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

Sameer Chavan¹

¹ Department of Mathematics & Statistics Indian Institute of Technology Kanpur Kanpur 208016, India

Studia Mathematica, Tome 203 (2011) no. 2, pp. 129-162

Résumé

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.

Détail

DOI : 10.4064/sm203-2-2

Keywords: cauchy dual operator given *t provides bounded unitary invariant closed left invertible hence special cases problems theory unbounded hilbert space operators related similar problems theory bounded hilbert space operators particular closed expansive finite dimensional cokernel shown admits cowen douglas decomposition only admits wold type decomposition see definitions below connection which even bounded enables establish interesting properties unbounded hyperexpansions their cauchy dual operators completeness eigenvectors hypercyclicity scalar multiples wandering subspace property particular certain cyclic hyperexpansions modelled forward shift mathscr reproducing kernel hilbert space analytic functions where complex polynomials form core mathscr however unlike unbounded subnormals *t never compact unbounded hyperexpansive turns out spectral theory unbounded hyperexpansions satisfactory unbounded subnormal operators