Analysis Seminar/Third Year Requirement: "K-spectral sets and Crouzeix’s Conjecture"

Abstract: Spectral sets and K-spectral sets, introduced by John von Neumann in 1951, offer a possibility to estimate the norm of functions of matrices in terms of the supremum norm of the function.  The concept of spectral sets is partially motivated by von Neumann’s inequality, which can be interpreted as saying that a Hilbert space operator T is a contraction if and only if the closed unit disk is a spectral set for T. In this expository talk, we discuss recent progress on a famous open problem in this area, the so-called Crouzeix's conjecture: the closure of the numerical range W(T) of every operator T is 2-spectral for T. We will begin by looking at some fundamental properties of W(T) and then analyze the recent Crouzeix-Palencia proof that W(T) is always $(1+\sqrt{2})$-spectral. Partial answers to the conjecture, the role of associated "extremal functions" and ways to extend the Crouzeix-Palencia proof to domains that do not necessarily contain W(T) will also be discussed.

Host: John McCarthy