# Journal of Operator Theory

Volume 32, Issue 1, Summer 1994 pp. 77-89.

Similarity, reducibility and approximation of the Cowen-Douglas operators**Authors**: Chun Lan Jiang

**Author institution:**Department of Mathematics, Jilin University, Changchun, 130023, The Peopleâ€™s Republic of China

**Summary:**An operator T on $\mathcal H$ is called strongly irreducible if T does not commute with any nontrivial idempotent operator. In this paper we obtain a characterization of the strongly irreducibility of Cowen-Douglas operators. For an analytic connected Cauchy domain and a positive integer n, we can find a strongly irreducible nice operator A in $\mathcal B_n (\Omega)$ â€” the class of Cowen-Douglas operators with index n. An operator A is called nice, if the commutant of either T or T* is a strictly cyclic Abelian algebra. Finally, we obtain a characterization of operators which can be uniquely written as an algebraic direct sum of strongly irreducible nice operators.

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