Similarity Classification of Cowen-Douglas Operators

Jiang, Chunlan

Canadian journal of mathematics, Tome 56 (2004) no. 4, pp. 742-775

Résumé

Let $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{L}\left( \mathcal{H} \right)$ denote the collection of bounded linear operators on $\mathcal{H}$ . An operator $A$ in $\mathcal{L}\left( \mathcal{H} \right)$ is said to be strongly irreducible, if ${{\mathcal{A}}^{\prime }}(T)$ , the commutant of $A$ , has no non-trivial idempotent. An operator $A$ in $\mathcal{L}\left( \mathcal{H} \right)$ is said to be a Cowen-Douglas operator, if there exists $\Omega $ , a connected open subset of $C$ , and $n$ , a positive integer, such that (a) $$\Omega \,\subset \,\sigma (A)\,=\,\left\{ z\,\in \,C|\,A-z\,\text{not}\,\text{invertible} \right\};$$ (b) $$\text{ran(A}-z\text{)}\,\text{=}\,\mathcal{H}\text{,}\,\text{for}\,z\,\text{in}\,\Omega \text{;}$$ (c) $${{\vee }_{z\in \Omega }}\,\ker (A-\,z)\,=\,\mathcal{H}\,\text{and}$$ (d) $$\dim\,\ker (A-z)\,=\,n\,\text{for}\,z\,\text{in}\,\Omega $$ In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the ${{K}_{0}}$ -group of the commutant algebra as an invariant.

Détail

DOI : 10.4153/CJM-2004-034-8

Mots-clés : 47A15, 47C15, 13E05, 13F05

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