The PIMS Postdoctoral Fellow Seminar: Sushil Singla

A common theme in mathematics is to discern properties of a mathematical structure by making certain mathematical "measurements" of that structure. In operator theory, this theme often translates to the problem of studying operators on high- or infinite-dimensional Hilbert spaces by analyzing their compressions (or measurement outcomes) to low- or finite-dimensional spaces. At the most basic level, one can consider the numerical range of an operator, which was introduced by Toeplitz in the early 20th century, as the numerical range is the set of measurement outcomes of an operator when the action of the operator is restriction to subspaces of $1$-dimension. Higher-dimensional analogues lead to the matrix range of an operator, or the matrix range of a $d$-tuple of operators. 

Hausdorff proved that the spatial numerical range of a single operator is a convex set. Although this is not also true for the spatial measurements of several operators, it is true if we allow for arbitrary representations of those operators. Therefore, the natural measurement outcome set for a $d$-tuple of Hilbert space operators is a compact convex set $K$ of $\mathbb C^d$. A noncommutative realization of $K$ is a compact matrix convex set $K_{\rm nc}$, which is a graded set of $d$-tuples of matrices such that the first level in the grading coincides with $K$ and such that $K_{\rm nc}$ is closed under matrix-convex combinations of its elements. 

In this talk, I will describe certain noncommutative realizations of three classical geometric objects: cubes, polydiscs, and prisms. I will describe various geometric and algebraic properties of these noncommutative realizations, and I will apply classical dilation theorems of Halmos and Mirman to give a complete description of the maximal noncommutative triangular prism in terms of joint unitary dilations. This is joint work with D.~Farenick, R.~Maleki, and S.~Medina Varela.