Szego Seminar: An introduction to Crouzeix's Conjecture 

Abstract: One of the most important results in operator theory is von Neumann's inequality which states for a fixed contraction A on a Hilbert space H, the unit disk is a 1-spectral set for A.  Now one may ask, if I sup over a different set, say a set containing the spectrum of A, will I achieve the same inequality? This leads us to a nice set called the Numerical Range of A, denoted W(A). In 2004, Crouzeix conjectured that W(A) is a 2-spectral set of A. Crouzeix then proved, along with Palencia, that W(A) is a 1+sqrt{2}-spectral set. However, the sharp constant of 2 is still an open question. We will dive into why this is an interesting question and some of the known results.

Host: Pooja Joshi