8 ADELE ZUCCHI

THEOREM

2.2.9. If 11 is a Blaschke function, then there exists a unique (up

to equivalence) inner function B^ having at each a G ft a zero with multiplicity

fi(a), and such that for all z G ft

log \B^(z)\ = -^2 V(°)9(z, o) + Hz),

aect

where h is a harmonic function on ft, continuous on ft with constant values on

every component of T.

We call /i the representing Blaschke function of B^.

DEFINITION 2.2.10. The functions B^ denned in Theorem 2.2.9 are called

(generalized) Blaschke products.

Clearly the product of two Blaschke products is a Blaschke product, and so

is their quotient if it is bounded.

THEOREM

2.2.11. ([13]) Any inner function 0 may be factored into 6 = BS,

where B is a Blaschke product having the same zeros of 6, and S is a singular

function. The factors are unique up to equivalence.

LEMMA

2.2.12. (i) If Sy andSv are singular functions, then

(ii) If B^ and B^ are Blaschke products, then B^B^ = B^^.

DEFINITION 2.2.13. A closed linear subspace M of

Hp(ft)

(weak*-closed if

p = oo) is said to be fully invariant if rf G M. for all / G M. and for all

r G R(il).

Since any function in ff°°(n) can be boundedly pointwise approximated on

ft by functions in i?(fi), we see that the fully invariant subspaces of

Hp(ft)

are

exactly those closed subspaces which are invariant under multiplication by any

function in H°°(fi).

If 6 is an inner function in ft, we denote by 6Hp(ft) the space of all / G Hp(ft)

which are multiples of 6. These are clearly fully invariant subspaces of Hp(ft).

The next theorem ([20] Theorem 1), which summarizes the results by Sarason,

Hasumi and Voichick, states that these are the only fully invariant subspaces.

THEOREM

2.2.14. Let 1 p oo and let M C

Hp(ft)

be a fully invari-

ant subspace of Hp(ft). Then there exists an inner function 9 such that M =

6Hp(ft).

Of course two inner functions 0 and 6' generate the same subspace if and only

if

6/ef

and 0'/O belong to

Hp{ft)

and thus to H°°(n), i.e., if and only if 0 =

6f.