Multipliers and local spectral theory

Laursen, Kjeld

doi:10.4064/-30-1-223-236

Banach Center Publ.

1994

DOI: 10.4064/-30-1-223-236

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Multipliers and local spectral theory

Kjeld Laursen¹

Abstract: The paper is in final form and no version of it will be published elsewhere. A portion of the material of this paper was presented to the Banach Centennial Semester in May 1992. In the preparation of the final version I have benefited greatly from many comments and suggestions from Michael M. Neumann and Jaroslav Zemánek. I want to express my appreciation of this help.

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Cited by 2 publications

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Simple Multipliers on Banach Modules

Bračič¹

2003

Glasgow Math. J.

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Abstract. In this paper we introduce simple multipliers, a special subclass of multipliers on a Banach module. We show that, from a local spectral point of view, these multipliers behave like multipliers on a commutative Banach algebra. Our definition of simple multipliers relies on the notion of point multipliers. These multipliers were studied earlier. However our approach gives new insight into this topic and therefore could be of some interest by itself.2000 Mathematics Subject Classification. Primary 46H25; Secondary 47B40. The goal of the present paper is to discuss the decomposability of multipliers between Banach modules. Since, for a general multiplier, we cannot say very muchfor instance, if A = ‫,ރ‬ then B A (X) = B(X) -it is quite clear that we have to confine ourselves to a special subclass of multipliers. In Section 4 we introduce simple multipliers on a Banach left module, which seem to be the suitable environment for our questions. Section 5 is devoted to the local spectral theory of simple multipliers. For instance, we extend to the class of simple multipliers on a left Banach module some results that are proven in [8], [16] and [23] only for algebras. Our definition of simple multipliers relies on the notion of point multipliers, which are studied in Sections 2 and 3. We believe that the content of these two sections are of some interest by itself. Introduction. A mapping T on a Banach algebraIt is assumed that the reader is familiar with the concept of Banach modules. We refer to [1, 18, 24], for details.

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Simple Multipliers on Banach Modules

Bračič¹

2003

Glasgow Math. J.

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Multipliers with closed range on regular commutative Banach algebras

Aiena

Laursen

1994

Proc. Amer. Math. Soc.

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Abstract. 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 injective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.

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Multipliers and local spectral theory

Cited by 2 publications

References 25 publications

Simple Multipliers on Banach Modules

Simple Multipliers on Banach Modules

Multipliers with closed range on regular commutative Banach algebras

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