Anti Lie-Trotter formula

TL;DR

This paper investigates the limits of certain matrix expressions as p approaches infinity, aiming to establish an explicit 'anti Lie-Trotter' formula, extending known results from the p approaching zero case.

Contribution

It introduces the study of the limits of matrix functions as p tends to infinity, providing explicit formulas in special cases and analyzing the geometric mean case for 2x2 matrices.

Findings

Limit of Z_p exists and an explicit formula is derived in a special case.

Limit of G_p is established for 2x2 matrices.

The paper proposes the concept of an anti Lie-Trotter formula as a counterpart to the classical formula.

Abstract

Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of $G_p:=(A^p\,\#\,B^p)^{2/p}$ as $p$ tends to $0$, where $X\,\#\,Y$ is the geometric mean of $X$ and $Y$. In this paper we study the complementary limit of $Z_p$ and $G_p$ as $p$ tends to $\infty$, with the ultimate goal of finding an explicit formula, which we call the anti Lie-Trotter formula. We show that the limit of $Z_p$ exists and find an explicit formula in a special case. The limit of $G_p$ is shown for $2\times2$ matrices only.