滿足Alg T=｛T｝'之線性變換

Abstract

For a bound linear operator T on a Hilbert space let {T}', {T}'' and Alg T denote the commutant, the double commutant and the weakly closed algebra generated by T and 1, respectively. Assume that T is a completely non-unitary contrction with a scalar-valued characteristics function Ψ(λ). In this note we show that the condition that |Ψ(e^it)|=1 on a set of positive Lebseque measure implies that Alg T={T}'. Moreover, if Ψ(λ) is assumed to be outer, then these two conditions are equivalent. Our main result generalizes the well-known fact that the compressions of the shift satisfy Alg T={T}'.