Although it is not typically viewed in this way, the classic Halmos dilation theorem of 1950 states that every selfadjoint contraction dilates to a symmetry (i.e., a selfadjoint unitary). Thus, every operator with numerical range in a line segment $L$ in the complex plane dilates to a normal operator with spectrum given by the end points of $L$. Mirman’s dilation theorem from 1968 can be viewed as an extension of the Halmos theorem, as it asserts every operator having its numerical range contained in a given triangle $T$ has a normal dilation with spectrum given by the vertices of $T$. That’s as far as it goes for single operators, although one can extend Mirman’s theory for $d$-tuples of selfadjoint operators by replacing $T$ with a simplex in $d$-dimensional real space.

From the modern perspective, the theorems of Halmos and Mirman characterise the maximal matrix convex set, which is a graded set, whose first level in the grading is a line segment or a simplex. In this lecture, I will consider what happens when we pair $T$ and $L$ to obtain a 3-prism P(3) in 3-dimensional space. It turns out that a pairing of the Halmos and Mirman theorems is possible. More generally, one can consider various noncommutative realisations of the $k$-prism P($k$) and their associated operator systems, and I will explain the some of the rich properties these operator systems possess. This is joint work with D. Farenick, R. Maleki, and S. Medina Varela.