The Essential Spectrum of the Essentially Isometric Operator

Mustafayev, H. S.

Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 145-158

Résumé

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.

Détail

DOI : 10.4153/CMB-2012-016-1

Mots-clés : 47A10, 47A53, 47A60, 47B07, Hilbert space, contraction, essentially isometric operator, (essential) spectrum, functionalcalculus