A solution to the Cauchy dual subnormality problem for 2-isometries · arXivDesk
1702.01264Feb 4, 2017The paper has 40 pages and 3 figures. The previous version of this manuscript has been divided into two parts. This is the first major part which focuses mainly on solving the Cauchy dual subnormality problem
A solution to the Cauchy dual subnormality problem for 2-isometries
Akash Anand, Sameer Chavan, Zenon Jan Jabłoński, Jan Stochel
The Cauchy dual subnormality problem asks whether the Cauchy dual operator T′:=T(T∗T)−1 of a 2
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-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′
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