In this section, we present some fundamental properties of reproducing kernel Hilbert spaces and we show that the Berezin transform acting on bounded operators is one-to-one. We start by some preliminaries.
Definition 1.1.1. Let H be a Hilbert space (over C) of complex valued functions on a set Ω ⊂ Cn. H is called a reproducing kernel Hilbert space (abbreviated by RKHS) if for every z ∈Ωthe linear evaluation map,δz :H −→C, defined byδz(f) = f(z)is continuous.
For a Hilbert spaceH we denote byh·,·ithe inner product onHand byk·kthe correspond-ing norm. We remark that H is a RKHS if and only if for everyz ∈ Ω there exists a unique functionKz ∈Hsuch that
f(z) = hf, Kzi, for allf ∈H. (1.1.1) Indeed, the necessity follows from the Riesz Representation Theorem and the sufficiency is direct:
|δz(f)|=|hf, Kzi| ≤ kfkkKzk.
Definition 1.1.2. The 2-variable function onΩ×Ωdefined by K(w, z) = Kz(w) is called the reproducing kernel ofH.
Note that, there are many Hilbert spaces of functions which are not RKHS. Indeed, whenΩ is an open connected subset of Cnthen the ordinary Lebesgue spaceL2(Ω)is not a RKHS. In fact, for anyz0 ∈Ωwe can construct a sequence{fn} ⊂Cc∞(Ω)such thatkfnkL2(Ω) = 1and fn(z0) = nfor alln∈Nand thus the evaluation map is not continuous.
19
20 CHAPTER 1. PRELIMINARIES The general theory of reproducing kernel Hilbert spaces was systematized in [6, 7] by N.
Aronszajn around 1948 and developed by L. Schwartz [150]. However, before N. Aronszajn there have been two trends in consideration of reproducing kernels. The first trend considered positive definite kernels, which were discovered by J. Mercer [133, 134] and E. H. Moore [135], and study the kernel itself. The kernels considered by J. Mercer were ,,continuous kernels of positive definite integral operators” [7]. They arose in the theory of integral equations as de-veloped by Hilbert and they were positive definite in the sense of E. H. Moore. Later on, E.
H. Moore [135, 136] considered such kernels on an abstract set and he discovered that to each positive definite kernel there corresponds a Hilbert space of functions to which these kernels has the reproducing kernel property (1.1.1). The second trend considered a Hilbert space of functions and the corresponding kernel is used as a tool in the study of these functions. In the investigation of S. Zaremba on boundary value problems for harmonic and biharmonic func-tions, he was the first to go through this trend and to find a reproducing kernel for the classes of these functions (c.f. [180]). Then the idea of reproducing kernels appeared in the work done by S. Bergman [36, 37]. He introduced reproducing kernels for the classes of harmonic and analytic functions and he called them kernel functions. In [44–46], S. Bergman and M. Schiffer used the original idea of Zaremba in applying the kernels to the solutions of boundary value problems. It turned out that these kernels are useful tools for solving boundary value problems of partial differential equations of elliptic type. Finally, N. Aronszajn [6] found that any repro-ducing kernel on a Hilbert space of functions is positive definite in the sense of E. H. Moore and thus formed the second link between these two trends. This link led to important results in the study of conformal mappings [38, 39, 41], integral equations [133], partial differential equations [40], invariant Riemannian metrics [42], in probability theory [121, 141, 142] and in other subjects in physics. For instant, in [70] J. Duchon formulated generalized smooth surface spline functions using the reproducing kernel Hilbert space technique in Sobolev space [1]. To-gether with the work of Lo´eve the theory of reproducing kernel Hilbert space has been used for a variety of applications, especially in data interpolation, signal processing, and smoothing (c.f.
[48, 140, 173]). Meanwhile, the notion of RKHS is quite used in numerous fields especially in solving boundary value problems [89–91], Tikhonov regularization [105, 159], image and video colorization [100].
In our study there are several reasons for introducing the abstract theory of reproducing kernel Hilbert spaces. The two main tasks in the thesis the Segal-Bargmann space and the Bergman spaces over bounded symmetric domains are special types of a RKHS. The corre-sponding kernels of these spaces share some fundamental properties. In particular, they can be calculated explicitly by finding an orthonormal system of functions in the corresponding space (c.f. Proposition 1.1.3 below). Moreover, the fact that the Berezin transform is one-to-one on bounded Toeplitz operators acting on a Bergman space follows from the fact that this transform is one-to-one on bounded operators acting on any RKHS of analytic functions whose kernel does not vanish on the diagonal ofΩ×Ω(c.f. Proposition 1.1.4 below).
Let us now present some basic properties of the reproducing kernelKin a RKHS. Applying
1.1. REPRODUCING KERNEL HILBERT SPACES 21 (1.1.1) to the functionKzatw, we get
K(w, z) =Kz(w) =hKz, Kwi forz, w ∈Ω.
Therefore, the reproducing kernel satisfies:
K(w, z) = K(z, w), kKzk=p
K(z, z), and |K(w, z)| ≤p
K(z, z)p
K(w, w).
Furthermore, for a fixed z0 ∈ Ωit follows thatK(z0, z0) = 0if and only iff(z0) = 0for all f ∈H. Again by the reproducing kernel property (1.1.1) it is easy to check that span{Kz}z∈Ω
is dense inH.
It is important to notice that if the RKHSH with kernelK is a subspace of a larger Hilbert space thenK determines the orthogonal projection toH.
Proposition 1.1.1. [151] Let(L,h·,·i)be a Hilbert space of complex valued functions defined onΩ. Equipped with the norm inhereted fromL, suppose there is a reproducing kernel Hilbert spaceH ⊂L. Then the orthogonal projectionP :L−→His given by
[P f](z) = hf, Kzi, f ∈L, z ∈Ω. (1.1.2) Proof. Letf ∈L, then[P f](z) =hP f, Kzi=hf, P Kzi=hf, Kzi.
The next proposition shows that the reproducing kernel of a separable RKHS can be de-scribed explicitly in terms of an orthonormal basis for the space.
Proposition 1.1.2. [151] Let H be a separable RKHS. Then for any complete orthonormal system{en}n∈Nwe have
K(w, z) =X
n∈N
en(w)en(z), (1.1.3)
where the series converges absolutely onΩ×Ω.
Proof. Since{en}n∈N is a complete orthonormal system then for anyz ∈ Ω we have Kz = P
n∈NhKz, enien, where the series converges inH. Hence Kz(w) =hKz, Kwi=X
n∈N
hKz, enihen, Kwi=X
n∈N
en(z)en(w).
The absolute convergence follows by Cauchy-Schwartz inequality and Parseval identity:
X
n∈N
|en(z)en(w)| ≤(X
n∈N
|hKz, eni|2)12(X
n∈N
|hen, Kwi|2)12 =kKzkkKwk. (1.1.4)
Hereafter, we confine our discussion to the case whereΩis an open connected domain in Cn.
22 CHAPTER 1. PRELIMINARIES Proposition 1.1.3. LetH be a RKHS of analytic functions onΩ. Then the kernelK(w, z) is analytic inwand anti-analytic inz. Moreover, H is separable and the convergence in (1.1.3) is uniform on compact sets ofΩ×Ω.
Proof. The first part of the proposition follows since Kz ∈ H is analytic and K(w, z) = K(z, w). Note that, for any dense sequence{zm}inΩthe corresponding sequence of functions {Kzm}is a total set inHthereforeHis separable. Moreover, ifS ⊂Ωis compact we know that for anyf ∈Hthe complex valued functionz 7−→f(z) = hf, Kziis analytic hence continuous on Ω. Therefore, the Hilbert space-valued function z 7−→ Kz is weakly continuous. So that, {Kz, z ∈ S} is weakly compact hence strongly bounded. This means thatsup{kKzk, z ∈ S}
is finite and the uniform convergence follows from (1.1.4).
The bounded operators on a RKHS admit an interesting representation via the so-called Berezin transform. This transform plays an important role in the theory of Bergman spaces as well as in the solution of the heat equation (c.f. Section 3). Let us start by recalling the definition of this transform.
Definition 1.1.3. Let H be a RKHS such that K(z, z) > 0for all z ∈ Ω. Let T be a linear operator acting on the linear space
LH := span{kz := 1
pK(z, z)Kz}z∈Ω,
of the normalized reproducing kernel functions with values inH. Then the Berezin transform of T, denoted byTe, is the complex valued function defined onΩby:
Te(z) = hT kz, kzi.
The fact that the Berezin transform acting on the bounded operators of a RKHS forms a representation of these operators is a special case of a more general statement given in the next proposition. It is sufficient to find a Hilbert subspaceH0inH such thatLH ⊂H0is dense then for any T ∈ L(H0, H) one hasTe = 0if and only if T = 0. This idea was used in [14] for the case of the Segal-Bargmann space to prove that the Berezin transform acting on Toeplitz operators with symbols having a certain growth at infinity is one-to-one. In the next chapter, we use a similar argument to that in [14] to prove that two Toeplitz operators with reasonable symbol growth at infinity commute on the space of holomorphic polynomials if and only if the Berezin transform of their commutator vanishes (c.f. Corollary 2.2.1).
Proposition 1.1.4. Let H be a RKHS of analytic functions onΩ. Suppose there is a Hilbert space H0 of analytic functions on Ω such thatLH ⊂ H0 is dense in H0. Then the mapping T −→ Te is an injective linear operator from L(H0, H) into the real-analytic functions in L∞(Ω).
Proof. Using the continuity ofT we have
|Te(z)|=|hT kz, kzi| ≤ kT kzk ≤ kTk,
1.1. REPRODUCING KERNEL HILBERT SPACES 23 henceTe∈L∞(Ω). Moreover, sinceK(w, z)is analytic inwconjugate-analytic inzthenkz is real-analytic but T is continuous henceTe(z)is real-analytic too. To prove the injectivity, we define the two-variable function onΩ×Ω
KT(z, w) := hT Kw, Kzi= (T Kw)(z).
SinceT Kw ∈H for allw∈Ωthe kernelKT is analytic inzbut
KT(z, w) =hKw, T?Kzi=hT?Kz, Kwi= (T?Kz)(w),
hence KT is anti-analytic inw. By assumption KT vanishes on the diagonal ofΩ×Ω. As a consequence (c.f. p. 371 in [124]), the map KT is identically zero onΩ×ΩhenceT Kw ≡ 0 for allw∈Ω. SinceLH ⊂H0 is dense andT is continuous onH0, it follows thatT = 0.
At the end of this section we give some classical examples of reproducing kernel Hilbert spaces.
Example 1.1.1. It is well known that the Peter-Wiener space P Wr which consists of square integrable functions on R such that their Fourier transformation vanishes outside the band [−r, r]is a RKHS (c.f. [179]). We generalize this fact to arbitrary dimensions and calculate the reproducing kernel of the corresponding space. For eachr > 0, we define the generalized Peter-Wiener spaceP Wr onRnto be (c.f. [178])
P Wr(Rn) := n
ϕ:Cn−→C|ϕis entire,ϕ|Rn ∈L2(Rn),s. t. for any >0,
∃ Cwith|ϕ(z)| ≤Ce(1+)|z|,∀z ∈Cn o
. According to the generalized Paley-Wiener theorem (c.f. [156, 178]) we know that
F(L2(B(0, r))) =P Wr(Rn),
where B(0, r) is the closed ball in Rn with radius r centered at the origin, and F is the n-dimensional Fourier transformation. Since F is an isometry, we can endow the space P Wr
with the following inner product
hf, giP Wr =hF−1f,F−1giL2(B(0,r)). It follows directly thatP Wr is a Hilbert space and for everyz ∈Cn
|ϕ(z)|= (2π)−n Z
B(0,r)
e−izξ(F−1ϕ)(ξ)dξ
≤(2π)−np
|B(0, r)|er|z|kF−1ϕkL2(B(0,r)) =CkϕkP Wr,
which shows thatP Wr is a RKHS. Moreover, for anyϕ∈P Wrand anyz ∈Cnthe identity ϕ(z) =hF−1ϕ, eiz(·)iL2(B(0,r)) =hϕ,Feiz(·)iP Wr,
24 CHAPTER 1. PRELIMINARIES shows that the reproducing kernel of P Wr is K(w, z) = F(eiz(·))(w). Using induction on n, one can prove that (c.f. [65])
F(eiz(·))(w) = ( r 2π)n2Jn
2(r|z−w|)
|z−w|n2 , whereJvis the Bessel function of the first kind of orderv.
Other examples of RKHS are the Bergman spaces, the Sobolev spacesHs(Rd)for alls > d2 [47], the Szeg¨o spacesAl2(∂Ω)and many others [151]. In the next section we discuss briefly the Bergman spaces over an open domain.