Patil Dattatraya J.

English

University of Szeged

Acta Sci. Math., 39 (1977), 135—138 t Mean Ergodic Theorem in reflexive spaces D. J. PATIL The mean ergodic theorem proved by LORCH [ 4 ] states that if T is a linear operator on a reflexive Banach space X with || TH S 1 then (1) I+T+T2 + ... + T"-1)x^Px, for each xZX, P being a projection onto the subspace {xSA'': Tx—x). BLUM and others in a series of papers [1, 2, 3] studied the question of the convergence (2) . J-(7^1 + 7 ^ + . . . + Tk„}x _ P x > where (k„) is a given subsequence of the positive integers and X is a Hilbert space. The definitive result due to these authors is that if X is a Hilbert space and || r | | S 1 then (2) holds for each x£X if for each z on the unit circle it is true that (3) -i-(z*i + z*« + ...+r*«)(-l-z) - 0. This result is the best possible in the sense that if (2) holds for each contraction T then (3) must follow. The methods used to prove these results depend heavily on the Hilbert space structure and do not apply in the case where X is not a Hilbert space. We prove below a theorem which enables us to obtain a condition on the subsequence (/c„) which is sufficient for the truth of (2) where T now acts on any reflexive Banach space. Since it involves no additional effort we have stated our theorem for a sequence of polynomials more general than the one appearing in (3). Theo rem. Let X be a reflexive Banach space, T a linear contraction on X, (Pk)T a sequence of complex polynomials and q(z)=(z—X1)...(z—Xn), A1=L, |A,| = 1, Received April 17, revised May 11, 1976. 136 D. J. Patil l^i^n; A^X.j if i^j. Suppose that (0 as fc-oo, (") SUpJpfc(r)|| < (iii) q(T)pk(T)x •— 0 as k -* x£ X. Then for each x€.X,pk(T)x—Px where P is the bounded projection onto the subspace {x£X\ Tx=x} such that the range of I—P is the closure of the range of I—T. P r o o f . In the following, for an operator S on a reflexive space B we will denote by R (S) and N(S) the closure of the range of S and the null space of S, respectively. We note the well-known result that if | | S | | s l , then (4) B = R(I-S)®N(I-S). We now claim that the following relations hold : (5) X = N(q(T))@R{q(T)), and (6) (Rq(T)), N((T-X2I)...(T-X„I)) g R(I-T). Assuming the truth of (5) and (6), we will prove the theorem. First, the relation (5) implies that pk(T)x converges for each xdX. This is so since for x£R(q(T)) and e>0, x=q(T)y+y' with | | / | |<e . By (iii) and (ii) we will then have that pk(T)x-0. If x£N(q(T)) then x = x 1 + . . .+x n with Tx—XiXi, (1 = i=«). Thus pk(T)x=pk(X1)x1 + ...+pk(X„)xn, and by the relations in (i), the sequence pk(T)x converges to xx. Next, if we also have the relation (6), then noting that N(q(T))=N(I-T)© ®N((T-X2I)...(T-XnI)) we have in view of the decomposition (4) that pk(T)x^Px where P is as in the statement of the theorem. We will now prove by induction on n that (7) X=N(I-T)®...@N(I-1„T)®Y, where (I-T)Y = ... = (I-XnT)Y=Y. This surely implies (5) and (6). Let us suppose that for n— I there exists such a Y=Yn_1. This Yn_x is necessarily invariant under T, and by (4), we have Y.-i = RV-^TlY^JeNil-^TlY^J. Now it is immediate that N(I-lnT)<gY„-1, thus N-(I-InT\Yn-1)=N(I-InT) and we only have to show that for Y„=R(/-InT\Yn_^ we have ( I - T ) Y „ = ... = = (/— lnT)Y„= Y„. The last equality is immediate, the others follow from the corresponding equalities for Yn_1} from the fact that N{I—1„T) is invariant under Mean ergodic theorem in reflexive spaces 137 T and from the boundedness of the projections defined by the decomposition of Thus the proof of (7) and therefore that of the theorem are complete. The following corollaries now follow directly from the theorem. These corollaries are stated in such a way that the conditions on the operator T and the sequence (pk) are independent of each other. For p{z) = 2anzn, set \\p\\A = 2\an\ and | |p |L=sup {|p(z)|: |z |^l}. 0 0 Coro l l a ry 1. Let X be a reflexive Banach space and T a linear contraction on X. Let (pk), q be as in the theorem and suppose that the relations (i) of the theorem hold. Suppose further that (")' SUp \\pk\\A < oo, k (iii)' || qpk\\A-0 as k-oa. Then pk(T)x-*Px (x(iX) where P is as in the theorem. Coro l l a ry 2. Let X be a reflexive Banach space and T a linear operator on X such that for every polynomial p, ||/>(T)[| = (|/>IL. Let (pk), q be as in the theorem and suppose that the relations (i) of the theorem hold. Suppose further that (ii)" sup (hi)" l l ? f t l U - 0 as Then pk(T)x-»Px (x£X) where P is as in the theorem. We now return to the problem discussed in the introduction. Let (k„) be a sub-sequence of the positive integers satisfying (3) and take p„(z)=— (zfcl + . . . +z*")„ n q(z)=zv— 1, v a positive integer. Then all the conditions except (iii)' of Corollary 1 are satisfied. The condition (iii)' will also be fulfilled if (8) lim -j- card (EN fl (EN+v)) = 1, It-*-OO J\ where EN= {kt,..., kN) and EN + v is the translate of EN by v. We can therefore: conclude that for a linear contraction J on a reflexive space X if a sequence (k„) satisfies (3) then the condition (8) is sufficient for the convergence of (2). The example in [2], p. 428 is of a sequence (kn) satisfying (3) and (8) with v=2. We note that any linear contraction T on a Hilbert space satisfies the hypothesis; (on T) of Corollary 2. However, as shown in [3], the conclusion of Corollary 2 holds under weaker hypothesis on (p„). Thus the Corollary 2 has significance only when, the reflexive space X is not a Hilbert space. 138 D. J. Patil: Mean ergodic theorem in reflexive spaces References {1] J. R. BLUM, B. EISENBERG and L. S. HAHN, Ergodic Theory and the measure of sets in the Bohr Group, Acta Sci. Math., 34 (1973), 17—24. 12] J. R. BLUM and B. EISENBERG, Generalized summing sequences and the mean ergodic theorem, Proc. Amer. Math. Soc., 42 (1974), 423^429. {3] J. R. BLUM and J. I. REICH, Mean Ergodic Theorem for families of contractions in Hilbert space, Proc. Amer. Math. Soc., to appear. 14] E. R. LORCH, Means of iterated transformations in reflexive Banach spaces, Bull. Amer. Math. Soc., 4 5 (1939) , 9 4 5 — 9 4 7 . DEPARTMENT OF MATHEMATICS T H E UNIVERSITY OF W I S C O N S I N - M I L W A U K E E MILWAUKEE, WISCONSIN 53201 

Mean ergodic theorem in reflexive spaces

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Mean ergodic theorem in reflexive spaces

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