Conjugations Preserving Toeplitz Kernels

Abstract We study conjugations in $$L^2(\mathbb {T})$$ L 2 ( T ) and their relation with kernels of Toeplitz operators on $$H^2(\mathbb H space. Such are a generalization model spaces. investigate properties an inequality between two unimodular functions defined the unit circle. This allows us to significantly strengthen previous theorems characterizing all $$M_z$$ M z -commuting -conjugations leaving invariant spaces setting.