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On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case. (English) Zbl 1241.47019
The concept of operator Cauchy dual for bounded operators was coined by the author inspired by some results of S. Shimorin [J. Reine Angew. Math. 531, 147–189 (2001; Zbl 0974.47014)]. For a closed left-invertible \(T\), he defines it in a similar fashion: the operator Cauchy dual \(T^{\prime}\) is \(T(T^*T)^{-1}.\) An operator \(T\) is expansive if \(\| x\| \leq \| Tx\| \) for all \(x \in \mathcal D(T),\) while it is 2-hyperexpansive if \(\| x\| ^2 -2\| Tx\| ^2+\| T^2x\| ^2<0\) for all \(x \in \mathcal D(T^2)\). The author showed in [Proc. Edinb. Math. Soc., II. Ser. 50, No. 3, 637–652 (2007; Zbl 1155.47024)] that the operator Cauchy dual to a 2-hyperexpansive bounded operator is a hyponormal contraction. From that, he obtained a version of the Berger-Shaw theorem for 2-hyperexpansions.
In the present paper, he shows that when \(T\) is a closed expansive operator with a finite-dimensional cokernel, then \(T\) admits a Cowen-Douglas decomposition if and only if \(T^{\prime}\) admits the Wold-type decomposition. (Both decompositions are the natural extensions to the unbounded case.) Another result, among many others, is that if \(T\) is 2-hyperexpansive and admits the Wold-type decomposition, then \(T^{\prime}\) admits a Cowen-Douglas decomposition.
Editorial remark: The author has pointed out to us that the notion of an operator Cauchy dual was actually introduced by S. Shimorin [loc. cit.], not by him.

MSC:
47B20 Subnormal operators, hyponormal operators, etc.
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
47A16 Cyclic vectors, hypercyclic and chaotic operators
47B33 Linear composition operators
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