Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

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• 作者：Aljaž Zalar ^{aljaz.zalar@imfm.si" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Free convexity ; Linear matrix inequality (LMI) ; Spectrahedron ; Completely positive ; Positivstellensatz ; Free real algebraic geometry
• 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：32-80
• 全文大小：800 K

文摘

This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=1fd026a2786fccf5876ac7d451a0a2f7" title="Click to view the MathML source">D_Lpan>pan class="mathContainer hidden">pan class="mathCode">$D_L$pan>pan>pan>, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) pan id="mmlsi2" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$L(X)=A_0\otimes I+{\sum }_{j=1}^g{A}_j\otimes X_j⪰0$pan>pan>pan>, where pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=6293fa1c93f172d34fc6a6bb00fc1a25" title="Click to view the MathML source">A_jpan>pan class="mathContainer hidden">pan class="mathCode">$A_j$pan>pan>pan> are self-adjoint linear operators on a separable Hilbert space, pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si4.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=9b1c280dd1e787a8a30a96d00eae4ab0" title="Click to view the MathML source">X_jpan>pan class="mathContainer hidden">pan class="mathCode">$X_j$pan>pan>pan> matrices and I   is an identity matrix. If pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=6293fa1c93f172d34fc6a6bb00fc1a25" title="Click to view the MathML source">A_jpan>pan class="mathContainer hidden">pan class="mathCode">$A_j$pan>pan>pan> are matrices, then pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si5.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=c6f33e87e9399faa5e8e83e2dafe8046" title="Click to view the MathML source">L(X)⪰0pan>pan class="mathContainer hidden">pan class="mathCode">$L(X)⪰0$pan>pan>pan> is called a linear matrix inequality (LMI) and pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=1fd026a2786fccf5876ac7d451a0a2f7" title="Click to view the MathML source">D_Lpan>pan class="mathContainer hidden">pan class="mathCode">$D_L$pan>pan>pan> a free spectrahedron. For monic LMIs, i.e., pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si6.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=9e7a11443c4246d688f701925732e81f" title="Click to view the MathML source">A₀=Ipan>pan class="mathContainer hidden">pan class="mathCode">$A_0=I$pan>pan>pan>, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion pan id="mmlsi185" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si185.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=29018bc997dc690d441a39f3eb5cf63f" title="Click to view the MathML source">D_L1⊆D_L2pan>pan class="mathContainer hidden">pan class="mathCode">$D_{L_1}\subseteq D_{L_2}$pan>pan>pan> from monic LMIs to monic LOIs  pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si8.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=9848d1b0d56293f99e9157a8941eb88a" title="Click to view the MathML source">L₁pan>pan class="mathContainer hidden">pan class="mathCode">$L_1$pan>pan>pan> and pan id="mmlsi9" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si9.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=a4f94183d94405e61c12caa0267b325b" title="Click to view the MathML source">L₂pan>pan class="mathContainer hidden">pan class="mathCode">$L_2$pan>pan>pan>. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=1fd026a2786fccf5876ac7d451a0a2f7" title="Click to view the MathML source">D_Lpan>pan class="mathContainer hidden">pan class="mathCode">$D_L$pan>pan>pan> and its projection, called a free Hilbert spectrahedrop. Further on, using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials F positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fejér–Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials F  . Finally, focusing on the algebraic description of the equality pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si10.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=d0779431dc1007a05caabc0b93927c9e" title="Click to view the MathML source">D_L1=D_L2pan>pan class="mathContainer hidden">pan class="mathCode">$D_{L_1}=D_{L_2}$pan>pan>pan>, we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs case by counterexamples.