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- W4297215457 abstract "The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if $T$ is a $2$-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then $T^{prime}$ is subnormal if and only if $T$ satisfies the (unperturbed) kernel condition. The second counterexample arises from a $2$-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if $T$ is a $2$-isometry satisfying the kernel condition or a quasi-Brownian isometry, then $T^{prime}$ is subnormal. We construct a $2$-isometric adjacency operator $T$ of a rooted directed tree such that $T$ does not satisfy the kernel condition, $T$ is not a quasi-Brownian isometry and $T^{prime}$ is subnormal." @default.
- W4297215457 created "2022-09-28" @default.
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- W4297215457 date "2017-02-04" @default.
- W4297215457 modified "2023-09-28" @default.
- W4297215457 title "A solution to the Cauchy dual subnormality problem for 2-isometries" @default.
- W4297215457 doi "https://doi.org/10.48550/arxiv.1702.01264" @default.
- W4297215457 hasPublicationYear "2017" @default.
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