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.
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