Kjeld Laursen

Introduction to Banach Algebras, Operators, and Harmonic Analysis

A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Introduction to Banach algebras, operators, and harmonic analysis / H. Garth Dales. .. [et al.]. p. cm. – (London Mathematical Society student texts ; 57) Includes bibliographical references and index.

Essential spectra through local spectral theory

Based on a nice observation of Eschmeier, this is a study of the use of local spectral theory in investigations of the semi-Fredholm spectrum of a continuous linear operator. We also examine the retention of the semiFredholm spectrum under weak intertwining relations; it is shown, inter alias, that if two decomposable operators are intertwined asymptotically by a quasiaffinity then they have identical semi-Fredholm spectra. The results are applied to multipliers on commutative semisimple Banach algebras.

Multipliers with closed range on regular commutative Banach algebras

Conditions equivalent with closure of the range of a multiplier T, defined on a commutative semisimple Banach algebra A, are studied. A main result is that if A is regular then T2A is closed if and only if T is a product of an idempotent and an invertible. This has as a consequence that if A is also Tauberian then a multiplier with closed range is invective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.

Automatic continuity of intertwining linear operators on Banach spaces

We extend the automatic continuity theory for linear operators θ:X→Y which intertwine two given bounded linear operatorsT∈LX andS∈LY on Banach spacesX andY, respectively. This is done both by relaxing the intertwining conditionSθ=θT and by enlarging the classes of operatorsT, resp.S, well beyond the decomposable operators. Among the operatorsS captured by these extensions are multipliers on commutative semi-simple Banach algebras.

Operators with finite chain length and the ergodic theorem

With a technical assumption (E-k), which is a relaxed version of the condition Tn/n O 0, n -n o0, where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition (X = (I T)kX D ker(I T)k), of closedness of (I T)kX, and of quasi-Fredholmness of I T. One consequence, still assuming (E-k), is that I T is semi-Fredholm if and only if I T is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras. 1. OPERATORS WITH FINITE CHAINS We begin with a few algebraic observations. Let S be a linear operator on the vector space X. If there is an integer n for which SnX = Sn+1X, then we say that S has finite descent and the smallest integer d(S) for which this equality occurs is called the descent of S. If there is an integer m for which ker Sm = kerSm+ , then S is said to have finite ascent and the smallest integer a(S) for which this equality occurs is called the ascent of S. If both a(S) and d(S) are finite, then they are equal [3], 38.3; we say that S is chain-finite and that its chain length is this common minimal value. Moreover ([3], 38.4), in this case there is a decomposition of the vector space

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.

Multipliers and local spectral theory

1. Brief introduction to multipliers. Multipliers are operators on Banach algebras with immediate appeal. Some of this appeal comes from the very properties that define them, emulating, as they do, the basic operation of “multiplication” in an algebra. Some of it has to do with the structure results they naturally give rise to. By explaining the situation in group algebras I can probably make it clear what I mean: The context in which we shall be working will be that of a commutative Banach algebra A. Any element a ∈ A gives rise to a multiplication operator Ta : A → A, defined by Tab := ab for all b ∈ A. Clearly each such Ta is a continuous linear operator. It is also obvious that because A is commutative Ta will commute with every other multiplication operator. Might this characterize multiplication operators? In other words, if T is a linear operator on A for which T (ab) = T (a)b for all a, b ∈ A, will T itself have to be a multiplication operator? If A has a unit 1, the answer is easily yes, because Ta = T (1a) = T (1)a, so that T = TT (1). However, as Wendel [W] and Helson [H] independently showed, around 1952, if we turn to the group algebra A := L(G), where G is a locally compact abelian group, the requirement on our sought-after operators, that of commuting with all multiplication operators, leads to a much larger class of operators: these operators turn out to coincide with the set of convolution operators Tμ, defined by the formula Tμa := μ∗a for all a ∈ L(G), as μ ranges over the set of complex Borel measures on G.

Maximal two-sided ideals in tensor products of Banach algebras

In [1 ] and [2] a construction of a bijection M3+->M1 X M2 is given, where MZi is the set of maximal modular (two-sided) ideals in the Banach algebraAi (i=1, 2, 3) and where A3=Al0,A2 is the greatest cross-norm tensor product of A1 and A2. In a recent correction [3] it is shown that there is indeed a closed 1-1 mapping MlXM2--+M3 when hull-kernel topologies are used. However, it is an open question when this mapping is surjective. In this note we show that the mapping is onto when one of the Banach algebras A1 and A2 is commutative. Also we give a correct proof of a theorem in [5], the original proof depended on [2]. The methods employed are adaptions of those in [6]. Suppose A1 and A2 are Banach algebras and suppose A1 is commutative. Let Sti be the set of maximal modular (two-sided) ideals. Each hEz-Tl is a continuous C-valued homomorphism and induces a homomorphismn

Introduction to Banach Algebras, Operators, and Harmonic Analysis: Harmonic analysis and amenability

A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Introduction to Banach algebras, operators, and harmonic analysis / H. Garth Dales. .. [et al.]. p. cm. – (London Mathematical Society student texts ; 57) Includes bibliographical references and index.

Introduction to Banach Algebras, Operators, and Harmonic Analysis: Banach algebras

A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Introduction to Banach algebras, operators, and harmonic analysis / H. Garth Dales. .. [et al.]. p. cm. – (London Mathematical Society student texts ; 57) Includes bibliographical references and index.