This article centers around the relation between the spectra of two Banach space operators that are linked by some intertwining condition such as quasi-similarity. Certain conditions from local spectral theory are shown to be both necessary and sufficient for these operators to have equal spectra, approximate point spectra, or surjectivity spectra. A key role is played by a localized version of Bishop’s classical property (β) and a related closed range condition. As an application to harmonic analysis, the measures on a locally compact abelian group that avoid the Wiener-Pitt phenomenon are characterized in terms of local spectral theory.

Let x0 be a nonzero vector in C. We show that a linear map Φ : Mn(C) → Mn(C) preserves the local spectral radius at x0 if and only if there is α ∈ C of modulus one and an invertible matrix A ∈ Mn(C) such that Ax0 = x0 and Φ(T ) = αATA−1 for all T ∈Mn(C).

Let x(0) be a nonzero vector in C-n. We show that a linear map Phi : M-n (C) --> M-n(C) preserves the local spectral radius at x(0) if and only if there is a alpha is an element of C of modulus one and an invertible matrix A is an element of M-n(C) such that Ax(0) = x(0) and Phi(T) = alpha AT A(-1) for all T is an element of M-n (C).

As shown by Mbekhta [9] and [10], the analytic core and the quasi-nilpotent part of an operator play a significant role in the local spectral and Fredholm theory of operators on Banach spaces. It is a basic fact that the analytic core is closed whenever 0 is an isolated point of the spectrum. In this note, we explore the extent to which the converse is true, based on the concept of support points. Our results are exemplified in the case of decomposable operators, Riesz operators, convolution operators, and semi-shifts.

The Cesàro operator is shown to be subdecomposable on the Bergman spaces \( A^{p} (\mathbb{D}) \) for \( p \geqq 2 \), extending a result of [12] to the case that p < 4. For \( A^{2} (\mathbb{D}) \), we show that Cesàro operator is in fact subscalar, but in contrast to the situation in the Hardy space, \( C |_{A^2} \) fails to be hyponormal.

We give a su-cient condition involving local spectra for an op- erator on a separable Banach space to be hypercyclic. Similar conditions are given for supercyclicity. These spectral conditions allows us to characterize the hyponormal operators with hypercyclic adjoints and those with supercyclic ad- joints.

We give conditions such that an operator given by the Dunford-Taylor functional calculus is supercyclic or hypercyclic. Indeed, we improve [15, Theorem 1].

For a pair of continuous linear operators T and S on complex Banach spaces X and Y, respectively, this paper studies the local spectral properties of the commutator C(S, T) given by C(S, T)(A): = SA−AT for all A∈L(X, Y). Under suitable conditions on T and S, the main results provide the single valued extension property, a description of the local spectrum, and a characterization of the spectral subspaces of C(S, T), which encompasses the closedness of these subspaces. The strongest results are obtained for quotients and restrictions of decomposable operators. The theory is based on the recent characterization of such operators by Albrecht and Eschmeier and extends the classical results for decomposable operators due to Colojoară, Foiaş, and Vasilescu to considerably larger classes of operators. Counterexamples from the theory of semishifts are included to illustrate that the assumptions are appropriate. Finally, it is shown that the commutator of two super-decomposable operators is decomposable.

This note centers around the class D(G) of decomposable measures on a locally compact abelian group G. This class is a large subalgebra of the measure algebra M(G), has excellent spectral properties, and is related to a number of concepts from commutative harmonic analysis. The discussion of D(G) in Section 1 is to illustrate this point. The main features of this class are collected in Theorem 1.1, which improves recent results from [19] and includes some new properties related to the involution of M(G). We shall present a different approach, which avoids previous tools like the hull-kernel topology [19], [23] or the spectral theory of several commuting operators [2], [10]. Theorem 1.1 is an immediate consequence of the spectral theory for multipliers on Banach algebras in Section 3. The emphasis is here on multipliers with the decomposition property (δ) from [3], which characterizes the quotients of decomposable operators. We show that multipliers with property (δ) behave very nicely and coincide with the strongly decomposable multipliers under fairly mild conditions on the underlying Banach algebra. Our results on multipliers require some new results on general local spectral theory in Section 2, which should be of independent interest. In particular, some basic results on decomposable operators from [8] and [28] will be extended to the more flexible case of quotients and restrictions of decomposable operators in the spirit of [3].