Location
Brown Hall 100
Start Date
7-20-2016 11:00 AM
End Date
20-7-2016 11:30 AM
Description
In this talk I will present a part of a recent joint work with Davidson, Dor-On, and Solel (a complementary talk will be given by Adam Dor-On in the {\em Multivariable Operator Theory} special session). If $A = (A_1, \ldots, A_d)$ is a tuple of operators on $H$ and $B = (B_1, \ldots, B_d)$ is a tuple of operators on $K$, then $B$ is said to be a {\em dilation} of $A$, denoted $A \prec B$, if $A_i = P_H B_i \big|_H$ for all $i$. For a long time it seemed that the name of the game was: given a commuting tuple of operators $A$, find a commuting tuple of {{\em normal} operators $B$ such that $A \prec B$ (usually with additional conditions on the joint spectrum $\sigma(B)$, and requiring the dilation to hold for powers as well). Quite recently, Helton, Klep, McCullough and Schweighofer changed the rules, and started dilating tuples of {\em noncommuting} operators to commuting tuples of normal operators. They showed that there is a universal constant $\vartheta_n$, such that given a tuple of $n \times n$ selfadjoint contractions $A$, there exists a tuple of commuting selfadjoints $B$, such that $\sigma(B) \subseteq [-1,1]^d$ and $\frac{1}{\vartheta_n} A \prec B$. This result had deep implications to spectrahedral inclusion problems. The constant $\vartheta_n$ behaves roughly like $\sqrt{n}$, and was shown to be the best constant possible. We were led to ask whether it is possible to obtain such a dilation result with a constant that does not depend on $n = \operatorname{rank}A$ (necessarily fixing $d$). Moreover, we sought a normal dilation $B$ with more precise control on the joint spectrum $\sigma(B)$. As a representative of our results, I will present the following theorem, as well as some applications. \vskip 5pt \noindent{\bf Theorem.} {\em Let $K$ be a convex set in $\mathbb{R}^d$ satsfying some reasonable conditions. Then for every $d$-tuple $A$ of selfadjoint operators with a joint numerical range contained in $K$, there is a $d$-tuple of commuting selfadjoint operators $B$ with joint spectrum $\sigma(B) \subseteq K$, such that } \[\frac{1}{d} A \prec B .\]
Dilations, inclusions of matrix convex sets, and completely positive maps
Brown Hall 100
In this talk I will present a part of a recent joint work with Davidson, Dor-On, and Solel (a complementary talk will be given by Adam Dor-On in the {\em Multivariable Operator Theory} special session). If $A = (A_1, \ldots, A_d)$ is a tuple of operators on $H$ and $B = (B_1, \ldots, B_d)$ is a tuple of operators on $K$, then $B$ is said to be a {\em dilation} of $A$, denoted $A \prec B$, if $A_i = P_H B_i \big|_H$ for all $i$. For a long time it seemed that the name of the game was: given a commuting tuple of operators $A$, find a commuting tuple of {{\em normal} operators $B$ such that $A \prec B$ (usually with additional conditions on the joint spectrum $\sigma(B)$, and requiring the dilation to hold for powers as well). Quite recently, Helton, Klep, McCullough and Schweighofer changed the rules, and started dilating tuples of {\em noncommuting} operators to commuting tuples of normal operators. They showed that there is a universal constant $\vartheta_n$, such that given a tuple of $n \times n$ selfadjoint contractions $A$, there exists a tuple of commuting selfadjoints $B$, such that $\sigma(B) \subseteq [-1,1]^d$ and $\frac{1}{\vartheta_n} A \prec B$. This result had deep implications to spectrahedral inclusion problems. The constant $\vartheta_n$ behaves roughly like $\sqrt{n}$, and was shown to be the best constant possible. We were led to ask whether it is possible to obtain such a dilation result with a constant that does not depend on $n = \operatorname{rank}A$ (necessarily fixing $d$). Moreover, we sought a normal dilation $B$ with more precise control on the joint spectrum $\sigma(B)$. As a representative of our results, I will present the following theorem, as well as some applications. \vskip 5pt \noindent{\bf Theorem.} {\em Let $K$ be a convex set in $\mathbb{R}^d$ satsfying some reasonable conditions. Then for every $d$-tuple $A$ of selfadjoint operators with a joint numerical range contained in $K$, there is a $d$-tuple of commuting selfadjoint operators $B$ with joint spectrum $\sigma(B) \subseteq K$, such that } \[\frac{1}{d} A \prec B .\]