On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

Abstract

The Cauchy dual operator T , given by T (T * T) −1 , provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below). This connection, which is new even in the bounded case, enables us to establish some interesting properties of unbounded 2-hyperexpansions and their Cauchy dual operators such as the completeness of eigenvectors, the hypercyclicity of scalar multiples, and the wandering subspace property. In particular, certain cyclic 2-hyperexpansions can be modelled as the forward shift F in a reproducing kernel Hilbert space of analytic functions, where the complex polynomials form a core for F. However, unlike unbounded subnormals, (T * T) −1 is never compact for unbounded 2-hyperexpansive T. It turns out that the spectral theory of unbounded 2-hyperexpansions is not as satisfactory as that of unbounded subnormal operators.