A joint similarity problem for n-tuples of operators on vector-valued Bergman spaces

We investigate n-tuples of commuting Foias–Williams/Peller type operators acting on vector-valued weighted Bergman spaces. We prove that a commuting n-tuple of such operators is jointly (completely) polynomially bounded if and only if it is similar to an n-tuple of contractions, if and only if each of the n operators is polynomially bounded.