The Essential Spectrum of the Essentially Isometric Operator

Abstract

Let T be a contraction on a complex, separable, infinite dimensional Hilbert space and let sigma(T) (resp. 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\sigma T(T) not equal phi, then for every f from the disc-algebra sigma(e)( f(T)) = f( sigma(e)(T)), where D is the open unit disc. In addition, if T lies in the class C-0. boolean OR C-.0, then sigma(e)( f(T)) = f( sigma(T) boolean AND Gamma), where Gamma is the unit circle. Some related problems are also discussed.