On operators with an absolute value condition

Abstract

Let U-fraktur sign denote the class of bounded linear Hilbert space operators with the property that

A2

≥

A

2. In this paper we show that U-fraktur sign-operators are finitely ascensive and that, for non-zero operators A and B, A ⊗ B is in U-fraktur sign if and only if A and B are in U-fraktur sign. Also, it is shown that if A is an operator such that p(A) is in U-fraktur sign for a non-trivial polynomial p, then Weyl's theorem holds for f(A), where f is a function analytic on an open neighborhood of the spectrum of A.