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

Let X X be a rearrangement-invariant Banach function space on the unit circle T \mathbb {T} and let H [ X ] H[X] be the abstract Hardy space built upon X X . We prove that if the Cauchy singular integral operator ( H f ) ( t ) = 1 π i ∫ T f ( τ ) τ − t d τ (Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau )}{\tau -t}\,d\tau is bounded on the space X X , then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator a I + b H aI+bH with a , b ∈ C a,b\in \mathbb {C} , acting on the space X X , coincide. We also show that similar equalities hold for the backward shift operator ( S f ) ( t ) = ( f ( t ) − f ^ ( 0 ) ) / t (Sf)(t)=(f(t)-\widehat {f}(0))/t on the abstract Hardy space H [ X ] H[X] . Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priloz̆en. 9 (1975), pp. 73-74] for the operator a I + b H aI+bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S S .