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The Essential Spectrum of the Essentially Isometric Operator
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 1 / 14 March 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 145-158
- Print publication:
- 14 March 2014
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- Article
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Let
$T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let 
$\sigma (T)\,(\text{resp}\text{.}\,{{\sigma }_{e}}(T))$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator; that is, 
${{I}_{H}}\,-\,T*T$ is compact. We show that if 
$D\backslash \sigma (T)\,\ne \,\varnothing $, then for every 
$f$ from the disc-algebra
$${{\sigma }_{e}}\left( f\left( T \right) \right)\,=\,f\left( {{\sigma }_{e}}\left( T \right) \right),$$where
$D$ is the open unit disc. In addition, if $T$ lies in the class 
${{C}_{0}}.\,\bigcup \,C{{.}_{0}}$, then
$${{\sigma }_{e}}\left( f\left( T \right) \right)\,=\,f\left( \sigma \left( T \right)\,\bigcap \,\Gamma \right),$$where
$\Gamma $ is the unit circle. Some related problems are also discussed.
