Fock space and Fock-Toeplitz operators Pedro Tiago Costa Soares 1 Fock space The work on Fock space has a long list of contributions, see e.g. [1, 8, 14, 36]. 1.1 Properties of the Fock space To begin we define the Fock space and we prove some elementary properties. Consider the measure 2 dµ(z) := e−|z| dA(z), where dA is the Lebesgue area measure in C. Let 1 p < + . A measurable function f : C C belong to the Fock space Fp(C) if f is entire and ≤ ∞ → 1/p p f p := f(z) dµ(z) < + . k k ZC | | ∞ The conjugate exponent of p is denoted by p′. If p = 1, let p′ := + . If1 <p< + , let p′ be the unique complex number such that 1/p + 1/p′ = 1. ∞ ∞ q 1/q′ It follows from the H¨older inequality that if 1 p q < + and if f F (C), then f p π f q. Therefore, we have that ≤ ≤ ∞ ∈ k k ≤ k k Fq(C) Fp(C) F1(C), 1 p q < + . ⊆ ⊆ ≤ ≤ ∞ In the following theorem we show that the point-evaluation functionals are continuous on the Fock space. As a consequence, the Fock space is a reproducing kernel Hilbert space. Theorem 1.1.2. Let 1 p< + and z C. If f Fp(C), then ≤ ∞ ∈ ∈ f(z) C f , (1) | |≤ z,pk kp where C > 0 is a constant which only depends on z and p. If K C is a nonempty compact, then z,p ⊆ exist a constant CK,p > 0, which only depends on K and p, such that sup f(z) CK,p f p. (2) z∈K{| |} ≤ k k We show that the Fock space is identified with a closed subspace of Lp(C,dµ). Hence, we have the following proposition. Theorem 1.1.4. If 1 p< + , then Fp(C) is a Banach space. ≤ ∞ The next proposition is helpful, for example to prove that the Taylor series of f converge weakly to f p Proposition 1.1.5. Let 1 <p< + and fn be a sequence in F (C). Then fn weakly converge to p ∞ f F (C) if and only if there exist C 0 such that fn p C and fn f, uniformly on the compacts subsets∈ of C. ≥ k k ≤ → The set of all polynomials in the complex variable z is denoted by P[z]. It is clear that P[z] is contained in the Fock space. In the next section we will prove that P[z] is dense in F2(C). Proposition 1.1.7. If 1 p< + , then P[z] Fp(C). ≤ ∞ ⊂ 1 1.2 Fock Kernel and Projection In this section we define a reproducing kernel Hilbert space. First, we check that the Fock space F2(C) is a reproducing kernel Hilbert space and we show some elementary properties. Such properties reamin valid in any reproducing kernel Hilbert space, see e.g. [3, 34]. We explicitly compute the reproducing kernel in the Fock space, which is said to be the Fock kernel. A space of functions H over a domain (open, connected and nonempty set) U C with complex output is said to be a reproducing kernel Hilbert space if H is a Hilbert space and if the⊂ point-evaluations functionals are continuous for each z U. It follows from the Riesz Theorem that exist a unique k : U U C such that if y U, then∈ × → ∈ f(y)= f(.),k(.,y) and k(.,y) H. h i ∈ From Theorem 1.1.4, we know that the Fock space F2(C) is a closed subspace of a Hilbert space. Acoording to the Theorem 1.1.2, the point-evaluations functionals are continuous on the Fock spaces. Therefore F2(C) is a reproducing kernel Hilbert space. Let z C. From the Riesz Theorem, we conclude ∈ that exist a unique κz such that 2 f(z)= f,κz = f(w)κz(w)dµ(w), f F (C). h i ZC ∈ F2 C It’s clear that the orthogonal of κz z∈C is trivial. Therefore, κz z∈C is dense in ( ). The Fock kernel is given by { } { } K : C C C, K(z,w) := κ (w). × → z The following properties hold in the Fock space. Proposition 1.2.2. If z,w C, then the Fock kernel respect the following properties: (a) K(z,.) F2(C); ∈ (b) K(z,w)=∈ K(w,z); 2 2 (c) K(z,z)= K(.,z) 2 = K(z,.) 2; (d) K(z,w) isk the uniquek reproductionk k kernel of the Fock space. The orthogonal projection from L2(C,dµ) to F2(C) is denoted by P and it is said to be the Fock projection. The Fock projection is a linear integral operator and its kernel is the Fock kernel. Proposition 1.2.3. Let f L2(C,dµ). Then ∈ Pf(z)= K(z,w)f(w)dµ(w), z C. ZC ∈ Using the Green formulas (see [14, page 77]), we calculate the inner product of a function f F2(C) with zn, for every n N . ∈ ∈ 0 Proposition 1.2.4. Let n N . If f F2(C), then f,zn = πf (n)(0). ∈ 0 ∈ h i It follows from the previous Proposition and from the Taylor development that P[z] is dense in F2(C). Moreover, the set of all monomials zn/√πn! (n N ) is an orthonormal base for F2(C). ∈ 0 Proposition 1.2.6. Let e be a orthonormal base of F2(C), with n N . The Fock kernel is given by n ∈ 0 ∞ K(z,w)= e (w)e (z), z,w C. j j ∈ Xj=0 Corollary 1.2.7. The Fock kernel is explicitly given by K(z,w)= ewz/π, where z,w C. ∈ 2 1.3 Minimization problems The properties of a reproducing kernel Hilbert space allow us to study the following minimization problems. Let w,α C. We will consider the following subset of F2(C) ∈ M := f F2(C): f(w)= α . w,α ∈ The set Mw,α is nonempty, since it contains the constant functions. Proposition 1.3.1. Let w,α C. Then there exist a unique f0 Mw,α such that inf f 2 = f0 2. ∈ ∈ f∈Mw,α k k k k Furthermore 2 K(w,z) 2 |w| f (z)= α = αewz−|w| and f = α √πe− 2 . 0 K(w,w) k 0k2 | | Let w ,α C, with j = 0, 1,..,n. Define j j ∈ M := f F2(C): f(w )= α , j = 0, 1,..,n . ∈ j j Note that M is nonempty, because there is a interpolation polynomial that respect the interpolation conditions and it belongs to the Fock space. Proposition 1.3.2. Let w ,α C, with j = 0, 1,..,n and M defined above. Then exist a unique j j ∈ F M such that inf f 2 = F 2. Moreover exist constants cj C such that ∈ f∈M k k k k ∈ n F (z)= cjK(z,wj ). (3) Xj=0 2 Fock-Toeplitz operators The Fock-Toeplitz operators have been studied in the last decades, e.g. [6, 8, 11, 23, 25]. 2.1 Properties of the Fock-Toeplitz Operator In this section we define the Fock-Toeplitz, Fock-Hankel and Weyl operators and we show some of its properties. Let (X, Ω,ν) be a measurable space with the σ finite measure ν and let g : X X be − 2 → a measurable function. The multiplication operator Mg in L (X,dν) is given by Mgf := fg, where f L2(X,dν). ∈ The Fock-Toeplitz operator Tg with symbol g is the composition of the projection P and the multi- plication operator M defined for each f F2(C) such that gf L2(C,dµ), i.e. g ∈ ∈ T : F2(C) F2(C), T := PM , g D⊂ → g g where is the set of function f F2(C) such that gf L2(C,dµ). First, we study the continuity of the Fock-ToeplitzD operator with symbols∈ z and z. ∈ Proposition 2.1.2. If f P[z], then ∈ 1 ∂ T f = f. z π ∂z k k−1 k It follows from the Proposition 1.2.4 that Tzz 2 = k z 2/π = z 2/π, for every k N. Then T is a bounded linear operator. On the otherk hand,k we havek thatk T zkk k= (k + 1) zk , for∈ k N . z k z k2 k k2 ∈ 0 So Tz is not continuous. From now on we will only consider Fock-Toeplitz operators with essentially bounded symbols. Proposition 2.1.3. Let g and gk be an essentially bounded functions and let λk be a complex numbers, k = 1, 2. Then (a) Tλ1g1+λ2g2 = λ1Tg1 + λ2Tg2 ; (b) Tg g ∞; k ∗ k ≤ k k (c) Tg = Tg . 3 Let a C. The normalized Fock kernel is denoted by k and it is given by ∈ a 2 |a| K(z,a) eaz− 2 ka(z) := = , z C. (4) K(a,a) π ∈ 2 p |a| +2iIm(az) Let ϕa be given by ϕa := e 2 . It is clear that ϕa an essentially bounded function. The Fock-Toeplitz operator with symbol ϕa is called the Weyl operator and it is denoted by Wa, i.e. W : F2(C) F2(C),W := T . (5) a → a ϕa The Weyl operator is explicitly given by Waf(z) = ka(z)f(z a), where z C. We extend the Weyl 2 C 2 C − ∈ ∗ operator to L ( ,dµ). The Weyl operator is unitary in L ( ,dµ) and its adjoint is given by Wa f(z)= √πk−a(z)f(z + a), where z C.