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

Elementary proofs of two theorems involving arguments of eigenvalues of a product of two unitary matrices