Isometric dilations of contractions on Banach spaces

Abstract

Throughout this note Z (resp. R, resp. R+, resp. N9 resp. C) is the set of all integer (resp. real, resp. nonnegative real, resp. nonnegative integer, resp. complex) numbers. Also G is a group, e e G its neutral element: K:G—>R+ a submultiplicat ive function (i.e., K{gh) ^ K(g)K(h) for all g,heG) with K(e) = 1; X a Banach space; the Banach algebra of all linear bounded operators on X and the identity. ^m{R) (m e N, m — oo) being the algebra of all m-times differentiable functions on R with the usual topology and Γ — {ze C; \z — 1}, Wm(Γ) is the algebra of all functions f:Γ-*C such that t->f(eu) belongs to unitary if it is 9fm(Γ)-scalar ([2], [4]).