Local spectral theory and spectral inclusions

Abstract

Introduction. Suppose that T and 5 are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and 5 in the sense that SA = A T, then a classical result due to Rosenblum implies that the spectra cr(T) and cr(S) must overlap, see [12]. Actually, Davis and Rosenthal [5] have shown that the surjectivity spectrum cr su (r) will meet the approximate point spectrum o" ap (S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see

SPECTRAL PROPERTIES OF k-QUASI-2-ISOMETRIC OPERATORS

“…Then p(T ) is a k-quasi-2-isometric operator for some nonconstant complex polynomial p, and hence p(T ) has SVEP. Therefore, T has SVEP by [13,Theorem 3.3.9].…”

Weyl's theorem for algebraically k-quasiclass A operators

“…Recall that semi-Fredholm operators are of Kato type [24,Theorem 4]. For more information on semi-Fredholm operators, semi-regular operators and Kato type operators, see [3,26].…”

SVEP for multipliers on a faithful commutative Banach algebra

“…Then T is an operator with SVEP (single-valued extension property) if, for every open set U ⊆ C, the only analytic function f : U → X that satisfies (T − λ)f (λ) = 0 (λ ∈ U ) is the zero function on U ( [3], Definition 1.2.9). We are concerned with the question whether all multipliers on a faithful commutative Banach algebra do have SVEP.…”

“…We are concerned with the question whether all multipliers on a faithful commutative Banach algebra do have SVEP. The question is stated as an open problem in [3] (Problem 6.2.1). In Section 3 we shall answer the question in the negative.…”