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 mmlsi1" class="mathmlsrc">mulatext 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">DLmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi>Dmi>mrow><mrow><mi>Lmi>mrow>msub>math>, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) mmlsi2" class="mathmlsrc">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">mg class="imgLazyJSB inlineImage" 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"> mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><mi>Lmi><mo stretchy="false">(mo><mi>Xmi><mo stretchy="false">)mo><mo>=mo><msub><mrow><mi>Ami>mrow><mrow><mn>0mn>mrow>msub><mo>⊗mo><mi>Imi><mo>+mo><msubsup><mrow><mo>∑mo>mrow><mrow><mi>jmi><mo>=mo><mn>1mn>mrow><mrow><mi>gmi>mrow>msubsup><msub><mrow><mi>Ami>mrow><mrow><mi>jmi>mrow>msub><mo>⊗mo><msub><mrow><mi>Xmi>mrow><mrow><mi>jmi>mrow>msub><mo>⪰mo><mn>0mn>math>, where mmlsi3" class="mathmlsrc">mulatext 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">AjmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msub><mrow><mi>Ami>mrow><mrow><mi>jmi>mrow>msub>math> are self-adjoint linear operators on a separable Hilbert space, mmlsi4" class="mathmlsrc">mulatext 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">XjmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msub><mrow><mi>Xmi>mrow><mrow><mi>jmi>mrow>msub>math> matrices and m>I   m> is an identity matrix. If mmlsi3" class="mathmlsrc">mulatext 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">AjmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msub><mrow><mi>Ami>mrow><mrow><mi>jmi>mrow>msub>math> are matrices, then mmlsi5" class="mathmlsrc">mulatext 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)⪰0mathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mi>Lmi><mo stretchy="false">(mo><mi>Xmi><mo stretchy="false">)mo><mo>⪰mo><mn>0mn>math> is called a linear matrix inequality (LMI) and mmlsi1" class="mathmlsrc">mulatext 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">DLmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi>Dmi>mrow><mrow><mi>Lmi>mrow>msub>math> a free spectrahedron. For monic LMIs, i.e., mmlsi6" class="mathmlsrc">mulatext 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=ImathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><msub><mrow><mi>Ami>mrow><mrow><mn>0mn>mrow>msub><mo>=mo><mi>Imi>math>, 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 mmlsi185" class="mathmlsrc">mulatext 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">DL1⊆DL2mathContainer hidden">mathCode"><math altimg="si185.gif" overflow="scroll"><msub><mrow><mi>Dmi>mrow><mrow><msub><mrow><mi>Lmi>mrow><mrow><mn>1mn>mrow>msub>mrow>msub><mo>⊆mo><msub><mrow><mi>Dmi>mrow><mrow><msub><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msub>mrow>msub>math> from monic m>LMIs m> to monic m>LOIs m>mmlsi8" class="mathmlsrc">mulatext 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">L1mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll"><msub><mrow><mi>Lmi>mrow><mrow><mn>1mn>mrow>msub>math> and mmlsi9" class="mathmlsrc">mulatext 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">L2mathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll"><msub><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msub>math>. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron mmlsi1" class="mathmlsrc">mulatext 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">DLmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi>Dmi>mrow><mrow><mi>Lmi>mrow>msub>math> 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 m>F m> 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 m>F m>. Finally, focusing on the algebraic description of the equality mmlsi10" class="mathmlsrc">mulatext 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">DL1=DL2mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll"><msub><mrow><mi>Dmi>mrow><mrow><msub><mrow><mi>Lmi>mrow><mrow><mn>1mn>mrow>msub>mrow>msub><mo>=mo><msub><mrow><mi>Dmi>mrow><mrow><msub><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msub>mrow>msub>math>, 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.