Similarity of parts to the whole for certain multiplication operators

Abstract

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