On the similarity of powers of operators with flag structure

Abstract:

Let $\mathrm {L}^2_a(\mathbb {D})$ be the classical Bergman space and let $M_h$ denote the operator of multiplication by a bounded holomorphic function $h$. Let $B$ be a finite Blaschke product of order $n$. An open question proposed by R. G. Douglas is whether the operators $M_B$ on $\mathrm {L}^2_a(\mathbb {D})$ similar to $\oplus _1^n M_z$ on $\oplus _1^n \mathrm {L}^2_a(\mathbb {D})$? The question was answered in the affirmative, not only for Bergman space but also for many other Hilbert spaces with reproducing kernel. Since the operator $M_z^*$ is in Cowen-Douglas class $B_1(\mathbb {D})$ in many cases, Douglas question can be reformulated for operators in $B_1(\mathbb {D})$, and the answer is affirmative for many operators in $B_1(\mathbb {D})$. A natural question occurs for operators in Cowen-Douglas class $B_n(\mathbb {D})$ ($n>1$). In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class $B_2(\mathbb {D})$, and give a negative answer.