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

Research output: Contribution to journalArticlepeer-review

Eugene Shargorodsky, Alexei Karlovich

Original languageEnglish
Pages (from-to)60-70
JournalProceedings of the American Mathematical Society
VolumeSer. B 9
Accepted/In press7 Dec 2021
Published22 Mar 2022

King's Authors

Abstract

Let $X$ be a rearrangement-invariant Banach function space on the unit circle
$\mathbb{T}$ and let $H[X]$ be the abstract Hardy space built upon $X$.
We prove that if the Cauchy singular integral operator
$(Hf)(t)=\frac{1}{\pi i}\int_{\mathbb{T}}\frac{f(\tau)}{\tau-t}\,d\tau$ is
bounded on the space $X$, then the norm, the essential norm, and the Hausdorff
measure of noncompactness of the operator $aI+bH$ with $a,b\in\mathbb{C}$,
acting on the space $X$, coincide. We also show that similar equalities hold
for the backward shift operator $(Sf)(t)=(f(t)-\widehat{f}(0))/t$ on the
abstract Hardy space $H[X]$. Our results extend those by Krupnik and Polonskii
\cite{KP75} for the operator $aI+bH$ and by the second author \cite{S21}
for the operator $S$.

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