For
α
>
0
\alpha >0
, the
α
\alpha
-Bloch space is the space of all analytic functions
f
f
on the unit disk
D
D
satisfying
\[
‖
f
‖
B
α
=
sup
z
∈
D
|
f
′
(
z
)
|
(
1
−
|
z
|
2
)
α
>
∞
.
\|f\|_{B^{\alpha }}=\sup _{z\in D}|f’(z)|(1-|z|^2)^{\alpha }>\infty .
\]
Let
φ
\varphi
be an analytic self-map of
D
D
. We show that for
0
>
α
,
β
>
∞
0>\alpha ,\beta >\infty
, the essential norm of the composition operator
C
φ
C_{\varphi }
mapping from
B
α
B^{\alpha }
to
B
β
B^{\beta }
can be given by the following formula:
\[
‖
C
φ
‖
e
=
(
e
2
α
)
α
lim sup
n
→
∞
n
α
−
1
‖
φ
n
‖
B
β
.
\|C_{\varphi }\|_e=\left (\frac {e}{2\alpha }\right )^{\alpha }\limsup _{n\to \infty } n^{\alpha -1}\|\varphi ^n\|_{B^{\beta }}.
\]