Hermitian Weighted Composition Operators on the Fock-type Space F 2 α(c N )

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1 Applied Mathematical Sciences, Vol. 9, 2015, no. 61, HIKARI Ltd, Hermitian Weighted Composition Operators on the Fock-type Space F 2 (C N ) Yong Ying Su Guangzhou Vocational College of Technology and Business Guangzhou, Guangdong, , China Zhi Jie Jiang School of Science, Sichuan University of Science and Engineering Zigong, Sichuan, , China Copyright c 2014 Yong Ying Su and Zhi Jie Jiang. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper we identify those weighted composition operators on Fock-type space that are Hermitian or Hermitian isometric. Mathematics Subject Classification: Primary 47B38, 46E10; Secondary 30D55 Keywords: Fock-type space, weighted composition operator, Hermitian operator, Hermitian isometric operator 1. Introduction Let z = (z 1,..., z N ) and w = (w 1,..., w N ) be two points in C N, z, w = N k=1 z kw k and z = z, z. Let B N = {z C N : z < 1} be the open unit ball, S = B N the boundary of the unit ball B N, dv (z) the Lebesgue volume measure on C N and H(C N ) the space of all holomorphic functions on C N (entire functions). For > 0 the Fock-type space F(C 2 N ) is the space of

2 3038 Yong Ying Su and Zhi Jie Jiang all entire functions f on C N for which f 2 = f(z) 2 e z 2 dv (z) <. π C N When = 1/2, F(C 2 N ) = F 2 (C N ) is called the Fock space. It is clear that F(C 2 N ) is a Hilbert space with the inner product f, g = f(z)g(z)e z 2 dv (z). π C N The problems for the Fock-type space, such as an interpolation sequence or a sampling set, have been studied(see, for example, [2, 13]), and several concrete operators on Fock-type or Fock space such as Toeplitz operators, Hankel operators and weighted composition operators have been considered(see, for example, [1, 7, 9, 16] ) and the references therein). Let ϕ : C N C N be an entire mapping and ψ H(C N ). The weighted composition operator W ϕ,ψ is defined by W ϕ,ψ f = ψ (f ϕ). If ψ 1 on C N, then W ϕ,ψ = C ϕ is called the composition operator. Although many papers discussed weighted composition operators over the past few decades(see,e.g.[10]- [12],[15]), weighted composition operators have usually arisen answering other questions related to operators on spaces of holomorphic functions, such as questions about multiplication operators or composition operators. For example, weighted composition operators arise in the characterization of commutators of analytic Toeplitz operators (see[4]) and in the adjoints of composition operators (see, for example [5]). Forelli [8] proved that the only isometry of Hardy space H p for p 2 is weighted composition operator. Recently, Carswell et al.[3] have determined when composition operators are bounded and compact on Fock space, and they have obtained the following result. Theorem A. Let ϕ : C N C N be an entire mapping. (a) If the operator C ϕ is bounded on F 2 (C N ), then ϕ(z) = Az + b, where A is an N N matrix and b is an N 1 vector. (b) If the operator C ϕ is compact on F 2 (C N ), then ϕ(z) = Az + b, where A < 1. Ueki [17] has given some necessary and sufficient conditions for weighted composition operators on Fock-type space F 2 (C) to be bounded and compact. Quite recently, Du[7] has obtained a complete description of Schatten class weighted composition operators on F 2 (C N ). By [6], we know that for composition operators on Hardy space H 2 the situation is trivial: the only Hermitian composition operators are induced by symbol ϕ(z) = rz with 1 r 1. In this paper we shall examine those weighted composition operators on Focktype space which are Hermitian and Hermitian isometric.

3 Hermitian weighted composition operators Main results We first prove several auxiliary lemmas, which will be used in the proofs of main results. Lemma 2.1. For every multi-index β let e β (z) = β β! zβ. Then {e β } β Γ forms an orthonormal basis of F 2 (C N ), where Γ is the set of all multi-indices. Proof. Let β = (β 1,..., β N ). We first calculate the norm e β of e β. By the definition of the norm on F(C 2 N ), we have that e β 2 β = z β 2 e z 2 dv (z) π β! C N β N = π β! (2π)N r 2β k+1 k e r2 k drk = = 1, k=1 π β β! (2π)N 1 2 N 0 N k=1 β k! β k+1 from which it follows that e β is a unit vector. Now we prove that the vectors in {e β } are mutually orthogonal. Let β and γ be two multi-indices and β γ. By integration in polar coordinates in [14], we have that e β, e γ = z β z γ e z 2 dv (z) π C ( N = 2N r 2N+ β + γ e r2 dr ζ β ζ γ dσ(ζ). π 0 Proposition in [14] shows that S ζβ ζ γ dσ(ζ) = 0. Hence we obtain e β, e γ = 0. Assume f(z) = β a βz β F(C 2 N ). Then f is in F(C 2 N ) if and only if f 2 = f, f = a β z β, β β By a simple calculation, we get that f, e γ e γ = γ β a β z β = β a β β! β e β, S a β 2 β! β <.

4 3040 Yong Ying Su and Zhi Jie Jiang which implies that n f, e β e β f 2 = β =0 β =n+1 β! a β e β β 2 = β =n+1 a β 2 β! β 0, as n. Hence for each f F(C 2 N ) we have that γ f, e γ e γ converges to f in the norm topology of F(C 2 N ). This completes the proof. Since F 2 (C N ) is a Hilbert space, the Riesz representation shows that it has the reproducing kernel function. Lemma 2.2. The reproducing kernel function of F 2 (C N ) is given by K w (z) = e z,w. Proof. By Theorem in [18] and Lemma 2.1, we have K w (z) = β e β (z)e β (w) = β from which the desired result follows. β β! zβ w β = e z,w, Lemma 2.3. Let ϕ : C N C N be an entire mapping and ψ H(C N ) and W ϕ,ψ be bounded on F(C 2 N ). Then Wϕ,ψ K w = ψ(w)k ϕ(w). Proof. For each z C N, we have This completes the proof. W ϕ,ψk w (z) = W ϕ,ψk w, K z = K w, W ϕ,ψ K z = W ϕ,ψ K z, K w = ψ(w)k z (ϕ(w)) = ψ(w)k ϕ(w) (z). Lemma 2.4. Let ϕ : C N C N be an entire mapping and ψ H(C N ). Then the bounded operator W ϕ,ψ : F 2 (C N ) F 2 (C N ) is Hermitian if and only if W ϕ,ψ K w = W ϕ,ψ K w for all w C N. Proof. If W ϕ,ψ is Hermitian, that is, W ϕ,ψ = W ϕ,ψ, then W ϕ,ψk w = W ϕ,ψ K w. Conversely, for w C N and f F 2 (C N ), we have W ϕ,ψ f(w) = W ϕ,ψ f, K w = f, W ϕ,ψk w = f, W ϕ,ψ K w = W ϕ,ψf, K w = W ϕ,ψf(w). It follows that W ϕ,ψ f = W ϕ,ψ f for each f F 2 (C N ), and then W ϕ,ψ = W ϕ,ψ. We are ready to investigate which combinations of weights ψ and entire mappings ϕ induce Hermitian weighted composition operators. It is not surprising that self-adjointness significantly restricts the possible symbols for the weighted composition operators.

5 Hermitian weighted composition operators 3041 Theorem 2.5. Let ϕ : C N C N be an entire mapping and ψ H(C N ). If the bounded operator W ϕ,ψ : F 2 (C N ) F 2 (C N ) is Hermitian on F 2 (C N ), then ψ(0) is real and ϕ(z) = Az + b and ψ(z) = ae z,b, where A is an N N real Hermitian matrix, a = ψ(0) and b = ϕ(0). Conversely, let a be a real number and A an N N real Hermitian matrix, and let b be an N 1 vector. If ϕ(z) = Az + b and ψ(z) = ae z,b, then W ϕ,ψ is Hermitian on F 2 (C N ). Proof. By Lemma 2.4, we have W ϕ,ψ K w = W ϕ,ψ K w for every w C N. Then using Lemma 2.3, we get ψ(z)k w (ϕ(z)) = ψ(z)e ϕ(z),w = ψ(w)k ϕ(w) (z) = ψ(w)e z,ϕ(w) (1) for all z, w C N. Particularly, letting w = 0 in (1), we get ψ(z) = ψ(0)e z,ϕ(0) for all z C N. Setting z = 0 implies that ψ(0) = ψ(0), so that ψ(0) is real. Defining a and b by a = ψ(0) and b = ϕ(0), we can write ψ as ψ(z) = ae z,b. Replacing ψ by ψ(z) = ae z,b in (1), we have e w,b e z,ϕ(w) = e z,b + ϕ(z),w. (2) Since using a simple calculation implies that e w,b = e w,b, by (2) we obtain Then e w,b + z,ϕ(w) = e z,b + ϕ(z),w. w, b + z, ϕ(w) = z, b + ϕ(z), w + 2πi k(z, w), k(z, w) N. (3) From (3), we obtain z, ϕ(w) b = ϕ(z) b, w + 2πi k(z, w), k(z, w) N. (4) Since b = ϕ(0), we can assume that ϕ(0) = 0. So, (4) becomes z, ϕ(w) = ϕ(z), w + 2πi k(z, w), (5) which implies that k(z, w) is continuous. From this and k(0, 0) = 0, it must have k(z, w) = 0 for each z, w C N. Let e j, j = 1,..., N, denote the ordered N-tuple that has 1 in the jth spot and 0 everywhere else. For fixed j, taking w = e j and letting ϕ(z) = (ϕ 1 (z),..., ϕ N (z)) in (5), we get ϕ j (z) = z 1 ϕ 1 (e j ) + + z N ϕ N (e j ). (6) Once again setting z = e k in (6), we obtain that ϕ j (e k ) = ϕ j (e k ), which shows that ϕ j (e k ) is real for each j, k = 1,..., N. So, ϕ(z) = Az. Once again by (5), we conclude that A = A. Therefore, ϕ(z) = Az +b, where A is real Hermitian and b = ϕ(0).

6 3042 Yong Ying Su and Zhi Jie Jiang Conversely, if a a real number, A an N N real Hermitian matrix, and b an N 1 vector, are such that ϕ(z) = Az + b and ψ(z) = ae z,b, then a straightforward calculation shows that W ϕ,ψ K w = W ϕ,ψ K w for all w, by Lemma 2.4, which means W ϕ,ψ is Hermitian on F 2 (C N ). As an application of Theorem 2.5, we have the following result. Corollary 2.6. Let ϕ : C N C N be an entire mapping. If the bounded operator C ϕ : F 2 (C N ) F 2 (C N ) is Hermitian, then ϕ(z) = Az, where A is an N N real Hermitian matrix. Conversely, if ϕ(z) = Az and A is an N N real Hermitian matrix, then C ϕ : F 2 (C N ) F 2 (C N ) is Hermitian. We begin considering when weights ψ and entire mappings ϕ give rise to Hermitian isometric weighted composition operators. Theorem 2.7. Assume there exists a point z 0 C N such that Az 0 + z 0 = b. If the bounded operator W ϕ,ψ : F 2 (C N ) F 2 (C N ) is Hermitian isometric, then ϕ(z) = Az and ψ(z) a, where A is an N N real Hermitian unitary matrix and a = ±1. Conversely, let A be an N N real Hermitian unitary matrix and a = ±1. If ϕ(z) = Az and ψ(z) a, then W ϕ,ψ : F 2 (C N ) F 2 (C N ) is Hermitian isometric. Proof. If W ϕ,ψ is Hermitian isometric, then it follows that for all f F 2 (C N ), (W ϕ,ψ ) 2 f = f. Then by Theorem 2.5, we have ψ(z)ψ(ϕ(z))f(ϕ(ϕ(z))) = a 2 e z+ϕ(z),b f(ϕ(ϕ(z))) = f(z). (7) Taking f = 1 in (7), we get a 2 e z+ϕ(z),b = 1. (8) Setting z = 0 in (8) shows that a = ±e 2 ϕ(0) 2. Replacing a by this value in (8), we see that ϕ(0) 2 + z + ϕ(z), b = 2k(z)πi, k(z) N, from which, b = ϕ(0) and ϕ(z) = Az + b, it follows that Az + z, b = 0. Noting that there is a point z 0 in C N such that Az 0 + z 0 = b, we obtain that b = ϕ(0) = 0. Hence ϕ(z) = Az, ψ(z) = a = ±1 and (7) becomes f(a 2 z) = f(z) for all f F(C 2 N ). Since f(z) = z k F(C 2 N ) for each k = 1,..., N, we conclude that A 2 = I. From this and Theorem 2.5, it follows that A is a real Hermitian unitary matrix. Conversely, if weighted composition operators W ϕ,ψ satisfying ϕ(z) = Az and ψ(z) a, where A is a real Hermitian unitary matrix and a = ±1, then W ϕ,ψ is Hermitian and Wϕ,ψ 2 = I, which means W ϕ,ψ is Hermitian isometric on F(C 2 N ). Acknowledgments. This work is supported by the National Natural Science Foundation of China (No ), the Sichuan Province University Key

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