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 $S$.
$\mathbb{T}$ 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 $S$.
Original language | English |
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Pages (from-to) | 60-70 |
Journal | Proceedings of the American Mathematical Society |
Volume | Ser. B 9 |
Publication status | Published - 22 Mar 2022 |