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 People'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 $\mathcal B_n (\Omega)$-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.

Keywords: Strongly irreducible operator, nice operator, Cowen-Douglas operators.

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