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

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 non-compactness 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 Polonskiĭ [Funkcional. Anal. i Priloz̆en. 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 $S$.