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 alttext="double-struck upper T"> T encoding="application/x-tex">\mathbb {T} and let H left-bracket X right-bracket"> H [ stretchy="false">] encoding="application/x-tex">H[X] abstract Hardy built upon . We prove that if Cauchy singular integral operator alttext="left-parenthesis f right-parenthesis left-parenthesis t equals StartFraction 1 Over pi i EndFraction Underscript double-struck T Endscripts tau minus d tau"> stretchy="false">( f stretchy="false">) t = 1 π i ∫ τ<!-- τ <mml:mo>−<!-- − <mml:mspace width="thinmathspace" /> d encoding="application/x-tex">(Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau )}{\tau -t}\,d\tau is bounded , then norm, essential Hausdorff measure of non-compactness alttext="a I plus b H"> a I + b encoding="application/x-tex">aI+bH with comma element-of C"> , ∈<!-- ∈ mathvariant="double-struck">C encoding="application/x-tex">a,b\in \mathbb {C} acting coincide. also show similar equalities hold for backward shift S ModifyingAbove With caret 0 slash t"> S ^<!-- ^ </mml:mover> 0 / encoding="application/x-tex">(Sf)(t)=(f(t)-\widehat {f}(0))/t Our results extend those by Krupnik Polonskiĭ [Funkcional. Anal. Priloz̆en. 9 (1975), pp. 73-74] second author [J. Funct. 280 (2021), p. 11] S"> encoding="application/x-tex">S