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

Oleksiy Karlovych, Eugene Shargorodsky

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Abstract

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 Hp, 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.

Original languageEnglish
Article number6
Number of pages17
JournalIntegral Equations And Operator Theory
Volume95
Issue number1
DOIs
Publication statusPublished - Mar 2023

Keywords

  • Banach function space
  • Coburn’s lemma
  • Fredholmness
  • Invertibility
  • Normal solvability
  • Toeplitz operator

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