Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Abstract

Let T = {T (t)}(t is an element of R) be a sigma(X, F)-continuous group of isometries on a Banach space X with generator A, where sigma(X, F) is an appropriate local convex topology on X induced by functionals from F subset of X*. Let sigma(A)(x) be the local spectrum of A at x is an element of X and r(A)(x) := sup{|lambda| : lambda is an element of sigma(A)(x)}, the local spectral radius of A at x. It is shown that for every x is an element of X and tau is an element of R, parallel to T(tau)x - x parallel to <= vertical bar tau vertical bar r(A)(x) parallel to x parallel to. Moreover if 0 <= tau r(A)(x) <= pi/2, then it holds that parallel to T(tau)x - T(-tau)x parallel to <= 2 sin (tau r(A)(x)) parallel to x parallel to. Asymptotic versions of these results for C-0-semigroup of contractions are also obtained. If T = {T(t)}(t >= 0) is a C-0-semigroup of contractions, then for every x is an element of X and tau >= 0, lim(t ->infinity) parallel to T (t + tau)x - T(t)x parallel to <= tau sup {vertical bar lambda vertical bar is an element of sigma(A)(x) boolean AND iR} parallel to x parallel to. Several applications are given.