Free Probability and Operators

Date: April 22, 2025

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Octavio Arizmendi, CIMAT

Title: The S-Transform in Free Probability

Abstract: An important analytical tool for computing the free multiplicative convolution of two probability measures is Voiculescu's S-transform. Several works have progressively extended its domain of applicability. It was first introduced by Voiculescu (1987) for distributions with nonzero mean and compact support, and later studied by Bercovici and Voiculescu (1993) in the case of probability measures on ℝ≥0 with unbounded support. Subsequently, Raj Rao and Speicher (2007) defined an S-transform for measures with zero mean and all moments finite. Their approach is combinatorial, using Puiseux series, and cannot be extended to the general case. Arizmendi and Pérez-Abreu (2009) considered the case of symmetric measures with unbounded support. In this talk, we will explain a recent result with Hasebe and Kitagawa, in which we extend Voiculescu's S-transform to the case of general probability measures on ℝ with possibly unbounded support. Our approach is analytical and is based on a reformulation using the T-transform, defined as the reciprocal of the S-transform. Unlike previous works, our construction does not require any assumptions on symmetry, mean, or boundedness of the support. This new definition generalizes previous approaches while preserving key properties such as the existence of the limit and non-vanishing in a natural domain.