Cowen–Douglas Operators and Shift Operators

Abstract

In this paper, we attempt to understand Cowen–Douglas operators by the way of basis theory and shift operator. For a Cowen–Douglas operator $T \in {\mathcal {B}}_{n}(\Omega )$ and a complex number $z_0 \in \Omega$, we show that there is a generalized basis $\{g_{k}\}_{k=0}^{\infty }$, such that the adjoint operator $(T-z_0)^*$ is the forward shift on $\{g_{k}\}_{k=0}^{\infty }$, and if $n \ge 2$, then $T-z_0$ never is a backward shift on any Markushevich basis. Moreover, we give a characterization of a Cowen–Douglas operator in ${\mathcal {B}}_{1}(\Omega )$ being a backward shift on some Markushevich basis. Also, we show that a multiplication operator defined on Bergman space with a M$\ddot{o}$bius transformation as its multiplier is a forward shift on some Markushevich basis.