Journal of the
Korean Mathematical Society
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J. Korean Math. Soc. 2005; 42(3): 529-541

Printed May 1, 2005

Copyright © The Korean Mathematical Society.

Weyl's theorems for posinormal operators

Bhagwati Prashad Duggal and Carlos Kubrusly

, Catholic University of Rio de Janeiro

Abstract

An operator $T$ belonging to the algebra $B(H)$ of bound-ed linear transformations on a Hilbert $H$ into itself is said to be $ posinormal$ if there exists a positive operator $P\in B(H)$ such that $TT^*=T^*PT$. A posinormal operator $T$ is said to be $ conditionally$ $totally$ $posinormal$ (resp., $totally$ $posinormal$), shortened to $T\in CTP$ (resp., $T\in TP$), if to each complex number $\lambda$ there corresponds a positive operator $P_{\lambda}$ such that $|(T-\lambda I)^*|^2=|P_{\lambda}^{\frac{1}{2}}(T-\lambda I)|^2$ (resp., if there exists a positive operator $P$ such that $|(T-\lambda I)^*|^2=|P^{\frac{1}{2}}(T-\lambda I)|^2$ for all $\lambda$). This paper proves Weyl's theorem type results for $TP$ and $CTP$ operators. If $A\in TP$, if $B^*\in CTP$ is $isoloid$ and if $d_{AB}\in B(B(H))$ denotes either of the $elementary$ $operators$ $\delta_{AB}(X)=AX-XB$ and $\triangle_{AB}(X)=AXB-X$, then it is proved that $d_{AB}$ satisfies Weyl's theorem and $d_{AB}^*$ satisfies $a$-Weyl's theorem.

Keywords: Weyl's theorems, single valued extension property, posinormal operators

MSC numbers: 47B47, 47A10, 47A11

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