Let
ϕ
\phi
be holomorphic and map the open unit disk into itself, and let
C
ϕ
:
f
→
f
∘
ϕ
{C_\phi }:f \to f \circ \phi
be the composition operator on
H
2
{H^2}
generated by
ϕ
\phi
. If
C
ϕ
{C_\phi }
is a compact operator then
(
1
)
ϕ
(
z
0
)
=
z
0
(1)\phi ({z_0}) = {z_0}
for some
z
0
ϵ
D
{z_0} \epsilon D
;
(
2
)
σ
(
C
ϕ
)
=
{
ϕ
′
(
z
0
)
n
:
n
=
0
,
1
,
2
,
…
}
∪
{
0
}
(2)\sigma ({C_\phi }) = \{ \phi ’{({z_0})^n}:n = 0,1,2, \ldots \} \cup \{ 0\}
.