Free Probability and Operators

Date: March 28, 2025

Time: 4:00PM - 5:00PM

Location: BLOC 306

Speaker: Michael Anshelevich, Texas A&M University

Title: Convergence of the product of exponents

Abstract: In a general Banach algebra, or even a matrix algebra, a product of exponents need not equal the exponential of the sum. Nonetheless, the Lie-Trotter formula famously asserts that alternating products of exponentials do converge to the exponential of the sum. We show that (in many circumstances) such behavior is typical: for almost all permutations of the factors, the products of exponentials converge. In a matrix algebra, the result holds if the norms of the matrices do not grow too fast. In a general Banach algebra, it holds if n matrices fall only into o(n / log n) distinct types. The methods involve elementary estimates and concentration inequalities. The results are an outcome of undergraduate projects with Austin Pritchett and Anh Nguyen.