Title: On the matrix range of random matrices
Abstract: In operator theory, one often attaches to an operator (or to a family of operators) certain invariants by which the operator can be understood. The spectrum $\sigma(A)$ of an operator $A$ is an obvious example. Another classical example, which is somewhat less familiar, is the numerical range of an operator: it is the set of all complex numbers obtained from the operator by applying a vector state to it. Unlike the spectrum, the notion of numerical range easily extends to families of operators: the numerical range of a d-tuple $A = (A_1, \ldots, A_d)$ is the collection of all tuples of complex numbers of the form $(\phi(A_1), \ldots, \phi(A_d))$, where $\phi$ is a state. It is a subset of the d-dimensional space, which carries some information about $A$.
My talk will be about a matrix-valued analogue of the numerical range, called the matrix range of an operator tuple. I will explain what is the matrix range, what it is good for, and then I will report on recent work in which we prove that there is a certain "universal" matrix range, to which the matrix ranges of a sequence of large random matrices tends to, almost surely. In particular, our results imply that the numerical range of a sequence of certain N-by-N random matrices tends in the Hausdorff metric to the closed disc as N goes to infinity, almost surely. The key novel technical aspect of our work is the continuity of the matrix range of a continuous field of operators, and a certain quantitative matrix valued Hahn-Babach type separation theorem.
Based on joint work with Malte Gerhold.