WEYL@S THEOREMS FOR POSINORMAL OPERATORS

Title & Authors
WEYL@S THEOREMS FOR POSINORMAL OPERATORS

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