3. FUNCTION THEORETIC BACKGROUND 15
3.5. Toeplitz kernels. Recall that the Toeplitz operator TU with a symbol
U
L∞(R)
is the map
TU :
H2

H2,
F P+(UF ),
where P+ is the Riesz projection, i.e. the orthogonal projection from L2(R) onto
the Hardy space H2 = H2(C+). As was mentioned before, via non-tangential
boundary values H2(C+) can be identified with a closed subspace of L2(R) which
makes such a projection correctly defined.
We will use the following notation for kernels of Toeplitz operators (or Toeplitz
kernels in H2):
N[U] = kerTU .
An important observation is that N[
¯
Θ] = if Θ is an inner function. Along with
H2-kernels, one may consider Toeplitz kernels N p[U] in other Hardy classes Hp,
the kernel N 1,∞[U] in the ‘weak’ space H1,∞ = Hp ∩L1,∞, 0 p 1, or the kernel
in the Smirnov class N +(C+), defined as
N
+[U]
= {f N
+
Lloc(R)
1
:
¯
U
¯
f N
+}
for N
+
and similarly for other spaces.
If Θ is a meromorphic inner function,
+
= N
+[
¯
Θ] can also be considered.
For more on such kernels see chapters 7 and 8.
3.6. Entire functions and de Branges’ spaces. Recall that an entire func-
tion F (z) is said to be of exponential type at most a 0 if |F (z)| =
O(ea|z|)
as
z ∞. The infimum of such a is the exponential type of F .
A classical theorem of Krein gives a connection between the Smirnov-
Nevanlinna class N
+(C+)
and the Cartwright class Ca consisting of all entire func-
tions F (z) of exponential type 2πa that satisfy
log |F (t)|
LΠ.1
An entire function F (z) belongs to the Cartwright class Ca if and only if
F (z)
S−2πa(z)
N
+(C+),
and
F #(z)
S−2πa(z)
N
+(C+),
where F
#(z)
= F (¯). z
As an immediate consequence one obtains a connection between the Hardy
space
H2(C+)
and the Paley-Wiener space PWa. Namely, an entire function F (z)
belongs to the Paley-Wiener class PWa if and only if
F (z)
S−2πa(z)

H2(C+),
F
#(z)
S−2πa(z)

H2(C+).
The definition of the de Branges spaces of entire functions may be viewed as
a generalization of the last definition of the Payley-Wiener spaces with
S−a(z)
re-
placed by a more general entire function. Consider an entire function E(z) satisfying
the inequality
|E(z)| |E(¯)|, z z C+.
Such functions are usually called de Branges functions. The de Branges space BE
associated with E(z) is defined to be the space of entire functions F (z) satisfying
F (z)
E(z)

H2(C+),
F
#(z)
E(z)

H2(C+).
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