When are the norms of the Riesz projection and the backward shift operator equal to one?

Oleksiy Karlovych, Eugene Shargorodsky

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection P on the Lebesgue space Lp lead to ‖P‖Lp→Lp =1/sin⁡(π/p) for every p∈(1,∞). Hence L2 is the only space among all Lebesgue spaces Lp for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces Lp. We prove that the norm of P is equal to one on the space X if and only if X coincides with L2 and there exists a constant C∈(0,∞) such that ‖f‖X=C‖f‖L2 for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.
Original languageEnglish
Article number110158
Number of pages29
JournalJournal Of Functional Analysis
Volume285
Issue number12
DOIs
Publication statusPublished - 15 Dec 2023

Keywords

  • Abstract Hardy space
  • Backward shift operator
  • Banach function space
  • Riesz projection

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