Similarity of parts to the whole for certain multiplication operators

Abstract:

We show that the Bergman shift $B$, multiplication by $z$ on the Bergman space ${A^2}$, is similar to its part $B\left | {_N} \right .$ if and only if $N = \varphi {A^2}$, where $\varphi$ is a finite product of interpolating Blaschke products. In addition, we show that $B$ is not unitarily equivalent to any of its parts. For the analytic Toeplitz operator ${T_f}$ on ${H^2}$, we obtain that ${T_f}$ is similar to each of its parts if and only if ${T_f}$ is unitarily equivalent to each of its parts if and only if $f$ is a weak-star generator of ${H^\infty }$.