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 $${\mathbb {T}}$$ T , let $$X'$$ X ′ be its associate space, and let H[X] and $$H[X']$$ H [ X ′ ] be the abstract Hardy spaces built upon X and $$X'$$ X ′ , respectively. Suppose that the Riesz projection P is bounded on X and $$a\in L^\infty {\setminus }\{0\}$$ a ∈ L ∞ \ { 0 } . We show that P is bounded on $$X'$$ X ′ . So, we can consider the Toeplitz operators $$T(a)f=P(af)$$ T ( a ) f = P ( a f ) and $$T({\overline{a}})g=P({\overline{a}}g)$$ T ( a ¯ ) g = P ( a ¯ g ) on H[X] and $$H[X']$$ 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 $$H^p$$ H p , $$1 1 < p < ∞ , and guarantee that T(a) has a trivial kernel or a dense range on H[X]. The first main result of the present paper is the following extension of Coburn’s lemma: the kernel of T(a) or the kernel of $$T({\overline{a}})$$ T ( a ¯ ) is trivial. The second main result is a generalisation of the Hartman–Wintner–Simonenko theorem saying that if T(a) is normally solvable on the space H[X], then $$1/a\in L^\infty $$ 1 / a ∈ L ∞ .