One dimensional perturbations of unitaries that are quasiaffine transforms of singular unitaries, and multipliers between model spaces

Abstract

It is shown that, under some natural additional conditions, an operator which intertwines one cyclic singular unitary operator with one dimensional perturbation of another cyclic singular unitary operator is the operator of multiplication by a multiplier between model spaces. Using this result, it is shown that if $T$ is one dimensional perturbation of a unitary operator, $T$ is a quasiaffine transform of a singular unitary operator, and $T$ is power bounded, then $T$ is similar to a unitary operator, and ${\sup }_{n\ge 0}\Vert T^{-n}\Vert \le (2({\sup }_{n\ge 0}\Vert T^n\Vert {)}^2+1)\cdot ({\sup }_{n\ge 0}\Vert T^n\Vert {)}^5$.

arXiv commentIn the previous version, a big misprint was: the condition of power boundedness of T in the emphasazing text on 2d page was forgotted. In this version tis condition is added. Also, the reference to an answer given by Fricain, Hartmann, Ross about existence of noninvertible multiplier with dense range is added