## Banach Journal of Mathematical Analysis

### Complete systems of unitary invariants for some classes of $2$-isometries

#### Abstract

We characterize the unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition. This relies on a model for such operators built on operator-valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator-valued unilateral weighted shifts in a fairly general context. We also provide a complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition. This is formulated purely in the language of graph theory—namely, in terms of certain generation branching degrees. Finally, we study the membership of the Cauchy dual operators of $2$-isometries in classes $C_{0\cdot }$ and $C_{\cdot 0}$.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 2 (2019), 359-385.

Dates
Accepted: 29 November 2018
First available in Project Euclid: 1 February 2019

https://projecteuclid.org/euclid.bjma/1548990189

Digital Object Identifier
doi:10.1215/17358787-2018-0042

Mathematical Reviews number (MathSciNet)
MR3927878

Zentralblatt MATH identifier
07045463

#### Citation

Anand, Akash; Chavan, Sameer; Jabłoński, Zenon Jan; Stochel, Jan. Complete systems of unitary invariants for some classes of $2$ -isometries. Banach J. Math. Anal. 13 (2019), no. 2, 359--385. doi:10.1215/17358787-2018-0042. https://projecteuclid.org/euclid.bjma/1548990189

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