Thompson-type formulae

TL;DR

This paper explores extensions of Thompson-type exponential formulae from finite-dimensional matrices to infinite-dimensional operators, including compact operators and operators in an embeddable II$_1$ factor, broadening the scope of the original result.

Contribution

It extends Thompson's finite-dimensional matrix exponential formula to infinite-dimensional operator settings, including compact operators and II$_1$ factors.

Findings

Extension to compact operators achieved

Application to operators in embeddable II$_1$ factors demonstrated

Broader class of operators now satisfies Thompson-type formulae

Abstract

Let X and Y be two nxn Hermitian matrices. In the article "Proof of a conjectured exponential formula" (Linear and Multilinear Algebra (19) 1986, 187-197) R. C. Thompson proved that there exist two nxn unitary matrices U and V such that $$ e^{i X}e^{i Y}=e^{i (UXU^*+VBV^*)}. $$ In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II$_1$ factor.