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

This paper is devoted to the study of operators satisfying the condition where 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 < .