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 hmlsrc">hImg" 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_LhContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">w>Dw>w>Lw>h>, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303675&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=8c1bbfcced632a57fa37781f38368b9e">height="21" width="259" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303675-si2.gif">hContainer hidden">hCode">h altimg="si2.gif" overflow="scroll">Lhy="false">(Xhy="false">)=w>Aw>w>0w>⊗I+w>∑w>w>j=1w>w>gw>w>Aw>w>jw>⊗w>Xw>w>jw>⪰0h>, where hmlsrc">hImg" 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_jhContainer hidden">hCode">h altimg="si3.gif" overflow="scroll">w>Aw>w>jw>h> are self-adjoint linear operators on a separable Hilbert space, hmlsrc">hImg" 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_jhContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">w>Xw>w>jw>h> matrices and I   is an identity matrix. If hmlsrc">hImg" 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_jhContainer hidden">hCode">h altimg="si3.gif" overflow="scroll">w>Aw>w>jw>h> are matrices, then hmlsrc">hImg" 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)⪰0hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">Lhy="false">(Xhy="false">)⪰0h> is called a linear matrix inequality (LMI) and hmlsrc">hImg" 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_LhContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">w>Dw>w>Lw>h> a free spectrahedron. For monic LMIs, i.e., hmlsrc">hImg" 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=IhContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">w>Aw>w>0w>=Ih>, 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 hmlsrc">hImg" 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_L2hContainer hidden">hCode">h altimg="si185.gif" overflow="scroll">w>Dw>w>w>Lw>w>1w>w>⊆w>Dw>w>w>Lw>w>2w>w>h> from monic LMIs to monic LOIs  hmlsrc">hImg" 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">L₁hContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">w>Lw>w>1w>h> and hmlsrc">hImg" 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">L₂hContainer hidden">hCode">h altimg="si9.gif" overflow="scroll">w>Lw>w>2w>h>. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron hmlsrc">hImg" 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_LhContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">w>Dw>w>Lw>h> 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 hmlsrc">hImg" 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_L2hContainer hidden">hCode">h altimg="si10.gif" overflow="scroll">w>Dw>w>w>Lw>w>1w>w>=w>Dw>w>w>Lw>w>2w>w>h>, 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.