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THE STRONG IRREDUCIBILITY OF A CLASS OF COWEN–DOUGLAS OPERATORS ON BANACH SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 94 / Issue 3 / December 2016
- Published online by Cambridge University Press:
- 16 August 2016, pp. 479-488
- Print publication:
- December 2016
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- Article
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Let
${\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ be the set of Cowen–Douglas operators of index 
$n$ on a nonempty bounded connected open subset 
$\unicode[STIX]{x1D6FA}$ of 
$\mathbb{C}$. We consider the strong irreducibility of a class of Cowen–Douglas operators 
${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ on Banach spaces. We show 
${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})\subseteq {\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ and give some conditions under which an operator 
$T\in {\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ is strongly irreducible. All these results generalise similar results on Hilbert spaces.
