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Let X be a Banach function space on the unit circle T , let X ′ be its associate space, and let H[X] and H [ X ′ ] be the abstract Hardy spaces built upon X and X ′ , respectively. Suppose that the Riesz projection P is bounded on X and a ∈ L ∞ \ { 0 } . We show that P is bounded on X ′ . So, we can consider the Toeplitz operators T (a) f = P (a f) and T (a ¯) g = P (a ¯ g) on H[X] and H [ X ′ ] , respectively. In our previous paper, we have shown that if X is not separable, then one cannot rephrase Coburn's lemma as in the case of classical Hardy spaces H p , 1 < p < ∞ , and guarantee that T(a) has a trivial kernel or a dense range on H[X]. The first main result of the present paper is the following extension of Coburn's lemma: the kernel of T(a) or the kernel of T (a ¯) is trivial. The second main result is a generalisation of the Hartman–Wintner–Simonenko theorem saying that if T(a) is normally solvable on the space H[X], then 1 / a ∈ L ∞ . [ABSTRACT FROM AUTHOR]

A well known lemma attributed to Coburn states that a Toeplitz operator with non-trivial kernel acting on the Hardy space must have dense range. We show that the range of a non-zero Toeplitz operator with non-trivial kernel must contain all polynomials and state this in a precise form. [ABSTRACT FROM AUTHOR]