On k-hyperexpansive operators

This paper considers the k-hyperexpansive Hilbert space operators T (those satisfying 0 p n np T ∗pT p 0, 1 n k) and the k-expansive operators (those satisfying the above inequality merely for n = k). It is known that if T is k-hyperexpansive then so is any power of T ; we prove the analogous result for T assumed merely k-expansive. Turning to weighted shift operators, we give a characterization of k-expansive weighted shifts, and produce examples showing the k-expansive classes are distinct. For a weighted shift W that is k-expansive for all k (that is, completely hyperexpansive) we obtain results for k-hyperexpansivity of back step extensions of W. In addition, we discuss the completely hyperexpansive completion problem which is parallel to Stampfli’s subnormal completion problem. © 2005 Elsevier Inc. All rights reserved.