Elementary Proofs Of Two Theorems Involving Arguments Of Eigenvalues Of A Product Of Two Unitary Matrices

TL;DR

This paper provides simple, accessible proofs for two theorems that establish bounds on the maximum argument of eigenvalues of a product of two unitary matrices, with implications for both finite and infinite-dimensional cases.

Contribution

It introduces elementary proofs for existing theorems on eigenvalue arguments, clarifying conditions for equality and enabling generalization to infinite-dimensional operators.

Findings

Elementary proofs for bounds on eigenvalue arguments.

Conditions for equality are clearly identified.

Proofs extend to infinite-dimensional unitary operators.

Abstract

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices --- one by Childs \emph{et al.} [J. Mod. Phys., \textbf{47}, 155 (2000)] and the other one by Chau [arXiv:1006.3614]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators.