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 is an element of L-infinity\{0}. We show that P is bounded on X'. So, we can consider the Toeplitz operators T(a)f = P(af) and T((sic))g = P((sic)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 < infinity, 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((sic)) 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 is an element of L-infinity.