn) be the dense set D ¼ > Dxn then G(v)({0})xn ¼ for all v [ D Hence Gvị{0}ị ẳ ;v [ D: Fix v [ D Consider the mapping h(z) ¼ (i z)21 Then h is a bijective onto Cn{0} Define E(v) by E(v)(M) ¼ G(v)(h(M)) From the above remark we have E(v)(C) ¼ I for almost all v, so E(v) is a random spectral measure We claims that E(v) is concentrated on R Indeed, suppose that l [ CnR Put z ¼ h(l) ! l ¼ i 1/z Since (Fl)21 is a generalized random normal operator we have (Fl)21x ¼ Rl(v)x a.s where Rl(v) is a normal operator for each v Put Bvị ẳ i lị2Rl vị ỵ i lÞ It is not hard to check that ðzI TvịịBvị ẳ BvịzI Tvịị ẳ I: Hence z ể s (T( v )) GvịhMị ẳ Next, put Hence if M , CnR ! hðMÞ > s ðTðvÞÞ ¼ B ! EðvÞðMÞ ¼ & D¼ u [ L0 Hị ẳ: lim n n ' lEvịdlịuvị exists a:s: : 2n We have to show that D ẳ DFị: Fixed v We have Tvịx ẳ zGvịdzịx ẳ s Tvịị ð ði zÞ21EðvÞðdzÞx: s ðTðvÞÞ There is n0(v) such that for each n n0(v) Tvịx ẳ 21 i zị Evịdzịx ẳ s Tvịị ẳ n 2n ði lÞ21EðvÞðdlÞx: ð ði zÞ21EðvÞðdzÞx s ðTðvÞÞ>R Stochastics: An International Journal of Probability and Stochastic Processes 1057 Hence for every v [ L0(H), n n0(v) Tvịvvị ẳ ðn ði lÞ21EðvÞðdlÞvðvÞ: 2n Let u [ D(F) There exists v [ L0(H) such that u ¼ Cv Hence for almost all v there is n0(v) such that if n n0(v) n lEvịdlịuvị ẳ n 2n Downloaded by [University of Otago] at 00:15 11 September 2014 ðn lEðvÞðdlÞCvðvÞ; 2n lEvịdlịTvịvvị ẳ n 2n lEvịdlị 2n ẳ n ! ði lÞ EðvÞðdlÞvðvÞ 21 2n ðn l EðvÞðdlÞvðvÞ: 2n i l Ð Noting that if E(v) is a spectral measure then R ðl=ði lÞÞEðvÞðdlÞx exists for each x [ H since jlj # ji lj From this, for each x [ H we have ðn lim n lEvịdlịx ẳ 2n l Evịdlịx: Ri2l Hence for almost all v n lim lEvịdlịuvị ẳ n l EðvÞðdlÞvðvÞ i l R 2n proving that u [ D Conversely, let u [ D For almost all v there is n0(v) such that if n n0(v) we have lịEvịdlịuvị é n é n ẳ 2n i lị21Evịdlị 2n i lịEvịdlịuvị én ẳ 2n Evịdlịuvị ẳ Evịẵ2n; nịuvị é n !iI Fị21 2n i lịEvịdlịuvị ẳ Evịẵ2n; nịuvị én !iI FịẵEvịẵ2n; nịuvị ẳ 2n i lịEvịdlịuvị én !FẵEvịẵ2n; nịuvị ẳ 2n lEvịdlịuvị én !Fun vị ẳ 2n lEvịdlịuvị; Tvị é n 2n i hence un vị ẳ Evịẵ2n; nịuvị [ DFị én We have limnun(v) ẳ u(v) and Fun vị ẳ 2n lEvịdlịuvị a:s Since u [ D there exists lim Fun a.s Because F is closed we conclude that u [ D(F) and Fuvị ẳ lim Fun vị ẳ lim n n 2n lEvịdlịuvị a:s: 1058 D.H Thang and N Thinh Finally, since for each spectral measure E ð n 2 ð n    lEðdlÞx ¼ l , EðdlÞx; x ;   m m we get that & ' ðn lEðvÞðdlÞuðvÞ exists a:s: u [ L0 Hị : lim n & ẳ u [ L0 Hị : 2n ỵ1 ' l kEðvÞðdlÞuðvÞ; uðvÞl , a:s: : Downloaded by [University of Otago] at 00:15 11 September 2014 21 A Acknowledgement This study was supported by National Foundation for Science and Technology Development (NAFOSTED) References [1] J Diestel and J.J Uhl, Vector Measures, AMS, Providence, RI, 1977 [2] A.A Dorogovstev, On application of Gaussian random operator to random elements, Teor Veroyatnost Primenen 30 (1986), pp 812– 814 [3] N Dunford and J.T Schwarts, Linear Operator Part II, Interscience Publishers, New York, 1963 [4] H.W Engl, M.Z Nashed, and M Zuhair, Generalized inverses of random linear operators in Banach spaces, J Math Anal Appl 83(2) (1981), pp 582– 610 [5] M.Z Nashed and H.W Engl, Random generalized inverses and approximate solution of random equations, in Approximate Solution of Random Equations, A.T Bharucha-Reid, ed., Elsevier/North-Holland, New York/Amsterdam, 1979, pp 149– 210 [6] D Nualard and E Pardoux, Stochastic calculus with anticipating integral, Prob Theory Relat Fields 78 (1988), pp 535– 581 [7] N Shahzad, Random fixed points of K-set- and pseudo-contractive random maps, Nonlinear Anal 57(2) (2004), pp 173– 181 [8] N Shahzad, Random fixed points of pseudo-contractive random operators, J Math Anal Appl 296(1) (2004), pp 302– 308 [9] N Shahzad, Random fixed point results for continuous pseudo-contractive random maps, Ind J Math 50(2) (2008), pp 263–271 [10] N Shahzad and N Hussain, Deterministic and random coincidence point results for f-nonexpansive maps, J Math Anal Appl 323 (2006), pp 1038– 1046 [11] A.V Skorokhod, Random Linear Operators, Reidel Publishing Company, Dordrecht, 1984 [12] V.S Sunder, Functional Analysis: Spectral Theory, Birkhauser Verlag, Boston/Berlin, 1998 [13] K.K Tan and X.Z Yuan, On deterministic and random fixed points, Proc Amer Math Soc 119 (1993), pp 849– 856 [14] D.H Thang, The adjoint and the composition of random operators on a Hilbert space, Stochas Int J Prob Stochas Process (formerly Stochastic and Stochastic Reports) 54 (1995), pp 53 –73 [15] D.H Thang, Series and spectral representations of random stable mappings, Stochas Int J Prob Stochas Process (formerly Stochastic and Stochastic Reports) 64 (1998), pp 33 – 49 [16] D.H Thang, Transforming random operators into random bounded operators, Random Oper Stochastic Equations 16 (2008), pp 293– 302 [17] D.H Thang and T.N Anh, On random equations and applications to random fixed point theorems, Random Oper Stochastic Equations 18 (2010), pp 199–212 Stochastics: An International Journal of Probability and Stochastic Processes 1059 Downloaded by [University of Otago] at 00:15 11 September 2014 [18] D.H Thang and T.M Cuong, Some procedures for extending random operators, Random Oper Stochastic Equations 17 (2009), pp 359– 380 [19] D.H Thang and N Thinh, Random bounded operators and their extension, Kyushu J Math 58 (2004), pp 257– 276 [20] R.U Verma, Stochastic approximation-solvability of linear random equations involving numerical ranges, J Appl Math Stoch Anal 10 (1997), pp 47 – 55  ... paper, we are concerned with generalized random linear operators on a separable Hilbert space Generalized random linear bounded operators, generalized random linear normal operators and generalized. .. generalized random linear self-adjoint operators are defined and investigated The spectral theorems for generalized random linear normal operators and generalized random linear self-adjoint operators are... obtained Keywords: random operator; generalized bounded random operator; generalized random normal operators; generalized random self-adjoint operators; random spectral measure 2000 Mathematics