V. G. Miller

Spectral subspaces of subscalar and related operators

For a bounded linear operator T E L(X) on a complex Banach space X and a closed subset F of the complex plane C, this note deals with algebraic representations of the corresponding analytic spectral subspace XT(F) from local spectral theory. If T is the restriction of a generalized scalar operator to a closed invariant subspace, then it is shown that X T (F) = E T (F) = ∩ λ ¬∈ F (λ-T) p X for all sufficiently large integers p, where E T (F) denotes the largest linear subspace Y of X for which (λ-T) Y = Y for all λ∈C \ F. Moreover, for a wide class of operators T that satisfy growth conditions of polynomial or Beurling type, it is shown that X T (F) is closed and equal to E T (F).

Local Spectral Theory for Multipliers and Convolution Operators

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].

Local spectral properties of commutators

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, Foias, 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.

On operators with closed analytic core

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.