The Coburn lemma and the Hartman-Wintner-Simonenko theorem for Toeplitz operators on abstract Hardy spaces

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(af) 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 ^∞.