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arXiv · 2509.08921

Operator realizations about a matrix-centre

Ali Karoobi·Robert T. W. Martin·Maximilian Tornes
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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, $Y$, in its domain, if and only if it is a uniformly analytic non-commutative function defined in a uniformly open neighbourhood of $Y$. This extends the finite-dimensional realization theory of non-commutative rational functions maximally -- to all uniformly analytic non-commutative functions.

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Ali Karoobi, Robert T. W. Martin, Maximilian Tornes. 2025-09-10. Operator realizations about a matrix-centre. https://arxiv.org/abs/2509.08921

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