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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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