Spatial numerical ranges of elements of Banach algebras

A. K. Gaur,T. Husain

doi:10.1155/s0161171289000785

Abstract

In this paper, the notion of spatial numerical range of elements of Banach algebras without identity is studied. Specifically, the relationship between spatial numerical ranges, numerical ranges and spectra is investigated. Among other results, it is shown that the closure of the spatial numerical range of an element of a Banach algebra without Identity but wlth regular norm is exactly its numerical range as an element of the unitized algebra. Futhermore, the closure of the spatial numerical range of a hermitian element coincides with the convex hull of its spectrum. In particular, spatial numerical ranges of the elements of the Banach algebraC0(X)are described.

Highlights

Let A be of the complex numbers, is called the numerical range of x, where A’ is the dual of A
A with respect to x, we see that VA(a) U Vx(A,a): x e S(A)} which is a bounded subset of the lall. complex numbers bounded by for any aeATEB(A) we have: VA(a) V(B(A), Ta). This Justifies the usage of the notion of spatial numerical range as defined by BonsaIl and Duncan [I]
When X Is a compact Hausdorff space, Stampfll and Williams [4] have shown that the numerical range of each element f of the Banach algebra C(X) of all continuous complex-valued functions on X Is the closed convex hull of its range, f(X)

Summary

INTRODUCTION

Let A be of the complex numbers, is called the numerical range of x, where A’ is the dual of A, (see [I], [2]). This definition is dependent on the identity of the Banach algebra. Wish to adopt this notion (Definition 2.1.) to define spatial numerical range for an element of an arbitrary Banach algebra without identity. Complex numbers bounded by for any aeATEB(A) (defined in the introduction) we have: VA(a) V(B(A), Ta) This Justifies the usage of the notion of spatial numerical range as defined by BonsaIl and Duncan [I]. THEOREM 2.3 Let A be a Banach algebra without identity. The corollary follows from [2]

SPATIAL NUMERICAL RANGES OF HERMITIAN ELEMENTS

Then there