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

Volume 203 / 2011

Sameer Chavan Studia Mathematica 203 (2011), 129-162 MSC: Primary 47B20, 47B32; Secondary 47A16, 47A70, 47B33. DOI: 10.4064/sm203-2-2

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.

Authors

  • Sameer ChavanDepartment of Mathematics & Statistics
    Indian Institute of Technology Kanpur
    Kanpur 208016, India
    e-mail

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