On essential norms of singular integral operators with constant coefficients and of the backward shift

Oleksiy Karlovych, Eugene Shargorodsky

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
11 Downloads (Pure)

Abstract

Let X be a rearrangement-invariant Banach function space on the unit circle T and let H[X] be the abstract Hardy space built upon X. We prove that if the Cauchy singular integral operator (Formula presented) is τ−t bounded on the space X, then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator aI + bH with a, b ∈ C, acting on the space X, coincide. We also show that similar equalities hold for the backward shift operator (Formula presented) on the abstract Hardy space H[X]. Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priložen. 9 (1975), pp. 73-74] for the operator aI + bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S.

Original languageEnglish
Pages (from-to)60-70
Number of pages11
JournalProceedings of the American Mathematical Society, Series B
Volume9
DOIs
Publication statusPublished - 22 Mar 2022

Keywords

  • abstract Hardy singular integral operator
  • backward shift operator
  • essential norm
  • measure of noncompactness
  • norm
  • Rearrangement-invariant Banach function space

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