Speaker
Douglas Farenick, University of Regina
Abstract
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.
Douglas Farenick, University of Regina
Abstract
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.