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This paper considers paired operators in the context of the Lebesgue Hilbert space L² 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. Inclusion relations between such kernels are considered in detail, and the results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.
Câmara et al. (Wed,) studied this question.