Operator realizations about a matrix-centre
Functional Analysis
2025-09-12 v1
Abstract
We develop a general theory of operator realizations, or ``linear representations" of analytic functions in several non-commuting variables about a matrix-centre. In particular we show that a non-commutative function has a matrix-centre realization about any matrix tuple, , in its domain, if and only if it is a uniformly analytic non-commutative function defined in a uniformly open neighbourhood of . This extends the finite-dimensional realization theory of non-commutative rational functions maximally -- to all uniformly analytic non-commutative functions.
Cite
@article{arxiv.2509.08921,
title = {Operator realizations about a matrix-centre},
author = {Ali Karoobi and Robert T. W. Martin and Maximilian Tornes},
journal= {arXiv preprint arXiv:2509.08921},
year = {2025}
}