The Essential Spectrum of the Essentially Isometric Operator

Canadian mathematical bulletin

H. S. Mustafayev¹ 

[1] Yuzuncu Yıl University, Faculty of Science, Department of Mathematics, 65080, VAN-TURKEY

关键词: Hilbert space;    contraction;    essentially isometric operator;    (essential) spectrum;    functional calculus;   

DOI  :  10.4153/CMB-2012-016-1

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $T$ be a contraction on a complex, separable, infinite dimensionalHilbert space and let $sigma left( Tight) $ (resp. $sigma _{e}left(Tight) )$ be its spectrum (resp. essential spectrum). We assume that $T$is an essentially isometric operator, that is $I_{H}-T^{ast }T$ is compact.We show that if $Ddiagdown sigma left( Tight) eq emptyset ,$ thenfor every $f$ from the disc-algebra, egin{equation*}sigma _{e}left( fleft( Tight) ight) =fleft( sigma _{e}left(Tight) ight) ,end{equation*}where $D$ is the open unit disc. In addition, if $T$ lies in the class $ C_{0cdot }cup C_{cdot 0},$ then egin{equation*}sigma _{e}left( fleft( Tight) ight) =fleft( sigma left( Tight)cap Gamma ight) ,end{equation*}where $Gamma $ is the unit circle. Some related problems are also discussed.

【 授权许可】

Unknown