Cowen-Douglas Operator and Shift on Basis

Abstract

In this paper we show a Cowen-Douglas operator $T\in {\mathcal{B}}_n(\Omega )$ is the adjoint operator of some backward shift on a general basis by choosing nice cross-sections of its complex bundle $E_T$. Using the basis theory model, we show that a Cowen-Douglas operator never be a shift on some Markushevicz basis for $n\ge 2$.

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• Operator algebras generated by left invertibles