The Invariant Subspaces of
Hl
and an Extension of Schwarz's Lemma
EDWIN J. AKUTOWICZ
Introduction. Let
Hp
(p ^ 1) denote as usual the Banach space of all func-
tions/(z) holomorphic in the unit disc |z| I such that the integral'
1 j
|/(re'9)|W
remains bounded as r tends to 1. The norm in the space
Hp
is the LT-norm of
the boundary function
limr^1/(re,a)
denoted
f(e1*):
We shall assume certain of the principal facts about these spaces to be known,
such as the factorisation into inner and outer factors, representation by the
Cauchy integral over the boundary values, etc. (for details consult K. Hoffman
W).
For/sif
1
, the inner factor of/shall be denoted by If and the outer factor
by Ofif=IfOf. The Blaschke factor of/shall be denoted by Bf and the singular
factor by !,:/ / = Bflf.
The closed, nonnull, linear subspaces S of
H1
which are invariant under multi-
plication by z (briefly, invariant subspaces) have been described as follows by
deLeeuw and Rudin [7] (also Helson [4]):
THEOREM.
Every invariant subspace S of
H1
is of the form
where s is an inner function essentially uniquely determined by the subspace S
(i.e., to within a constant factor of modulus
unity).2
The proof suggested (Rudin [8, p. 432]) would give the generating element s
1
Here, and in the sequel, an unadorned integral sign refers to integration over the interval
-71 g & g 71 .
- This is of course also true for
Hp,
p 1, but these cases are simpler.
1
http://dx.doi.org/10.1090/pspum/011/9995
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