Lifting Commuting 3-Isometric Tuples

TL;DR

This paper extends the theory of 3-isometric operators from single-variable to multi-variable settings, characterizing their structure via positivity conditions and exploring spectral properties of sub-Jordan tuples.

Contribution

It generalizes the characterization of 3-isometries as restrictions of Jordan operators to multiple variables, introducing new conditions and analyzing spectral aspects.

Findings

Multi-variable 3-isometries characterized by positivity conditions.

Additional hypotheses are necessary in the multi-variable setting.

Results on joint spectrum of sub-Jordan tuples and 3-symmetric operators.

Abstract

An operator $T$ is called a 3-isometry if there exists operators $B_1(T^*,T)$ and $B_2(T^*,T)$ such that \[Q(n)=T^{*n}T^n=1+nB_1(T^*,T)+n^2 B_2(T^*,T)\] for all natural numbers $n$. An operator $J$ is a Jordan operator of order $2$ if $J=U+N$ where $U$ is unitary, $N$ is nilpotent order $2$, and $U$ and $N$ commute. An easy computation shows that $J$ is a $3$-isometry and that the restriction of $J$ to an invariant subspace is also a $3$-isometry. Those $3$-isometries which are the restriction of a Jordan operator to an invariant subspace can be identified, using the theory of completely positive maps, in terms of a positivity condition on the operator pencil $Q(s).$ In this article, we establish the analogous result in the multi-variable setting and show, by modifying an example of Choi, that an additional hypothesis is necessary. Lastly we discuss the joint spectrum of sub-Jordan tuples and derive results for 3-symmetric operators as a corollary.