Let denote a complex, infinite-dimensional, separable Hilbert space, and for any such Hilbert space , let () denote the algebra of bounded linear operators on . We show that for any co-analytic, right-invertible T in (), αT is hypercyclic for every complex α with $\left|\alpha \right|>{\beta}^{-1}$, where $\beta \equiv in{f}_{\left|\right|x\left|\right|=1}\left|\right|T*x\left|\right|>0$. In particular, every co-analytic, right-invertible T in () is supercyclic.

Let T be a spherical 2-expansive m-tuple and let ${T}^{}$ denote its spherical Cauchy dual. If ${T}^{}$ is commuting then the inequality
$\genfrac{}{}{0pt}{}{{\sum}_{\left|\beta \right|=k}{(\beta !)}^{-1}{\left({T}^{}\right)}^{\beta}\left({T}^{}\right){*}^{\beta}\le (k+m-1}{k){\sum}_{\left|\beta \right|=k}{(\beta !)}^{-1}\left({T}^{}\right){*}^{\beta}{\left({T}^{}\right)}^{\beta}}$
holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.

We introduce and discuss a class of operators, to be referred to as operators close to isometries. The Bergman-type operators, 2-hyperexpansions, expansive p-isometries, and certain alternating hyperexpansions are main examples of such operators. We establish a few decomposition theorems for operators close to isometries. Applications are given to the theory of p-isometries and of hyperexpansive operators.

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)....

Motivated by some structural properties of Drury-Arveson d-shift, we investigate a class of functions consisting of polynomials and completely monotone functions defined on the semi-group ℕ of non-negative integers, and its operator-theoretic counterpart which we refer to as the class of completely hypercontractive tuples of finite order. We obtain a Lévy-Khinchin type integral representation for the spherical generating tuples associated with such operator tuples and discuss its applications.

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