STOCHASTIC APPROXIMATION-SOLVABILITY OF LINEAR RANDOM EQUATIONS INVOLVING NUMERICAL

The random (or stochastic) approximation-solvability, based on a projection scheme, of linear random operator equations involving the theory of the numerical range of a bounded linear random operator is considered. The obtained results generalize results with regard to the deterministic approximation-solvability of linear operator equations using the Galerkin convergence method.


Introduction
The theory of random operator equations originated from a desire to develop deter- ministic operator equations that were more application-oriented, with a special desire to deal with various natural systems in applied mathematics, since the behavior of natural systems is governed by chance.As attempts were made by many scientists and mathematicians to develop and unify the theory of random equations employing concepts and methods of probability theory and functional analysis, the Prague School of probabilists under Spacek initiated a systematic study using probabilistic operator equations as models for various systems.This development was further ener- gized by the survey article by Bharucha-Reid [2] on various treatments of random equations under the framework of functional analysis.For details on random opera- tor equations, consider the work of nharucha-Reid [1-3], Hans [5], Saaty [8], and others.
Engl and Nashed [4] studying the deterministic projection schemes of the approxi- mation-solvability of linear operator equations, considered a stochastic projection 48 RAM VERMA scheme in a Hilbert space setting, and established the existence of the best-approxi- mate solutions by a selective approach.Our aim has been to apply the theory of a random numerical range to the random (or stochastic) approximation-solvability of linear random operator equations.Among the obtained results, is a generalization of a result of Zarantonello [14] regarding the numerical range of the classical type.Non- linear analogs of the results involving A-regular random operators using the random version of the Zarantonello numerical range [14] can be discovered.For more informa- tion about A-regular operators, please study [10].
Consider a complete measure space (W, F, #).Let X be a separable (real or com- plex) Hilbert space with inner product /'," }and I1" I1" Le B(X)denote the -alge- bra or Borel fields of Borel subsets of X.Let f:WX be a mapping such that f-I(B) is in F whenever B is in B(X), that is, f is a random variable in X. Giving this definition is equivalent to stating that a random variable with values in X is a Borel measurable function.An operator T: W x XX is said to be a random opera- tor if {w in W: T(w,x)is in B} is in F for all x in X and for all B in B(X).An operator T: W x XX is measurable if it is measurable with respect to the r-algebra x B(X).A random operator T is continuous at x, if for each w in W, T(w,. is continuous at x, that is, if implies T(w, xn)T(w,x for all w in W. A measurable mapping f: WX is called a random fixed point of a random operator T: W x X X if, for all w in W, T(w, f(w))= f(w).
Definition 1.1" Let T:W x XX be a linear random operator.A random (stochastic) numerical range of T, denoted N[T(w)], is defined for all w in W and u in X by NIT(w)] {{T(w, u), u}: I I u I I 1}.
N[T(w)] is a random version of the classical numerical range, and it does have properties similar to those of the classical numerical range.
Definition 1.2: (The Moore-Penrose inverse).Let T: XY be a bounded linear operator from a Hilbert space X to another Hilbert space Y.Let TIN denote the restriction of T to the orthogonal complement of the null space of T. The generalized inverse or the Moore-Penrose inverse) of T, denoted by T +, is the uni- que linear extension of {TIN }-1 SO that its domain is D R T (R) R and its null space isN =R74.

T + T-r"
If P M denotes the orthogonal projector of X onto a closed subspace M of X and if Q denotes the orthogonal projector of Y onto R, then T+T-PN and I-TT + -Q D T +.On one hand, when R r is not closed, D r+ isadenselinear manifold of Y and T + is unbounded; and on the other hand, if R T is closed, then by the open mapping theorem, T + is bounded and D Y.For more on generalized inverses, see [6].Generalized nverses of hnear operators seem to have nice applications in analysis, statistics, prediction and control theory.As most of these applications are related to the least-squares property that the generalized inverses possess in Hilbert spaces, T + is characterized by the following extremal property.
Let u be defined so that u T + y for y in R r R. Then u minimizes I I Tx y I I over x in D T and has the smallest norm among all other minimizers.
Definition 1.3: (The least-squares solution): Let T be a linear operator form a Hilbert space X into another Hilbert space Y.We call u in X a least-squares solution of the operator equation Tx y for a fixed y in Y if inf{ I I y I1 is in x) I I Tu-y II. in addition, u in X is a least-squares solution of minimal norm, it is called a best-approximate solution of Txy.
Note that when T is a bounded linear operator, Tx-y has a best-approximate solution if and only if y is in RT(R)R.Furthermore, T+y for y in D + -R T (R) R is the unique best-approximate solution of minimal norm.As a resTult, if R T is closed, a best-approximate solution exists for every y in Y; however, if R T is not closed, a best-approximate solution does not exist if the orthogonal projection of y onto R T is not in R T.
Now let us recall the random version of a best-approximate solution.For random operator T" W XY, we consider the operator equation T(., x) y, where y is a fixed element of Y; or more generally, we consider the equation T(.,x(.))y(.). (1.1) A mapping u: W---,X satisfying the relation inf{ I I y(w) [I-(w) is in x} I I T(w,u(w))-y(w)II for all w in W, is said to be a wide-sense best-approximate solution of (1.1).A wide- sense best-approximate solution which is also measurable is called a random best- approximate solution.
Next, we recall some auxiliary results crticial to the work at hand.
Lemma 1.1: [4] Let X,Y and Z be separable Hilbert spaces, and let T: W x X---Y be a continuous random operator, U: W x Z---X be a random operator, and z: W--+X be measurable.Then (i) w-+T(w,z(w)) is measurable.
Lemma 1.3: [4] Let S'W---2 x and P:W X--X be such that for all w in W, q(w) is a closed subspace of X and P(w,.) is the orthogonal projection onto q(w).
Then the following are equivalent.
(i) S is measurable.(ii) P is a continuous random operator.(iii) There exists a sequence of measurable functions Ul,U2,...:W--X, such that for all wE W, {Ul(W),U2(W),...} is an orthonormal basis of S(w) (with the understanding that some ui(w may be zero).(iv) There is a sequence of measurable functions el,e2,...:W--+X such that for all w in W, cl{span{el (W), e2(w), .})Lemma 1.4: [7, Theorem 2.2] Let (W,F,#) be a complete measure space, and let X and Y be separable Hilbert spaces.Let T be a.s. a bounded random linear operator from WxX into Y.Let T +(w) denote a.s. the generalized inverse ofT(w).Then (a) a.s.for each , with 0 < < 2/[I T(w) [[ 2, a(I-aT*(w)T(w))kT*(w)y k=0 converges to T +(w)y for each y in DT+(w); (b) T +(w) is a random linear operator from W x Y into X; (c) for each Y-valued generalized random variable y(w) RAM VERMA such that a.s.y(w) is in D T +(w)y(w) is an X-valued generalized random variable.
T + (w)' Lemma 1.5: [15, Theorem 18El: Let A:X---X be a continuous linear operator on a Hilbert space X over the field K (real or complex).Suppose that there is a constant c > 0 such that (Au, u) >_ c I I u I I 2 for all u in X.
Then, for each given f in X, the operator equation Au f (u in X) has a unique solution.

Stochastic Projection Schemes
In this section we consider stochastic projection schemes based on the deterministic projection schemes in Hilbert spaces.Let X and Y be separable Hilbert spaces.The approximation-solvability of a deterministic linear operator equation of the form Tx y (x in X, yinY)  (2.1) and corresponding approximate equations of the form where Tn: QnT, is based on a projection scheme IX 1 {Xn, Yn, Pn, Qn}.Here X n and Yn are, respectively, subspaces of X and Y; and Pn and Qn are orthogonal projectors onto X n and Yn respectively.A very frequent choice of the approximation scheme involves Yn: TXn" Definition 2.1: (Stochastic projection scheme) Let T:W x X--,Y be a linear random operator.
II 1 {Xn, Yn, Pn, Qn} is a stochastic projection scheme if Xn: W---*2 x and Yn: W---*2Y are measurable, if for all n in N and w in W, Xn(w and Yn(w) are closed subspaces of X and Y respectively, and if Pn(w) and Qn(w) are orthogonal projectors onto Xn(w and Yn(w) respectively.
Furthermore, let Tn: -Qn T. Note that if II 1 is a stochastic projection scheme and T is a bounded linear random operator, then Pn, Qn and T n are random operators.
We need to recall the following lemma [4] regarding the measurability of func- tions.
Lemma 2.1: Let II 1 {Xn, Yn, Pn, Qn} be a stochastic projection scheme, let y: W---Y be measurable, let T: W x XY be a bounded linear random operator; and let T,(w,x): Q,(w,T(w,x)). (2.3) If, for some k in N, Tk(w, ) Qk(w, y(w)) (2.4) is solvable in Xk(w for all w in W, then there is a measurable function Xk:W---X such that, for all w in W, Xk(W is in Xk(w and Tk(W, Xk(W)) Qk(W,y(w)).

Random Operator Equations
As random operator equations differ from their deterministic counterparts only in the aspect of the measurability of solutions, one general approach to establishing the mea- surability of the solutions is as follows.First of all, represent a solution by a conver- gent approximation scheme (use iterative or projectional methods), and then establish the measurability of approximations.Having done so, apply the following lemma on limits.
Lemma 3.1: Let {Xn} be sequence of measurable functions from W to X converg- ing (weakly or strongly) to x.Then x is measurable.
We remark that, when the equation considered has a nonunique solution, one may not expect measurability of all solutions even in very simple cases.Consider the following example, where a random operator equation has a nonmeasurable solution.
Example 3.1: Let T" W x R--,R be defined by T(w,x)x 2-1.Let E be a non- measurable subset of W, that is, E is not in F. Then the real-valued random varia- ble x: W-R, defined by Now, we turn our attention to the stochastic approximation-solvability of a linear random operator equation of the form T(w, x)-(w)x y(w), (3.1) where T: W X--,X is a bounded linear random operator on W x X, : W--R + is a random variable, and y: W--X is a measurable function.We let T,X: T-AI.The symbols "--*" and ,,w_.,,represent strong convergence and weak convergence respective- ly.First, we consider the result where the strong monotonicity of the operator T,X is quite restrictive.
Theorem 3.1: Let T: W X--X be an everywhere define linear random operator on W X, where X is a separable Hilbert space.Let a number ) be at a positive dis- tance d inf{lA-'1:' in NIT(w)]} from the numerical range NIT(w)] of T. Let {en} be a sequence of measurable func- tions from W to X such that, for all w in W, {en(w)} is linearly independent and complete in X.Let y: W--,X be measurable and let, for all w in W and n in N, Xn(w)" -span{el(W),...,en(w)} with orthogonal projector Pn(w).Then 1I 2 {Xn, Pn} is a stochastic projection scheme.Suppose, in addition, that Tn'-Pn(Thi), that is, Tn(w,x)" .Pn(w,T(w,x)) -,kPn(w,x).Then, for an r in q such that, for all n >_ r and for each corresponding x n in Xn(w), there exist measur- able functions X, Xr, X r _[.1,'" "W--+X such that, for all w in W, xn(w is the unique o utio to x(w), the unique solution of T(w, x)-Ax y(w).
Proof: By Lemma 1.2, II 2 {Xn, Pn} is a stochastic projection scheme.Since T is random, T.x is also random.Since d > 0, for all w in W and x E X, It follows that (T-AI)(w,z(w)) y(w) and z(w) x(w).
Finally, since Xr, X r -t-1 ''''W--X are measurable functions by Lemma 2.1, and since xn(w)x(w) for all w in W and n in N, it follows from Lemma 3.1 that the limit x(w) is measurable.This completes the proof.
We note that d and r are independent of w in Theorem 3.1, and this indepen- dence is rather restrictive; but it guarantees the existence of approximate solutions for all w in W from the index r onward, tIowever, in the next theorem we allow d and r to depend on w.As one cannot guarantee, then, the solvability of a projectional equation for all w in W for any given n, we define the functions of the form x n as best-approximate solutions of the projectional equation.Once inequality (3.3) is established for specific w, this best-approximate solution will, in fact, be a solution of the projectional equation for this w.Theorem 3.2: Let T: W x X---X be an everywhere defined linear random opera- tor.Let a real-valued random variable A:W--R + (with (w) > O) be at a positive dis- tance d(w) from the numerical range N[T(w)] of T, where d:W--.N + is a random variable.Let y: W---X be measurable, and let II 2 {xn, Pn} and T n as in Theorem 3.1.Then, for all w in W and r(w) in N and for all n>_r(w) and Xn(W in Xn(W), Furthermore, there exist measurable functions x, Xl x2, W---X with the following properties. (i) xn(w is in Xn(W for all n in N and w in W. (ii) Tn(w, Xn(W)) Pn(w,y(w)) for all w in W and n >_ r(w).(iii) For all w in W, x(w) is the unique solution of T)(w,x)y(w).(iv) For all w in W and n in N, xn(w is the best-approximate solution of Tn(w)Pn(w x) P,(w, y(w)) in X.