One-dimensional perturbations of unitaries that are quasiaffine transforms of singular unitaries, and multipliers between model spaces
Volume 251 / 2020
Studia Mathematica 251 (2020), 1-29
MSC: Primary 30J05; Secondary 30H10, 47B15, 47A55, 47B99, 47B35.
DOI: 10.4064/sm180114-19-11
Published online: 7 August 2019
Abstract
It is shown that, under some natural additional conditions, a transformation which intertwines a cyclic singular unitary operator with a 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 a one-dimensional perturbation of a unitary operator and also a quasiaffine transform of a singular unitary operator, and $T$ is power bounded, then $T$ is similar to a unitary operator. Moreover, $$ \sup_{n\geq 0}\|T^{-n}\|\leq\Bigl(2\Bigl(\sup_{n\geq 0}\|T^n\|\Bigr)^2+1\Bigr)\cdot\Bigl(\sup_{n\geq 0}\|T^n\|\Bigr)^5. $$
