On the similarity of powers of operators with flag structure

Abstract

Let L a 2 ( D ) \mathrm {L}^2_a(\mathbb {D}) be the classical Bergman space and let M h M_h denote the operator of multiplication by a bounded holomorphic function h h . Let B B be a finite Blaschke product of order n n . An open question proposed by R. G. Douglas is whether the operators M B M_B on L a 2 ( D ) \mathrm {L}^2_a(\mathbb {D}) similar to ⊕ 1 n M z \oplus _1^n M_z on ⊕ 1 n L a 2 ( D ) \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 ∗ M_z^* is in Cowen-Douglas class B 1 ( D ) B_1(\mathbb {D}) in many cases, Douglas question can be reformulated for operators in B 1 ( D ) B_1(\mathbb {D}) , and the answer is affirmative for many operators in B 1 ( D ) B_1(\mathbb {D}) . A natural question occurs for operators in Cowen-Douglas class B n ( D ) B_n(\mathbb {D}) ( n > 1 n>1 ). In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class B 2 ( D ) B_2(\mathbb {D}) , and give a negative answer.