Cowen-Douglas Operator and Shift on Basis

Abstract

In this paper we show a Cowen-Douglas operator T ∈ Bn(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 ≥ 2.