Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras

Abstract

The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a whole. The authors develop various applications, in particular to the study of Banach algebras where the numerical range provides an important link between the algebraic and metric structures.

Numerical Range on Weighted Hardy Spaces as Semi Inner Product Spaces

“…Later, Duncan et al [4] determined the range of values of the numerical index. More precisely, they proved that As an interesting open problem is to compute the numerical index of H p (β).…”

Properties of lush spaces and applications to Banach spaces with numerical index 1

“…Again, by (2), y * n 6. Let us show that g n = e n , g * n = e * n + y * n form the biorthogonal system we need.…”

“…By (2), e * n 6 for every n ∈ N. Now, we use the separability of Y to take a norming sequence with norming tails {y…”

“…In the complex case, it is a deep result that n(X) is always bigger or equal than 1/e. Classical references here are the monographs from the 1970's [2,3]. The state-of-the-art on numerical indices may be found in the recent survey [9] and references therein.…”

Finite-dimensional Banach spaces with numerical index zero

“…In the complex case, it is a celebrated result due to H. Bohnenblust and S. Karlin [2] (see also [8]) that n(X) 1/ e, so the numerical radius is always an equivalent norm. Classical references on this topic are the monographs by F. Bonsall and J. Duncan [3,4]. For recent results we refer the reader to [6,10,11,15,16,18] and the survey paper [13].…”