Location
Crow 206
Start Date
7-21-2016 4:00 PM
End Date
21-7-2016 4:20 PM
Description
In the talk we will study algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set $D_L$, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) $L(X)=A_0\otimes I+\sum_{j=1}^g A_{j}\otimes X_j\succeq 0,$ where $A_j$ are self-adjoint linear operators on a separable Hilbert space, $X_j$ matrices and $I$ is an identity matrix. If $A_j$ are matrices, then $L(X)\succeq 0$ is called a linear matrix inequality (LMI) and $D_L$ a free spectrahedron. For monic LMIs, i.e., $A_0=I$, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough \cite{HKM2,HKM1}. We extend the characterization of the inclusion $D_{L_1}\subseteq D_{L_2}$ from monic \emph{LMIs} to monic \emph{LOIs} $L_1$ and $L_2$.Using this characterization in a separation argument, we obtain a certificate for matrix-valued nc polynomials $F$ positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Finally, focusing on the algebraic description of the equality $D_{L_1}=D_{L_2}$, we remove the assumption of boundedness from the description in the LMIs case and present counterexamples for the extension to LOIs case. \begin{thebibliography}{1} \bibitem{HKM2} J.W.\ Helton, I.\ Klep, S.\ McCullough: The convex Positivstellensatz in a free algebra, \textit{Adv.\ Math.}\ \textbf{231} (2012) 516--534. \bibitem{HKM1} J.W.\ Helton, I.\ Klep, S.\ McCullough: The matricial relaxation of a linear matrix inequality, \textit{Math.\ Program.}\ \textbf{138} (2013) 401--445. \bibitem{Z} A. Zalar, Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets. \textit{Preprint} \url{http://arxiv.org/abs/1602.00765.} \end{thebibliography}
Operator positivstellensätze for noncommutative polynomials positive on matrix convex sets
Crow 206
In the talk we will study algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set $D_L$, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) $L(X)=A_0\otimes I+\sum_{j=1}^g A_{j}\otimes X_j\succeq 0,$ where $A_j$ are self-adjoint linear operators on a separable Hilbert space, $X_j$ matrices and $I$ is an identity matrix. If $A_j$ are matrices, then $L(X)\succeq 0$ is called a linear matrix inequality (LMI) and $D_L$ a free spectrahedron. For monic LMIs, i.e., $A_0=I$, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough \cite{HKM2,HKM1}. We extend the characterization of the inclusion $D_{L_1}\subseteq D_{L_2}$ from monic \emph{LMIs} to monic \emph{LOIs} $L_1$ and $L_2$.Using this characterization in a separation argument, we obtain a certificate for matrix-valued nc polynomials $F$ positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Finally, focusing on the algebraic description of the equality $D_{L_1}=D_{L_2}$, we remove the assumption of boundedness from the description in the LMIs case and present counterexamples for the extension to LOIs case. \begin{thebibliography}{1} \bibitem{HKM2} J.W.\ Helton, I.\ Klep, S.\ McCullough: The convex Positivstellensatz in a free algebra, \textit{Adv.\ Math.}\ \textbf{231} (2012) 516--534. \bibitem{HKM1} J.W.\ Helton, I.\ Klep, S.\ McCullough: The matricial relaxation of a linear matrix inequality, \textit{Math.\ Program.}\ \textbf{138} (2013) 401--445. \bibitem{Z} A. Zalar, Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets. \textit{Preprint} \url{http://arxiv.org/abs/1602.00765.} \end{thebibliography}