Thompson-type formulae

Abstract

Let X and Y be two n × n 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 n × n unitary matrices U and V such that e i X e i Y = e i U XU * +V BV * .In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II 1 factor. 1

A Survey of the Matrix Exponential Formulae with Some Applications

“…([4]) Let A,B be Hermitian operators whose spectra are disjoint from each other. Let f be any function in L1 (R) such that f (λ) = 1 λ whenever λ ∈ Spec(A)−Spec(B).Then the solution of the equation AX −XB = Y can be expressed as X = +∞ −∞ e −itA Ye itB f (t)dt. (14.4) Proof.…”

“…Antezana et al have made extensions of this result to compact operators as well as to operators in an embeddable II 1 factor[1], and use it to study the optimal path in unitary group[2].…”