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] = KΘ 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, KΘ

+

= 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+).