This paper considers paired operators in the context of the Lebesgue Hilbert
space on the unit circle and its subspace, the Hardy space H2. 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
H2, is derived. The results are applied to describing the kernels of
finite-rank asymmetric truncated Toeplitz operators.Comment: 26 page