Mapping Theorems and the Numerical Radius

Abstract

Introduction and notationIn this paper we gather together and examine a variety of problems from the theory of numerical ranges. In § 1 we look at relations between the spectrum and the numerical range, extending results of Rota ([12]) and Hildebrandt ([9]) in Hilbert space theory. For an element a of a complex unital Banach algebra there is the well-known inequality e-1 1|a|| ^v{a) ^ \\a\\;and in § 2 we investigate similar inequalities for real unital Banach algebras; a topic initiated by Bonsall and Duncan ([2]) and developed by Lumer ([10]). The next two sections are concerned with the numerical index. Section 3 deals with relations between the index and the power inequality and also spaces with index 1, while § 4 consists of some examples showing how the index can vary under certain mappings. The numerical range of idempotents is the subject of the final section; the main results being concerned with extremal situations.Given a real or complex Banach space X we shall write X' for the dual space, X x for the closed unit ball of X and S(X) for the set {x e X: \\x\\ = 1}. We shall write B(X) for the algebra of all bounded linear operators on X and |-| for the operator norm on B(X).Let A be a real or complex unital Banach algebra with identity 1. For a E S(A), the set of support functionals for a is defined by D(a) = {feA': \\f\\ = 1 =/(a)}. For a £ A the numerical range of a is defined by V(a) = {f(a):feD(l)} and the numerical radius of a by v(a) = sup{|A|: Ae V{a)}.We shall write Sp(a) and p(a) for the spectrum and spectral radius of a, respectively.Let X be a real or complex Banach space. B(X) is a unital Banach algebra with respect to | • | and so each operator T e B(X) has a numerical

The Polynomial Numerical Index of a Banach Space

“…McGregor [18] obtained necessary and sufficient conditions such that a finite-dimensional normed space has numerical index 1. The disc algebra is another example of a Banach space with numerical index 1 [8,Theorem 3.3]. Crabb et al .…”