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[ ¯ = 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 + ∩ L1 loc (R) : ¯ ¯ ∈ N + } for N + and similarly for other spaces. If Θ is a meromorphic inner function, K+ Θ = N + [ ¯ 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 (¯). 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 ∈ 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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