OPERATORS OF CLASS Co

7

DEFINITION

2.2.3. A nonzero function 0 in H°°(Q) is said to be inner if |0|

is constant almost everywhere on each component of T.

A function F in Hp(ft) is said to be outer if for all z G ft we have

log|F(s)| = - i - j T l o g | F ( O I ^ « , z ) d s .

An inner function is said to be trivial if it is invertible. The invertible inner

functions form a finitely generated group under multiplication. An outer function

has no zeros. The product of two inner (or outer) functions is inner (or outer),

and so is their quotient if it is bounded. An inner function is outer if and only

if it is trivial.

It is known [13, 20] that the factorization of a function in Hp into a product

of an inner and an outer function carries over to

HP(Q).

The following result

follows from Theorem 4.7.3 in [13].

THEOREM

2.2.4. Each function f G

Hp(ft)

has a factorization f = OF, where

0 is inner and F is an outer function in HP(Q). Iff is not identically zero, then

0 and F are uniquely determined up to a trivial inner factor.

DEFINITION

2.2.5. Given two inner functions 0 and 0', we say that 0 is equiv-

alent to 0' {0 = 0') if there exists a trivial inner function I/J such that 0 =

ip0f.

Then, if 1 denotes the constant function in i72(ft) with values equal to 1, 0

is trivial if and only if 0 = 1.

DEFINITION

2.2.6. An inner function S with no zeros is called a singular

function.

THEOREM

2.2.7. (i) Given any positive measure v onT singular relative to

arc-length, there exists a unique (up to equivalence) singular function Su such

that for all z E Q

log|Sv(s)| = / |^(C, z)dv{Q) + h(z),

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

every component of T.

(ii) If S is a singular function, then there exist a unique positive measure v on

T singular relative to arc-length, such that S = Su.

We call v the representing measure of Su.

It is known ([13] Proposition 4.7.1) that if a function in #°°(fJ) is not iden-

tically zero, and if {an}^L1 is the sequence of its zeros repeated according to

multiplicity, then for each z G O we have J^^Li g(z, an) oo, and the conver-

gence is uniform on compact subsets of f2—{an}'^=1. Moreover ([18] Theorem 3.1)

the uniform convergence on compact subsets of U — {an}^°=1 of Y^?=i9(zian)

is equivalent to the convergence of ^2^=i dist(an,T).

DEFINITION

2.2.8. A function // : ft — N = {0,1,2,...} is said to be a

Blaschke function if the series YlaeQ ^(

a

)

dist(a

T) is convergent.