GENERALIZED RIESZ POINTS FOR PERTURBATIONS OF TOEPLITZ OPERATORS

Inthisnoteweconsider “generalizedRieszpoints”forcom-pactandquasinilpotentperturbationsofToeplitzoperatorsactingontheHardyspaceoftheunitcircle. 1. IntroductionIf M is a subset of C, write isoM, accM, and ∂M for the isolated points,the accumulation points, and the boundary of M, respectively. Let X be aninfinite dimensional complex Banach space and write B(X) for the set of allbounded linear operators acting on X. We recall ([1], [5], [6]) that an operatorT ∈ B(X) is Fredholm if T(X) is closed and both T −1 (0) and X/T(X) arefinite dimensional. If T ∈ B(X) is Fredholm we can define the indexof T byindex(T) = dimT −1 (0)−dimX/T(X). An operator T ∈ B(X) is called Weylif it is Fredholm of index zero. The essential spectrum σ e (T) and the Weylspectrum ω(T) of T ∈ B(X) are defined by(1) σ e (T) = {λ ∈ C: T −λI is not Fredholm}and(2) ω(T) = {λ ∈ C: T −λI is not Weyl}.If T ∈ B(X) we write(3) π left (T) = {λ ∈ C: (T −λI) −1 (0) 6= {0}}for the set of all eigenvalues of T,(4) π left0 (T) = {λ ∈ iso σ(T) : 0 < dim(T −λI)