Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2004; 41(4): 617-627

Printed July 1, 2004

Copyright © The Korean Mathematical Society.

On operators with an absolute value condition

In Ho Jeon and B. P. Duggal

Ewha Women's University, UAEU

Abstract

Let $\frak A$ denote the class of bounded linear Hilbert space operators with the property that $|A^2|\ge |A|^2$. In this paper we show that $\frak A$-operators are finitely ascensive and that, for non-zero operators $A$ and $B$, $A\otimes B$ is in $\frak A$ if and only if $A$ and $B$ are in $\frak A$. Also, it is shown that if $A$ is an operator such that $p(A)$ is in $\frak A$ 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 $A$.

Keywords: class A operator, polynomially class A operator, tensor product, Weyl's theorem

MSC numbers: 47B20