/. Austral. Math. Soc. (Series A) 53 (1992), 9-16 Af-COHYPONORMAL POWERS OF COMPOSITION OPERATORS SATISH K. KHURANA and BABU RAM (Received 7 May 1990) Communicated by W. Moran Abstract Let Tj , i = 1, 2 be measurable transformations which define bounded composition operators C T on I? of a (T-finite measure space. Let us denote the Radon-Nikodym derivative of 1 i m o T~ with respect to m by h i ,, i = 1, 2 . The main result of this paper is that if C^ and C£ are both A/-hyponormal with h t < M 2 (h 2 o T 2 ) a.e. and h 2 < M 2 (h l o T x ) a.e., then for all positive integers m, n and p , [(C™ C^- ) p ]* is M p <m+ "' -hyponormal. As a consequence, we see that if C£ is an A/-hyponormal composition operator, then (C^)" is M n -hyponormal for all positive integers n . 1991 Mathematics subject classification (Amer. Math. Soc): primary 47 B 20, secondary 47 B 38. 1. Introduction Let (X,^2, m) be a cr-finite measure space and let T be a measurable transformation from X into itself. Let L 2 = L 2 (X,^2,m). Then the composition transformation C T is defined by C T f = f o T for every / e L 2 . If C T happens to be a bounded operator on L 2 , then we call it the composition operator induced by T. C T is a bounded linear operator on L precisely when (i) m o T~ x is absolutely continuous with respect to m and (ii) h = dm o T~ l /dm is in L°°(X, £, m). Let R{C T ) denote the © 1992 Australian Mathematical Society 0263-6115/92 $A2.00 + 0.00 9 of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700035345 Downloaded from https://www.cambridge.org/core. IP address: 104.140.183.212, on 24 Apr 2020 at 00:25:33, subject to the Cambridge Core terms