Deddens Algebras and Shift

Abstract

We consider Deddens algebras associated to operators of the form S−λI, where S is the unilateral shift and λ is a complex number. We show that such an algebra properly contains the commutant of S and that it is always weakly dense in \({{\mathcal L}({\mathcal H})}\). Yet, it contains no rank one operators, unless λ = 0, in which case it equals \({{\mathcal L}({\mathcal H})}\).