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 DL, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) 
, where Aj are self-adjoint linear operators on a separable Hilbert space, 80dd1e787a8a30a96d00eae4ab0" title="Click to view the MathML source">Xj matrices and I is an identity matrix. If Aj are matrices, then 8046" title="Click to view the MathML source">L(X)⪰0 is called a linear matrix inequality (LMI) and DL a free spectrahedron. For monic LMIs, i.e., A0=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 85" class="mathmlsrc">85.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=29018bc997dc690d441a39f3eb5cf63f" title="Click to view the MathML source">DL1⊆DL2 from monic LMIs to monic LOIs L1 and L2. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron DL 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 DL1=DL2, 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.
, where Aj are self-adjoint linear operators on a separable Hilbert space, 80dd1e787a8a30a96d00eae4ab0" title="Click to view the MathML source">Xj matrices and I is an identity matrix. If Aj are matrices, then 8046" title="Click to view the MathML source">L(X)⪰0 is called a linear matrix inequality (LMI) and DL a free spectrahedron. For monic LMIs, i.e., A0=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 85" class="mathmlsrc">85.gif&_user=111111111&_pii=S0022247X16303675&_rdoc=1&_issn=0022247X&md5=29018bc997dc690d441a39f3eb5cf63f" title="Click to view the MathML source">DL1⊆DL2 from monic LMIs to monic LOIs L1 and L2. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron DL 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 DL1=DL2, 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.