Bounded Generalized Random Linear Operators

VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 Bounded Generalized Random Linear Operators Nguyen Thinh * Department of Mathematics VNU University of Science 334 Nguyen Trai Hanoi Vietnam Received 3 May 27 Revised 3 June 27; Accepted 5 September 27 Abstract: In this paper we are concerned with bounded generalized random linear operators. It is shown that each bounded generalized random linear operator can be seen as a set-valued random variable. The properties of some special bounded generalized random linear operators and the random resolvent set of generalized random linear operators are investigated. Keywords: Random linear operator random bounded linear operator generalized random linear mapping bounded generalized random linear operator set-value random variable random resolvent set random regular value. AMS Subject Classification 2: Primary 6H25; Secondary 6H5 6B 45R5.. Introduction Let be separable Banach spaces and ( ƑP) be a probability space. By a random mapping (or a random operator) from to we mean a rule that assigns to each element x a -valued random variable (r.v.). Mathematically a random mapping defined from to is simply a mapping A: L L stands for the space of all -valued random variables (r.v. s). If where S = [ab] is a interval of the real line then F t t a b F is said to be a -valued stochastic process. The interest in random mappings has been arouse not only for its own right as a random generalization of deterministic mappings as well as a natural generalization of stochastic processes but also for their widespread applications in other areas. Research in theory of random mappings has been carried out in many directions including random linear mappings which provide a framework of stochastic integral infinite random matrix (see e.g. [2 5 ] [4-9]) random fixed points of random operators semi groups of random operators and random operator equations (e.g. [3] [6] []-[6] and references therein). Tel.: 84-983888 Email: thinhj@gmail.com https//doi.org/.2573/2588-24/vnumap.49 95

96 N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 Under the original definition a random mapping F : S L is a rule that transforms each deterministic input x S into a random output Fx. Taking into account that inputs may be also : S L where S is a random a generalized random mapping is defined as a mapping subset of L. A random mapping L which is linear and bounded is called a random linear bounded operator (see [ 4 5 7]) and a generalized random mapping L L which is strongly linear and bounded is called a bounded generalized random linear operator. In Section 2 the one-one corresponding between random linear bounded operators and bounded generalized random linear operators is discussed. It is shown that every random linear bounded operator from to admits a unique extension which is a bounded generalized random L L : L L is a bounded linear operator from. Reversely if generalized random linear operator then the mapping $ restricted on will be a random linear bounded operator. By [7] the random mapping A : L is a random linear bounded operator if and only if there exists almost surely uniquely a mapping T from to set of all linear bounded operators from Ax T x a.s. Thus a random linear bounded to such that for each x we have operator from can be regarded as a family T indexed by satisfying for each x the T x is measurable. mapping Section 3 is concerned with a different form of random linear bounded operators and bounded generalized random linear operators. Theorem 3. 3.2 show that a random linear bounded operator (or a bounded generalized random linear operator) from to can be regarded as a measurable setvalued mapping from Q to set of all linear bounded operators from to. As an application the properties of some special bounded generalized random linear operators and random resolvent set of generalized random linear operators are investigated (Theorem 3.3 3.4 3.6). 2. Random bounded operators and bounded generalized random linear operators In this section some definitions and typical results on random bounded operator bounded generalized random linear operator are listed and discussed. For more details we refer the reader to [4 7 8 9]. Throughout this paper ( ƑP) is a complete probability space are separable Banach spaces. A measurable mapping from ( Ƒ) into (ß()) is called a -valued random variable. The set of all -valued random variables is denoted by L random variables which are equal almost surely. L L. We do not distinguish two - is a metric space under the metric of convergence in probability. If a sequence (u n ) in converges to u in probability then we write p lim n

N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 97 Definition 2.. ([4 7]) - By a random mapping A from to we mean a mapping from into L - By a random linear mapping A from to we mean a mapping from into L for every 2 x x2 we have A x x A x A x 2 2 2 2 a.s. - A random linear mapping A: L satisfying is said to be a random opeartor if it is continuous and is said to be bounded (or random bounded operator) if there exists a real-valued random variable k such that for each x Ax k x a.s. () Noting that the exceptional set in () may depend on x.a random bounded operator is a random operator but in general a random operator needs not be bounded. For examples of random operators random bounded operators and unbounded random bounded operators we refer to [4 7 9]. It is easy to prove the following Theorem which is a little bit more general than a result in [7]. Theorem 2.2. A random mapping A : L there is an almost surely uniquely mapping T : L is a random bounded operator if and only if such that for each x E Ax T x a.s. (2) For the sake of convenience we denote the a.s. uniquely determined mapping T(w) in the Theorem above by [A](w). So for each x we have A Ax x a.s. Definition 2.3.. Let be a subset of L defined on with values in we mean a mapping : L of is denoted by Ɗ. 2. A subset L. By a generalized random mapping. As usual the domain is said to be a random linear subspace if for every u u2 M 2 L we have u 2u2. 3. Let L (g.r.l.o) defined on with values in we mean a strongly linear mapping : L if u u L then 2 2 be a random linear subspace. By a generalized random linear operator 2 2 2 2 2 i.e. u u u u (3) 4. A generalized random linear operator : L L a.s. u L. exist a random variable k s.t. u k u is said to be bounded if there

98 N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 It should be noted that the notion of g.r.l.o. has been introduced in [8] where are Hilbert spaces. If : L L operator : L A: L : L L each u L u Au is a bounded generalized random linear operator then the restricted is a random bounded linear operator. Reversely if is a random bounded linear operator then by [7] A admits uniquely an extention which is a bounded generalized random linear operator and moreover for a.s. Combining this with Theorem 2.2 it is easy to have the following Theorem. Theorem 2.4. A generalized random mapping : L L is a bounded g.r.l.o. if and only if there is an almost surely uniquely mapping T : L() such that for each ul u T u a.s. (4) For the sake of convenience we denote the a.s. uniquely determined mapping T( ) as in the ul we have Theorem above by [ ]( ). So for each u u a.s. The set-valued analysis is used as main technique in proofs in the next chapter. Next we list some notions and typical results relating to set-valued r.v. to be used later on. Let (E d) be a separable metric space. Denote 2 E the collection of all subsets of E ß(E) the set of all Borel measurable sets in (E d). A mapping F : 2 E is called a set-valued function. A r.v. f : f F E is said to be a measurable selections of F if Definition 2.5. ([7] Definition.) Let F : Ω 2 E \ be a set-valued function. (a) F is said to be strongly measurable if for every C E closed we have F (C ) = {ω S : F (ω) C = } Ƒ (b) F is said to be measurable or set-valed random variable if for every C E open we have F (C ) = {ω T : F (ω) C = } Ƒ (c) If F is measurable then it is called a set-valued random variable. (d) F is said to be graph measurable if Gr(F ) = {[ω x] T E : x F (ω)} Ƒ ß(E). Theorem 2.6. [7] (Theorem.35 Proposition 2.3) Let F : Ω 2 \ s.t. F(w) is closed set for every ω Ω. Then the following statements are all equivalent.

N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 99. For every C ß(E) F (C) Ω; 2. F is strongly measurable; 3. F is measurable; 4. For every x E the mapping ω d(x F (ω)) is measurable; 5. F is graph measurable. f of measurable selections of F s.t. for every 6. There exists a sequence nn w F w fn wn. Such a sequence (f n ) is called dense measurable selections of F. Given a set-valued function F : Ω 2 E \{ } we denote SF = {f L (Ω E) : f (ω) F (ω) a.s.}. Theorem 2.7. ([7] implied from Theorem 3.9) Let F G : Ω 2E \ are closed set-valued r.v. s. If S F = S G then F( ) = G( ) a.s. 3. Main results Let : L L be a bounded g.r.l.o. It is known that in general the mapping [Φ] : Ω L( ) ω [Φ](ω) is not Borel-measurable. However the following Theorem shows that this mapping is measurable in term of set-valued measurable. Let A : D(A) be an arbitrary mapping. Denote Gr(A) ={[x y] : x Ɗ(A)} 2. Note that x is a separable Banach space under the norm x y x y x y Theorem 3.. Let Φ : L set of all mapping from Φ to (needs not be measurable). Then Φ is a bounded g.r.l.o. if and only if there is an almost surely uniquely mapping T : Ω L( ul ) such that for each Φu(ω) = T (ω)u(ω) a.s. (5) and the mapping : Gr T Gr T 2 is a closed set-valued r.v. Because of the corresponding between random bounded operator and bounded g.r.l.o. the Theorem above is equivalent to the Theorem below. Theorem 3.2. A : set of all mapping from Ω to (needs not be measurable). Then A is a random bounded operator if and only if there is an almost surely uniquely mapping T : Ω L( ) such that for each x

N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 Ax(ω) = T (ω)x a.s. (6) and the mapping : Gr T Gr T 2 is a closed set-valued r.v. Proof. Sufficiency condition: Let A be a random bounded operator and (x n) be a condense sequence of. Put T = [A]. We construct a sequence of mapping ( n ) from Ω to as follows: w x T x n n n Axnis measurable so n is also measurable. Now we will check that for each the sequence ( n (u)) n is a dense set in Gr(T( )). Indeed let be an arbitrary element of Gr(T( )) then there exists x such that = (xt( )x). Since (x n ) is dense in then there exists a subsequence x converging to x. The boundedness of T(u) implies that the sequence n' converges to T x and thus n' xn ' T xn ' converges to x T T x n' x. So ( n ) are dense measurable selections of Gr(T). By Theorem 2.6 the closed set-valued mapping Gr(T) is measurable. Necessity condition: assume T : Ω L( ) is mapping such that (6) holds and the closed setvalued mapping Gr(T) is measurable. By Theorem 2.2 it remains to prove Ax is measurable. By Theorem 2.6 there exists a measurable sequence ( n ) such that for every Ω n (ω) = (u n (ω) T (ω)u n (ω)) and ( n (ω)) is dense in Gr(T( )). The measurability of n leads to the measurability of u n and Au n = T ( )u n ( ). Let x and fix ω Ω. There exist a subsequence n' of n that converges to x T x. If then un' converges to x. This implies the sequence un n' n' n' u T u is dense in for every u. Now let x and fix e we will construct a random variables v such that v(u) x e for every w and T( )v ( ) is also measurable. Indeed for each n we define a set by induction: B : u x e and n n n Cn Bn \ k Bk. It is not difficult to verify that C n are disjoint measurable sets and Cn. Let v n C un it is easy to see that v(u) x e for n every and T( )v n ( ) is measurable. From this we can construct a sequence of measurable random variables v n such that v n ( ) x /n for every and T( )v n ( ) is measurable. Combining with the boundedness of T( ) we can conclude that T ( )x is measurable. The theorem is proved completely. Theorem 3.3. Let : L L for almost surely the mapping L is injective. Proof. For each let : x : x. Observe that for each be a bounded g.r.l.o. If is an injective mapping then N x x. Put N Gr. It is

N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 obvious that the closed set-valued mapping is a graph measurable. Thus the closed set- N is graph measurable. By Theorem 2.6 there exists a sequence valued mapping un L such that for every N un Since n un un a.s. and is injective u n = a.s. So there exists a measurable set s.t. P and un Hence. In other words for almost surely the for every N for every mapping : is injective. Denote c the set of all closed linear operators : Theorem 3.4. Let : L L : L L T : such that T D T be a g.r.l.o. If is injective surjective and is a bounded g.r.l.o. then there exists an almost surely uniquely mapping T c. For almost surely u E Q the mapping : L. 2. The mapping : Gr T Gr T 2 is a closed set-valued r.v. T T is injective surjective and 3. u T ua. s. u u L : u T a. s. (7) (8) Gr vl v Gr T (9) : Proof.. The mapping : L L injective by Theorem 3.3 the mapping L is a bounded g.r.l.o. Since is is injective for almos. For each put T then T c since L obviuos that T : T T L. It is is injective surjective and and 2. By Theorem 3. the closed set-valued mapping Gr : 2

2 N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 Gr is measurable. Thus it is easy to see that the closed set-valued mapping Gr : 2 Gr Gr T transposed is also measurable. 3. It is easy to verify () (2) and (3). The a.s. uniqueness of T is implied from (3) and Theorem 2.7. Definition 3.5.. Let : L L be a g.r.l.o. is said to be a random regular value of $ if the mapping ID is injective L surjective and the mapping ID : L L the identity mapping on L is a bounded g.r.l.o. Where ID is 2. The set of all random regular values of is called random resolvent set of and is denoted by Theorem 3.6. Let : L L is not empty then there is an almost surely uniquely mapping U :. The mapping : 2 Gr T Gr U be a g.r.l.o. If the random resolvent set of c such that () 2. Is a closed set-valued r.v u L : u T U a. s. () u T u a. s. u (2) Gr v L v Gr T and we have : L T : (4) (3)

N. Thinh / VNU Journal of Science: Mathematics Physics Vol. 33 No. 3 (27) 95-4 3 Proof. Assume L L uniquely mapping U :. Put ID. Then is injective surjective and is a bounded g.r.l.o. By Theorem 3.4 there exists an almost surely U such that c. For almost surely w the mapping : L. 2. The mapping : 2 Gr U Gr U is a closed set-valued r.v. 3. u L : u U a. s. u U u a. s. u Gr v L v Gr U : U U is injective surjective and Now for each w G Q put T( ) = A(u)id - U( ) where id is the identity mapping on. It is not difficult to verify () () (2) (3) (4). Acknowledgments This work was supported by Vietnam National University under Grant QG.4.2 References [] R. Carmona and J.Lacroix Spectral theory of random Schrdinger operators Birkhuser Boston MA 99. [2] A.A. Dorogovtsev On application of gaussian random operator to random elements Theor. veroyat. i priment 3:82-84 986. [3] A.A. Dorogovtsev Semigroups of finite-dimensional random projections Lith. Math. J. 5(3):33-34 2. [4] N. Dunford and J.T. Schwarts Linear operators Part II Interscience Publishers Nework 963. [5] H.W. Engl M.Z. Nashed and M. Zuhair Generalized inverses of random linear operators in banach spaces J. Math. Anal. Appl. 83:582-6 98. [6] H.W. Engl and W. Romisch Approximate solutions of nonlinear random operator equations: Convergence in distribution Pacific J. Math. 2:55-77 985. [7] S. Hu and N.S. Papageorgiou Handbook of multivalued analysis. Vol. I Mathematics and Its Applications Kluwer Academic Publishers Dordrecht 997. [8] M. Ledoux and M. Talagrand Probability in Banach spaces Springer-Verlag 99.

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