On the Characterization of Spectral Operators

Abstract

Introduction.This paper is mainly concerned with the characterization problem for bounded spectral operators (see [5]).We restrict the discussion to scalar type operators with real spectrum, which we call pseudo-hermitian operators (p.h.).The criteria are given in terms of certain properties of the exponential group generated by the operator.The simplest criterion of this kind is for Hubert space: a bounded linear operator S is p.h. if and only if || e"5| ;S M < oo for all real /.This statement is false for Banach spaces, and even for reflexive Banach spaces.In the latter case, we give additional necessary conditions on the group e"s, which together with its uniform boundedness are both necessary and sufficient for S to be p.h.Another useful criterion is the following: a bounded linear operator S on a reflexive Banach space is p.h. if and only if for every/in LX(R) (R the real line), we have || $Rf(t)e't!idt|| ^ M \\ f\n, where the norm on the left is the operator norm, / is the Fourier transform of/ and || | oe is the sup norm.Unlike the general theorems in [5] about the spectrality of an operator, the characterizations obtained in this paper are easily applied to analytically given operators on concrete Banach spaces.An example is discussed in 4.Applications are discussed at the end of 5.2 deals with an improvement of a theorem of Foguel [7] about the resolutions of the identity of sums and products of commuting spectral operators.1 contains preliminaries.The characterization problem for spectral operators is discussed in 3 and 5.1. Preliminaries.The term "operator" is used for bounded linear operators on X into X, where X is some fixed Banach space.We refer to [5] for definitions and properties of spectral operators.If A is a Banach algebra with unit and xeA, then a(x), p(x), r(x) and R(X;x) denote respectively the spectrum, the resolvent set, the spectral radius, and the resolvent of x.We write r+(x) = sup (Re X ; X e a(x)) and s +(x) = sup (Im X ; X e a(x)).Similarly, r ~(x) and s~(x) are defined with inf instead of sup.Complementation is denoted by a prime.The commutativity of x, ye A is expressed by the symbol x ^y.