Extremal problems for operators in banach spaces arising in the study of linear operator pencils

Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition \\A\\ = max{p(AB) : \\B\\ = 1}, where rho stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V-spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as L,(0, 1) and C,, are non-V-spaces if p 0 2; and that the spaces 1, are V-spaces if and only if 1 < p < infinity. The authors pose and discuss some related open problems.