On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case
Volume 203 / 2011
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 $\mathscr F$ in a reproducing kernel Hilbert space of analytic functions, where the complex polynomials form a core for $\mathscr F$. 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.
