  This paper is concerned with paired operators in the context of the Lebesgue
Hilbert space on the unit circle and its subspace, the Hardy space. By
considering when such operators commute, generalizations of the Brown--Halmos
results for Toeplitz operators are derived. Further, the kernels of such
operators are described, giving results on invariant and nearly-invariant
subspaces, together with a generalization of Coburn's theorem on Toeplitz
kernels.
