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

摘要

Let T = {T(t)}_(t∈?) be a σ(X, F)-continuous group of isometries on a Banach space X with generator A, where σ(X, F) is an appropriate local convex topology on X induced by functionals from F ? X~*. Let σ A_(x) be the local spectrum of A at x ∈ X and r A_(x):=sup{{pipe}λ{pipe}:λ ∈ σ A_(x)}, the local spectral radius of A at x. It is shown that for every x ∈ X and τ ∈ ?, {double pipe}T(τ)x-x{double pipe} ≤ {pipe}τ{pipe} r A_(x) {double pipe}x{double pipe}. Moreover if 0 ≤ τr A_(x) ≤ π/2, then it holds that {double pipe}T(τ)x-T(-τ)x{double pipe} ≤ 2sin (τ r A_(x)) {double pipe}x{double pipe}. 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 ∈ X and τ ≥ 0,Several applications are given.