On k-hyperexpansive operators

This paper considers the k-hyperexpansive Hilbert space operators T (those satisfying Sigma(0 <= p <= n) ((n)(p))T*T-p(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. (c) 2005 Elsevier Inc. All rights reserved.