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Abstract

The paper deals with Toeplitz and Hankel operators acting between distinct Hardy type spaces over the unit circle $\mathbb {T}$. We characterize possible symbols of such operators and prove general versions of the Brown–Halmos theorem and the Nehari theorem. A lower bound for the Kuratowski measure of noncompactness of a Toeplitz operator is also found. Our approach allows handling Hardy spaces associated with arbitrary rearrangement invariant spaces, but the main part of the results are new even for the classical $H^p$ spaces.