Cowen–Douglas Operators and Shift Operators.

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 ∈ B n (Ω) and a complex number z 0 ∈ Ω , we show that there is a generalized basis { g k } k = 0 ∞ , such that the adjoint operator (T - z 0) ∗ is the forward shift on { g k } k = 0 ∞ , and if n ≥ 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 B 1 (Ω) being a backward shift on some Markushevich basis. Also, we show that a multiplication operator defined on Bergman space with a M o ¨ bius transformation as its multiplier is a forward shift on some Markushevich basis. [ABSTRACT FROM AUTHOR]