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 si1" class="mathmlsrc">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_Lclass="mathContainer hidden">class="mathCode">

si1.gif" overflow="scroll"> D_{L}

, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) si2" class="mathmlsrc">title="View 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">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">

titlethesi

class="mathContainer hidden">class="mathCode">

si2.gif" overflow="scroll"> L (X) = A_{0} \otimes I + \sum_{j = 1}^{g} A_{j} \otimes X_{j} ⪰ 0

, where si3" class="mathmlsrc">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_jclass="mathContainer hidden">class="mathCode">

si3.gif" overflow="scroll"> A_{j}

are self-adjoint linear operators on a separable Hilbert space, si4" class="mathmlsrc">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_jclass="mathContainer hidden">class="mathCode">

si4.gif" overflow="scroll"> X_{j}

matrices and I is an identity matrix. If si3" class="mathmlsrc">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_jclass="mathContainer hidden">class="mathCode">

si3.gif" overflow="scroll"> A_{j}

are matrices, then si5" class="mathmlsrc">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)⪰0class="mathContainer hidden">class="mathCode">

si5.gif" overflow="scroll"> L (X) ⪰ 0

is called a linear matrix inequality (LMI) and si1" class="mathmlsrc">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_Lclass="mathContainer hidden">class="mathCode">

si1.gif" overflow="scroll"> D_{L}

a free spectrahedron. For monic LMIs, i.e., si6" class="mathmlsrc">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=Iclass="mathContainer hidden">class="mathCode">

si6.gif" overflow="scroll"> A_{0} = I

, 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 si185" class="mathmlsrc">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_L₂class="mathContainer hidden">class="mathCode">

si185.gif" overflow="scroll"> D_{L_{1}} \subseteq D_{L_{2}}

from monic LMIs to monic LOIs si8" class="mathmlsrc">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">L₁class="mathContainer hidden">class="mathCode">

si8.gif" overflow="scroll"> L_{1}

and si9" class="mathmlsrc">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">L₂class="mathContainer hidden">class="mathCode">

si9.gif" overflow="scroll"> L_{2}

. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron si1" class="mathmlsrc">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_Lclass="mathContainer hidden">class="mathCode">

si1.gif" overflow="scroll"> D_{L}

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 si10" class="mathmlsrc">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_L₂class="mathContainer hidden">class="mathCode">

si10.gif" overflow="scroll"> D_{L_{1}} = D_{L_{2}}

, 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.

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