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Toeplitz operators with non-trivial kernels and non-dense ranges on weak Hardy spaces

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Eugene Shargorodsky, Alexei Karlovich

Original languageEnglish
Title of host publicationToeplitz Operators and Random Matrices
Subtitle of host publication In Memory of Harold Widom
Editors Estelle Basor, Albrecht Böttcher, Torsten Ehrhardt, Craig Tracy
PublisherBirkhaeuser Publishing Ltd
Pages463-476
Number of pages14
Accepted/In press17 Feb 2022
Published2 Jan 2023

King's Authors

Abstract

The well known Coburn lemma can be stated as follows: a nonzero Toeplitz operator $T(a)$ with symbol $a\in L^\infty(\mathbb{T})$ has a trivial kernel or a dense range on the Hardy space $H^p(\mathbb{T})$ with $p\in(1,\infty)$. We show that an analogue of this result does not hold for the Hardy-Marcinkiewicz (weak Hardy) spaces $H^{p,\infty}(\mathbb{T})$ with $p\in(1,\infty)$: there exist continuous nonzero functions $a:\mathbb{T}\to\mathbb{C}$ depending on $p$ such that $\operatorname{dim} \left(\operatorname{Ker} T(a)\right) = \infty$ and $\operatorname{dim} \left(H^{p,\infty}(\mathbb{T})/\operatorname{clos}_{H^{p,\infty}(\mathbb{T})}\big(\operatorname{Ran} T(a)\big)\right) = \infty$.

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