In this paper we study the range of the isometry on
H
p
{H^p}
arising from an inner function which is zero at zero by composition. The range of such an isometry is characterized as a closed subspace
M
\mathfrak {M}
of
H
p
{H^p}
(weak-
∗
^ \ast
closed for
p
=
∞
p = \infty
) satisfying the following: (i) the constant function 1 is in
M
\mathfrak {M}
; (ii) if
f
∈
M
f \in \mathfrak {M}
and
g
∈
H
∞
∩
M
g \in {H^\infty } \cap \mathfrak {M}
, then
f
g
∈
M
fg \in \mathfrak {M}
; (iii) if
f
∈
M
f \in \mathfrak {M}
has inner-outer factorization
f
=
χ
⋅
F
f = \chi \cdot F
, then
χ
\chi
is in
M
\mathfrak {M}
; (iv) if
{
B
α
:
α
∈
A
}
\{ {B_\alpha }:\alpha \in \mathcal {A}\}
is a collection of inner functions in
M
\mathfrak {M}
, then the greatest common divisor of
{
B
α
:
α
∈
A
}
\{ {B_\alpha }:\alpha \in \mathcal {A}\}
is also in
M
\mathfrak {M}
; and (v) if
f
∈
M
,
B
∈
M
f \in \mathfrak {M},B \in \mathfrak {M}
, where
B
B
is inner and
B
¯
⋅
f
∈
H
p
\bar B \cdot f \in {H^p}
, then
B
¯
⋅
f
∈
M
\bar B \cdot f \in \mathfrak {M}
. The proof makes use of the fact that there exists a projection onto such a subspace satisfying the axioms of an expectation operator, which for
p
=
2
p = 2
, is simply the orthogonal projection. This characterization is applied to give an equivalent formulation of a conjecture of Nordgren concerning reducing subspaces of analytic Toeplitz operators.