A note on the characterization of spectral operators

Abstract

We are concerned with the characterization problem for bounded spectral operators. We restrict the discussion to scalar type operator with real spectrum. The simplest criterion of this kind o\is for Hilbert space. We give additional necessary conditions on the group e, which together with its uniform boundedness are both necessary and sufficient for S to be pseudo-hermitian operators. A bounded linear operator S on a reflexive Banach space is pseudo-hermitian operators if and only it for every f in L1(R), we have ~~, where the norm on the left is the operator norm, f is the Fourier transform of f and ~~ is the sup norm. 스칼라형 작용소의 스펙트럼으로 제한한 스펙트럴 작용소의 특성화에 관하여 생각할 때, 그룹 e 에 필요조건을 첨가함으로써 유계선형작용소 S가 pseudo-hermitian 작용소가 되기 위한 필요 충분한 조건은 L1(R)의 원소 f 가 ~~~이다.