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Noncommutative Partial Convexity Via \(\Gamma \)-Convexity

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Abstract

Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials \(\Gamma \), a free set \({{\mathcal {K}}} \) is called \(\Gamma \)-convex if for all \(X\in {{\mathcal {K}}} \) and isometries V satisfying \(V^*\Gamma (X)V=\Gamma (V^*XV)\), we have \(V^*XV\in {{\mathcal {K}}} .\) We establish an Effros–Winkler Hahn–Banach separation theorem for \(\Gamma \)-convex sets; they are delineated by linear pencils in the coordinates of \(\Gamma \) and the variables x.

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Notes

  1. See also the MATLAB toolbox, https://set.kuleuven.be/optec/Software/bmisolver-a-matlab-package-for-solving-optimization-problems-with-bmi-constraints.

  2. Explicitly, if \(X\in {\mathcal {K}}(n)\) and \(Y\in {\mathcal {K}}(m)\), then \(X\oplus Y\in {\mathcal {K}}(n+m);\) if U is a \(n\times n\) unitary, then \(U^*XU=(U^*X_1 U,\dots ,U^*X_{\texttt {g}}U)\in {\mathcal {K}}(n)\); and if \({{\mathcal {K}}} \subset {\mathbb {C}}^n\) is k dimensional reducing subspace for X, then \(X|_{{{\mathcal {K}}} }\in {\mathcal {K}}(k).\)

  3. See for instance the MATLAB toolbox, https://set.kuleuven.be/optec/Software/bmisolver-a-matlab-package-for-solving-optimization-problems-with-bmi-constraints.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, USA

    Michael Jury, Scott McCullough & James Eldred Pascoe

  2. Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia

    Igor Klep

  3. Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, USA

    Mark E. Mancuso

Authors
  1. Michael Jury
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  2. Igor Klep
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  3. Mark E. Mancuso
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  4. Scott McCullough
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  5. James Eldred Pascoe
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Correspondence to Scott McCullough.

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M. Jury: Research Supported by NSF Grant DMS-1900364

I. Klep: Supported by the Slovenian Research Agency Grants J1-8132, N1-0057 and P1-0222. Partially supported by the Marsden Fund Council of the Royal Society of New Zealand.

M. E. Mancuso: Research partially supported by NSF Grant DMS-1565243

S. McCullough: Research supported by NSF Grants DMS-1361501 and DMS-1764231

J. E. Pascoe: Partially supported by NSF MSPRF DMS 1606260.

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Jury, M., Klep, I., Mancuso, M.E. et al. Noncommutative Partial Convexity Via \(\Gamma \)-Convexity. J Geom Anal 31, 3137–3160 (2021). https://doi.org/10.1007/s12220-020-00387-1

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