Operator realizations of non-commutative analytic functions

TL;DR

This paper explores operator-based realizations of non-commutative analytic functions, extending classical finite-dimensional models to infinite-dimensional cases, and characterizes entire and meromorphic functions via properties of their realizations.

Contribution

It introduces a framework for infinite-dimensional realizations of NC functions, characterizes entire and meromorphic NC functions through realization properties, and defines a universal skew field of fractions.

Findings

Infinite-dimensional realizations characterize broader classes of NC functions.

Entire NC functions correspond to realizations with compact, quasinilpotent operators.

Global extension of NC power series linked to realizations with specific operator properties.

Abstract

A realization is a triple, $(A,b,c)$, consisting of a $d-$tuple, $A= (A =_1, \cdots, A_d )$, $d\in \mathbb{N}$, of bounded linear operators on a separable, complex Hilbert space, $\mathcal{H}$, and vectors $b,c \in \mathcal{H}$. Any such realization defines a (uniformly) analytic non-commutative (NC) function in an open neighbourhood of the origin, $0:= (0, \cdots , 0)$, of the NC universe of $d-$tuples of square matrices of any fixed size via the formula $h(X) = I \otimes b^* ( I \otimes I =_{\mathcal{H}} - \sum X_j \otimes A_j ) ^{-1} I \otimes c$. It is well-known that an NC function has a finite-dimensional realization if and only if it is a non-commutative rational function that is defined at $0$. Such finite realizations contain valuable information about the NC rational functions they generate. By considering more general, infinite-dimensional realizations we study, construct and characterize more general classes of uniformly analytic NC functions. In particular, we show that an NC function is (uniformly) entire, if and only if it has a jointly compact and quasinilpotent realization. Restricting our results to one variable shows that an analytic Taylor-MacLaurin series extends globally to an entire or meromorphic function if and only if it has a realization whose component operator is compact and quasinilpotent, or compact, respectively. This then motivates our definition of the set of global uniformly meromorphic NC functions as the (universal) skew field (of fractions) generated by NC rational expressions in the (semi-free ideal) ring of NC functions with jointly compact realizations.