Let
W
(
D
)
=
{
f
(
z
)
=
Σ
n
=
0
∞
a
n
z
n
|
|
|
f
|
|
1
=
Σ
n
=
0
∞
|
a
n
|
>
+
∞
}
W(D) = \{ f(z) = \Sigma _{n = 0}^\infty {a_n}{z^n}|\;||f|{|_1} = \Sigma _{n = 0}^\infty |{a_n}| > + \infty \}
,
f
(
z
)
f(z)
a function in
W
(
D
)
W(D)
for which
f
(
0
)
=
1
f(0) = 1
, and
M
f
{M_f}
the operator of multiplication by
f
(
z
)
f(z)
on
W
(
D
)
W(D)
. It is shown that if
k
k
and
m
m
are integers for which
0
≤
m
≤
k
−
1
0 \leq m \leq k - 1
and
X
k
m
X_k^m
is the closed subspace of
W
(
D
)
W(D)
spanned by
{
z
n
k
+
i
|
n
=
0
,
1
,
…
;
i
=
0
,
1
,
…
,
m
}
\{ {z^{nk + i}}|n = 0,1, \ldots ;i = 0,1, \ldots ,m\}
, then
M
f
{M_f}
is bounded below on
X
k
m
⇔
f
(
z
)
X_k^m \Leftrightarrow f(z)
does not have
k
−
m
k - m
distinct zeros in any set of the form
{
w
i
z
0
|
0
≤
i
≤
k
−
1
;
|
z
0
|
=
1
}
\{ {w^i}{z_0}|0 \leq i \leq k - 1;|{z_0}| = 1\}
, where
w
w
is a primitive
k
k
th root of unity.