Circular free spectrahedra

• Published in: Journal of mathematical analysis and applications Vol. 445; no. 1; pp. 1047 - 1070
• Main Authors: Evert, Eric, Helton, J. William, Klep, Igor, McCullough, Scott
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2017
• Subjects: Circular domain; Free convexity; Invariant polynomial; Linear matrix inequality (LMI); Matrix convex set; Spectrahedron; Free convexity; Circular domain; Linear matrix inequality (LMI); Spectrahedron; Invariant polynomial; Matrix convex set
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2016.07.011

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by eit: up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials.