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

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., $\mathbf{\text{47}}$, 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.