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