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
.