Int. J. Pure Appl. Sci. Technol., 2(2) (2011), pp. 46-53

International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in

Research Paper

On Dvoretsky's Theorem and Norms of Elementary On Dvoretsky's Theorem and Norms of Elementary On Dvoretsky's Theorem and Norms of Elementary On Dvoretsky's Theorem and Norms of Elementary OperatorsOperatorsOperatorsOperators N. B. Okelo 1,*

1 School of Biological and Physical Sciences Bondo University College, Box 210, Bondo, Kenya

* Corresponding author, e-mail: bnyaare@yahoo.com

(Received: 19-11-2010; Accepted: 29-11-2010)

Abstract:Let H be a complex Hilbert space and B(H) the algebra of bounded linear operators on H. An operator T : B(H) B(H) is an elementary operator if T has a representation T (x) =∑n i=1 aixbi where ai, bi are fixed in B(H). In this paper we study the structural properties of elementary operators. Specifically, we utilize the Dvoretsky's theorem and its application in establishing the norm of a symmetrized two-sided multiplication operator acting on a C*-algebra B(H). We also include results on the norms of matricial operators.

Keywords: Norms, Elementary operators, C*-algebra.

1. Introduction

Properties of elementary operators have been investigated in the past two decades and there are many excellent surveys and expositions on them. Elementary operators on C*-algebras were extensively examined by Mathieu [4]. Curto [2] gave an exhaustive overview of spectral properties of elementary op- erators, Fialkow [2] comprehensively discussed their structural properties (with an emphasis on Hilbert space aspects and methods), and Bhatia and Rosenthal [2] dealt with their applications to operator equations and linear algebra. Mathieu [5] surveyed some recent topics in the computation of the norm of elementary operators, and elementary operators on the Calkin algebra. In [7], Okelo, Agure and Ambogo characterized the norm-attainable operators. Through all these studies, it has emerged that for general operators, a full description of their properties is rather intricate since these are often intimately interwoven with the structure of the underlying algebras. Therefore, there is no known general formula describing the norm of an arbitrary elementary operator (see [1-14] and references therein). In this paper we determine the norms of elementary operators but in a

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