A study of hermitian operators on banach algebras

Abstract

The concept of Hermitian Operators on Banach spaces was first introduced by I. Vidav in 1956 and their theory received fresh vitality when in 1961 G. Lumer introduced the concept of semi-inner- products on linear spaces. Moreover, when G. Lumer (1961) and F. L. Bauer (1962) extended the concepts of numerical ranges and hermitian operators from Hilbert spaces to general normed linear spaces showing the fact that the numerical range is an effective tool for relating algebraic and geometric properties of Banach algebras u and specifically of their dual spaces u*, and more interestingly, when their results appeared to be the smoothest and most satisfactory theory including a bulk of applications in different branches of Mathematics, an innovating outlook has been evolved and then this field suggests a very wide scope of doing research. Let u be a complex Banach algebra with identity e and dual space u*. The numerical range , V (u), of an element u of u is a non-void compact convex subset of the complex field C defined by V (u) = { f (u) : fEu*, f(e) = 1 = ||f|| }. An element u in u is called Hermitian if V(u) is a subset of the real field TR. If a continuous linear operator H regarded as an element of the Banach algebra, BL (u), of all continuous linear operators on u, is a hermitian element of BL (u) in the above sense, then H is called a Hermitian Operator on u. Hermitian operators are interesting objects in operator theory; and the study of their structure and norm is very fascinating because of many varied and powerful properties of these operators. Furthermore, many powerful technical results showing some closed link of hermitian operators with denivations, automorphisms, primitive ideals, analytic semigroups, approximation properties etc., have stimulated us immensely to pursue this study and investigation. OBJECTIVES OF THE STUDY : Our primary work is an investigation of the extent to which certain properties of hermitian elements of Banach algebras are passed on to the hermitian elements of their projective...