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



.