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

Akash Anand; Jan Stochel; Sameer Chavan; Zenon Jan Jab{\l}o\'nski

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

Akash Anand, Sameer Chavan, Zenon Jan Jab{l}oński, Jan Stochel

Functional Analysis · Submitted 8 Jun 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized.

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arXiv ID1806.03229 (version 1)

Pith ID7CBU3EQX, Pith's permanent record of this paper and its review

CitationAkash Anand, Jan Stochel, Sameer Chavan, Zenon Jan Jab{\l}o\'nski. "Complete systems of unitary invariants for some classes of $2$-isometries." arXiv:1806.03229 (2018). https://pith.science/paper/7CBU3EQX

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.08 · mentions code or data availability 0.02.

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The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It 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. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The membership of the Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot 0}$ is also studied.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized.	result 0.67	0.85	0.77	asserted 1.00
s1	It 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.	method 1.00	0.81	0.10	asserted 1.00
s2	A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided.	result 0.75	0.93	0.56	asserted 1.00
s3	It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees.	method 0.67	0.81	0.12	asserted 1.00
s4	The membership of the Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot 0}$ is also studied.	contribution 0.88	0.87	0.05	asserted 0.97

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Original abstract

The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It 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. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The membership of the Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot 0}$ is also studied.

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2 arXiv papers cite this in 4 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Cites it as background 1 paper

It has been extensively studied by several authors (see, for example, [ref] , [ref] , [ref] , [ref] , [ref] ).

The reflexivity of hyperexpansions and their Cauchy dual operators · Introduction · 1.00

What it cites · 20 sentences without a confident label

Uses the method or tool of 1 paper

In Section [ref], using the model for $2$-isometries satisfying the kernel condition (see [ref] ), we answer the question of when two such operators are unitarily equivalent (see Theorem [ref] and Lemma [ref]).

A solution to the Cauchy dual subnormality problem for 2-isometries · Introduction · 0.81

Cites as support 1 paper

It follows from [ref] and [ref] that $T:=$ satisfies the kernel condition.

A solution to the Cauchy dual subnormality problem for 2-isometries · Unitary equivalence of $2$-isometric weighted shifts on directed trees satisfying the kernel condition · 0.97

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