The Coburn Lemma and the Hartman–Wintner–Simonenko Theorem for Toeplitz Operators on Abstract Hardy Spaces

Abstract

Let X be a Banach function space on the unit circle {mathbb {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 ain L^infty {setminus }{0}. We show that P is bounded on X'. So, we can consider the Toeplitz operators T(a)f=P(af) and T({overline{a}})g=P({overline{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<infty , 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({overline{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/ain L^infty .