King's College London

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 $S$.