This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space \(H^2\) . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly \(S^*\) -invariant subspace of \(H^2\) , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.
