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

摘要

It is shown that, under some natural additional conditions, an operator whichintertwines one cyclic singular unitary operator with one dimensionalperturbation of another cyclic singular unitary operator is the operator ofmultiplication by a multiplier between model spaces. Using this result, it isshown that if $T$ is one dimensional perturbation of a unitary operator, $T$ isa quasiaffine transform of a singular unitary operator, and $T$ is powerbounded, then $T$ is similar to a unitary operator, and $sup_{ngeq0}|T^{-n}|leq(2(sup_{ngeq 0}|T^n|)^2+1)cdot(sup_{ngeq 0}|T^n|)^5$.