PROCEEDINGS OF THE AMERICAN MATHEMATICAL Volume J7, Number 1, ttfj ".' I : ERGODIC THEOREMS OF WEAK MIXING TYPE LEE K. JONES AND MICHAEL LIN» Abstract. Given a linear contraction T on a Banach space X and x G X, the convergence Vx'ex'N-*? Kx*,T<x}\^^0 i=\ is shown to be equivalent to the convergence sup AT"1 2 «Jc*, r*>x>| -» 0 lU'iKi y-i for every subsequence with fc// bounded. A sufficient condition is that, for some {/i,}, T"'x -* 0 weakly. The following theorem gives a characterization of weak mixing for operators in general Banach spaces, generalizing the well-known results for measure preserving transformations. For more details see [2]. Theorem. Let T be a linear operator on a Banach space with supn>0||T"|| < oo, and let x G X. Then the following conditions are equivalent. if) For every x* G X* we have lim^^ A-1 2JIi \(x*,TJx}\ = 0. (a) suF> n-x 2 KxT.TJxX-^O. ||x*||<1 J-' (iii) For every subsequence [kA with positive lower density (i.e., sup kJj < oo) wehaveXimN^\\N-x 2^i Tk'x\ = 0. (iv) For every subsequence [kj] with positive lower density, sup N~x 2 \<x*,TkJx)\-R=^0. \\x*\\<\ 7=1 Proof, (i) -» (ii): We first assume ||T|| < X. Let B = {x* G X* : \\x*\\ < 1). B is a compact Hausdorff space in the w*-topology, and T*iB) G B. Since T* is w*-continuous on B, the operator Afix*) =/(T*x*) is a contraction of C(5). Forg(x*) = |<x*,x>|, we have by (i) that A"1 2/-1 AJg —* 0 pointwise. Hence for p G C(ß) satisfying A* ¡i = 'u we have <p,g) = 0, Received by the editors March 21, 1975. AMS (MOS) subjectclassifications (1970).Primary 47A35; Secondary 28A65,60J05. Key words and phrases. Ergodic theorems, weak mixing. 1 Research supported in part by NSF grant GP 34118. © American Mathematical Society 1976 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use