Circular free spectrahedra

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• 作者：Eric Evert^a ; ^{eevert@ucsd.edu" class="auth_mail" title="E-mail the corresponding author} ; J. William Helton^a ; ¹ ; ^{helton@math.ucsd.edu" class="auth_mail" title="E-mail the corresponding author} ; Igor Klep^b ; ² ; ^{igor.klep@auckland.ac.nz" class="auth_mail" title="E-mail the corresponding author} ; Scott McCullough^c ; ³ ; ^{sam@math.ufl.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Linear matrix inequality (LMI) ; Spectrahedron ; Matrix convex set ; Free convexity ; Circular domain ; Invariant polynomial
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：1047-1070
• 全文大小：555 K

文摘

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303225&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303225&_rdoc=1&_issn=0022247X&md5=85672eb03c489381e257ed127e1dcbce" title="Click to view the MathML source">e^itclass="mathContainer hidden">class="mathCode">$e^{it}$: 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.