Bishop's property (beta), a commutativity theorem and the dynamics of class A(s, t) operators

摘要

Given a Hilbert space operator A is an element of B(H) with polar decomposition A = U vertical bar A vertical bar, the class A(s, t), 0 < s, t <= 1, consists of operators A is an element of B(H) such that vertical bar A*vertical bar(2t) <= (vertical bar A*vertical bar(t)vertical bar A vertical bar(2s)vertical bar A*vertical bar(t))(1/t+s). Every class A(s, t) operator is paranormal; prominent amongst the subclasses of A(s, t) operators are the class A(1/2, 1/2) consisting of w-hyponormal operators and the class A(1,1) consisting of (semi-quasihyponormal [16, p. 93], or) class A operators. Our aim here is threefold. We prove that A(s, t) operators satisfy: (i) Bishop's property (a), thereby providing a proof of Theorem 3.1), and (ii) a Putnam-Fuglede commutativity theorem, thereby answering a question posed in [18. Conjecture 2.4]; we prove also an extension of [3, Theorem 3.4] to prove that (iii) if an A(s, t) operator is weakly supercyclic then it is a scalar multiple of a unitary operator. (C) 2015 Elsevier Inc. All rights reserved.