Cowen-Douglas Operator and Shift on Basis

In this paper we show a Cowen-Douglas operator $T \in \mathcal{B}_{n}(Ω)$ 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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