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The Cauchy dual subnormality problem for cyclic \(2\)-isometries. (English) Zbl 1501.47034

A bounded linear operator \(T\) on a complex Hilbert space \(H\) is called \(2\)-isometry if \[0=I-2T^*T+{T^*}^2T^2.\] Obviously, any \(2\)-isometry operator \(T\) is left invertible and thus \(T^*T\) is invertible. The so-called Cauchy dual operator of \(T\) is then given by \(T^{'}:=T(T^*T)^{-1}\). The Cauchy dual subnormality problem asks whether the Cauchy dual operator \(T^{'}\) of any \(2\)-isometry operator \(T\) is subnormal. Recently, it was shown in [the authors, J. Funct. Anal. 277, No. 12, Article ID 108292, 51 p. (2019; Zbl 1476.47018)] that this problem has a negative solution. In the paper under review, the authors show that it has a negative solution even in the class of cyclic \(2\)-isometric operators. The counterexample is implemented with the help of a weighted composition operator on \(L^2\) space over a directed graph with a circuit.

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
44A60 Moment problems

Citations:

Zbl 1476.47018
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References:

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