Joint similarity to operators in noncommutative varieties

TL;DR

This paper addresses the problem of joint similarity of operator tuples within noncommutative varieties, extending classical concepts to multivariable noncommutative operator theory.

Contribution

It provides solutions to joint similarity problems in noncommutative varieties associated with free holomorphic functions and noncommutative polynomials, generalizing classical multivariable operator theory.

Findings

Characterization of joint similarity in noncommutative varieties

Extension of Agler's hypercontraction concepts to noncommutative settings

New criteria for operator tuple equivalence in noncommutative operator spaces

Abstract

In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated with positive regular free holomorphic functions in $n$ noncommuting variables and with sets of noncommutative polynomials in $n$ indeterminates, where $B(\cH)$ is the algebra of all bounded linear operators on a Hilbert space $\cH$. In particular, if $f=X_1+...+X_n$ and $\cP=\{0\}$, the elements of the corresponding variety can be seen as noncommutative multivariable analogues of Agler's $m$-hypercontractions.