International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 4, Pages 633-640

Spatial numerical ranges of elements of Banach algebras

A. K. Gaur1 and T. Husain2

1Department of Mathematics, Duquesne University, Pittsburgh 15282, PA, USA
2Department of Mathematics and Statistics, McMaster University, Ontario, Hamilton L8S 4K1, Canada

Received 30 August 1988; Revised 19 January 1989

Copyright © 1989 A. K. Gaur and T. Husain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 algebra C0(X) are described.