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<p>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, <i>Y</i>, in its domain, if and only if it is a uniformly analytic non-commutative function defined in a uniformly open neighbourhood of <i>Y</i>. This extends the finite–dimensional realization theory of non-commutative rational functions maximally – to all uniformly analytic non-commutative functions.</p>

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Operator Realizations About a Matrix-Centre

  • 期刊论文

  • 发表日期：

    2026年7月27日

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  • Ali Karoobi,
  • Robert T. W. Martin,
  • Maximilian Tornes

摘要

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.