Operators with slowly growing resolvents towards the spectrum

Abstract

A closed densely defined operator H, on a Banach space X, whose spectrum is contained in R and satisfies

(z −H)−1

≤ c hziα |=z|β ∀ z 6∈ R (0.1) for some α , β ≥ 0; c > 0, is said to be of (α, β)−type R . If instead of (0.1) we have

(z −H)−1

≤ c |z|α |=z|β ∀ z 6∈ R, (0.2) then H is of (α, β)0−type R . Examples of such operators include self-adjoint operators, Laplacian on L1(R), Schro¨dinger operators on Lp(Rn) and operators H whose spectra lie in R and permit some control on

eiHt

. In this paper we will characterise the (α, β)−type R operators. In particular we show that property (0.1) is stable under dialation by real numbers in the interval (0,1) and perturbation by positive reals. We will also show that is H is of (α, β)−type R then so is H2.