Extremal Problems for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils

Abstract

This paper is devoted to the study of operators satisfying the condition $$ ||A||\, = \max \{ \rho (AB):||B||\, = 1\} , $$ where ρ stands for the spectral radius; and Banach spaces in which all operators satisfy this condition. Such spaces are called V−spaces. The present paper contains partial solutions of some of the open problems posed in the first part of the paper. The main results: (1) Each subspace of l p (1 < p < ∞) is a V−space. (2) For each infinite dimensional Banach space X there exists an equivalent norm |||·||| on X such that the space (X, |||·|||) is not a V−space. (3) Let X be a separable infinite dimensional Banach space with a symmetric basis. If X has the V-property, then X is isometric to l p , 1 < p < ∞.