2025 CMS Summer Meeting
Quebec City, June 6 - 9, 2025
Recent progress in matrix, graph and operator theory / Progrès récents dans la théorie des matrices, graphes et opérateurs
Org: Ludovick Bouthat (Laval), Steve Kirkland (University of Manitoba) and Hermie Monterde (University of Manitoba)
[PDF]
Org: Ludovick Bouthat (Laval), Steve Kirkland (University of Manitoba) and Hermie Monterde (University of Manitoba)
[PDF]
- LUDOVICK BOUTHAT, Université Laval
- DOUG FARENICK, University of Regina
Matrix convexity and unitary dilations of Toeplitz-contractive d-tuples [PDF]
- DOUG FARENICK, University of Regina
-
A
well-known theorem of P.R. Halmos concerning the existence of unitary dilations for
contractive linear operators acting on Hilbert spaces is recast as a result
for $d$-tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition.
Such operator $d$-tuples satisfying this matrix-positivity condition are called, herein,
Toeplitz-contractive, and a characterisation of the Toeplitz-contractivity condition is presented.
The matrix-positivity condition leads to definitions of new distance-measures in several variable operator theory,
generalising the notions of norm, numerical radius, and spectral radius to $d$-tuples of operators (commuting, for the spectral radius)
in what appears to be a novel, asymmetric way.
Toeplitz contractive operators form a matrix convex set, and a scaling constant $c_d$ for inclusions of
the minimal and maximal matrix convex
sets determined by a stretching of the unit circle $S^1$ across $d$ complex dimensions is shown to exist.
- AVLEEN KAUR, University of British Columbia
- MATTHEW KREITZER, University of Guelph
- PIETRO PAPARELLA, University of Washington Bothell
- SARAH PLOSKER, Brandon University
- PAUL SKOUFRANIS, York University
Non-Commutative Majorization [PDF]
- MATTHEW KREITZER, University of Guelph
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The notion of majorization of one self-adjoint $n \times n$ matrix by another is a very useful concept in matrix/operator theory. For example, a classical theorem of Schur and Horn states that a diagonal matrix $D$ is majorized by a self-adjoint matrix $B$ if and only if a unitary conjugate of $B$ has the same diagonal as $D$. Some equivalent characterizations of $A$ being majorized by $B$ include
there existing a doubly stochastic matrix that maps the vector or eigenvalues of $B$ to the the vector or eigenvalues of $A$, tracial inequalities involving convex functions of $A$ and $B$, and there exists a mixed unitary quantum channel that maps $B$ to $A$.
Given the prevalence of qunatum information theory, the following is an interesting question in the context of matrix/operator theory: given $m$-tuples $A_1$, $\ldots$, $A_m$ and $B_1$, $\ldots$, $B_m$ of $n \times n$ matrices, can a mathematical condition be given for when there exists a unital quantum channel $\Phi$ such that $\Phi(B_k) = A_k$ for all $k$. In this talk, we answer this question using non-commutative Choquet Theory as developed by Davidson and Kennedy.
This talk is based on joint works with Kennedy.
- PRATEEK VISHWAKARMA, Universite Laval
- HARMONY ZHAN, Worcester Polytechnic Institute
- XIAOHONG ZHANG, University of Montreal
- HARMONY ZHAN, Worcester Polytechnic Institute