Applied Mathematical Sciences, Vol. 9, 2015, no. 61, 3037 - 3043
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.136
Hermitian Weighted Composition Operators
on the Fock-type Space Fα2(CN )
Yong Ying Su
Guangzhou Vocational College of Technology and Business
Guangzhou, Guangdong, 511442, China
Zhi Jie Jiang
School of Science, Sichuan University of Science and Engineering
Zigong, Sichuan, 643000, China
c 2014 Yong Ying Su and Zhi Jie Jiang. This article is distributed under the
Copyright 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 = (z1 , ..., zN ) p
and w = (w1 , ..., wN ) be two points in CN , hz, wi =
PN
hz, zi. Let BN = {z ∈ CN : |z| < 1} be the open
k=1 zk w k and |z| =
unit ball, S = ∂BN the boundary of the unit ball BN , dV (z) the Lebesgue
volume measure on CN and H(CN ) the space of all holomorphic functions on
CN (entire functions). For α > 0 the Fock-type space Fα2 (CN ) is the space of
3038
Yong Ying Su and Zhi Jie Jiang
all entire functions f on CN for which
α N Z
2
2
kf k =
|f (z)|2 e−α|z| dV (z) < ∞.
π
CN
When α = 1/2, Fα2 (CN ) = F 2 (CN ) is called the Fock space. It is clear that
Fα2 (CN ) is a Hilbert space with the inner product
α N Z
2
hf, gi =
f (z)g(z)e−α|z| dV (z).
π
CN
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 : CN → CN be an entire mapping and ψ ∈ H(CN ). The weighted
composition operator Wϕ,ψ is defined by Wϕ,ψ f = ψ · (f ◦ ϕ). If ψ ≡ 1 on CN ,
then Wϕ,ψ = Cϕ 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 6= 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 : CN → CN be an entire mapping.
(a) If the operator Cϕ is bounded on F 2 (CN ), then ϕ(z) = Az + b, where A
is an N × N matrix and b is an N × 1 vector.
(b) If the operator Cϕ is compact on F 2 (CN ), then ϕ(z) = Az + b, where
kAk < 1.
Ueki [17] has given some necessary and sufficient conditions for weighted composition operators on Fock-type space Fα2 (C) to be bounded and compact.
Quite recently, Du[7] has obtained a complete description of Schatten class
weighted composition operators on Fα2 (CN ). 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.
Hermitian weighted composition operators
3039
2. 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
s
α|β| β
z .
eβ (z) =
β!
Then {eβ }β∈Γ forms an orthonormal basis of Fα2 (CN ), where Γ is the set of
all multi-indices.
Proof. Let β = (β1 , ..., βN ). We first calculate the norm keβ k of eβ . By the
definition of the norm on Fα2 (CN ), we have that
α N α|β| Z
2
2
keβ k =
|z β |2 e−α|z| dV (z)
π
β! CN
N Z ∞
α N α|β|
Y
2
N
(2π)
rk2βk +1 e−αrk drk
=
π
β!
k=1 0
=
N
1 Y βk !
(2π) N
β!
2 k=1 αβk +1
α N α|β|
π
N
= 1,
from which it follows that eβ is a unit vector.
Now we prove that the vectors in {eβ } are mutually orthogonal. Let β and
γ be two multi-indices and β 6= γ. By integration in polar coordinates in [14],
we have that
α N Z
2
heβ , eγ i =
z β z̄ γ e−α|z| dV (z)
π
CN
Z
α N Z ∞
γ
2N +|β|+|γ| −αr2
dr ζ β ζ dσ(ζ).
= 2N
r
e
π
0
S
R β γ
Proposition 1.4.8 in [14] shows that S ζ ζ dσ(ζ) = 0. Hence we obtain
heβ , eγ i = 0.
P
Assume f (z) = β aβ z β ∈ Fα2 (CN ). Then f is in Fα2 (CN ) if and only if
DX
E X
X
β!
kf k2 = hf, f i =
aβ z β ,
aβ z β =
|aβ |2 |β| < ∞.
α
β
β
β
By a simple calculation, we get that
r
X
γ
hf, eγ ieγ =
X
β
aβ
β!
eβ ,
α|β|
3040
Yong Ying Su and Zhi Jie Jiang
which implies that
r
∞
n
∞
2 X
X
X
β! β!
2
aβ
hf, eβ ieβ − f = eβ =
|aβ |2 |β| → 0,
|β|
α
α
|β|=n+1
|β|=0
|β|=n+1
P
as n → ∞. Hence for each f ∈ Fα2 (CN ) we have that γ hf, eγ ieγ converges
to f in the norm topology of Fα2 (CN ). This completes the proof. Since Fα2 (CN ) is a Hilbert space, the Riesz representation shows that it has
the reproducing kernel function.
Lemma 2.2. The reproducing kernel function of Fα2 (CN ) is given by Kw (z) =
eαhz,wi .
Proof. By Theorem 4.1.5 in [18] and Lemma 2.1, we have
X
X α|β|
Kw (z) =
eβ (z)eβ (w) =
z β wβ = eαhz,wi ,
β!
β
β
from which the desired result follows.
Lemma 2.3. Let ϕ : CN → CN be an entire mapping and ψ ∈ H(CN ) and
∗
Wϕ,ψ be bounded on Fα2 (CN ). Then Wϕ,ψ
Kw = ψ(w)Kϕ(w) .
Proof. For each z ∈ CN , we have
∗
∗
Wϕ,ψ
Kw (z) = hWϕ,ψ
Kw , Kz i = hKw , Wϕ,ψ Kz i
= hWϕ,ψ Kz , Kw i = ψ(w)Kz (ϕ(w))
= ψ(w)Kϕ(w) (z).
This completes the proof . Lemma 2.4. Let ϕ : CN → CN be an entire mapping and ψ ∈ H(CN ). Then
the bounded operator Wϕ,ψ : Fα2 (CN ) → Fα2 (CN ) is Hermitian if and only if
∗
Wϕ,ψ Kw = Wϕ,ψ
Kw for all w ∈ CN .
∗
∗
Proof. If Wϕ,ψ is Hermitian, that is, Wϕ,ψ = Wϕ,ψ
, then Wϕ,ψ Kw = Wϕ,ψ
Kw .
N
2
N
Conversely, for w ∈ C and f ∈ Fα (C ), we have
∗
Wϕ,ψ f (w) = hWϕ,ψ f, Kw i = hf, Wϕ,ψ
Kw i
∗
= hf, Wϕ,ψ Kw i = hWϕ,ψ
f, Kw i
∗
= Wϕ,ψ
f (w).
∗
∗
It follows that Wϕ,ψ f = Wϕ,ψ
f for each f ∈ Fα2 (CN ), and then Wϕ,ψ = Wϕ,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.
Hermitian weighted composition operators
3041
Theorem 2.5. Let ϕ : CN → CN be an entire mapping and ψ ∈ H(CN ). If
the bounded operator Wϕ,ψ : Fα2 (CN ) → Fα2 (CN ) is Hermitian on Fα2 (CN ),
then ψ(0) is real and ϕ(z) = Az + b and ψ(z) = aeαhz,bi , 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) = aeαhz,bi , then Wϕ,ψ
is Hermitian on Fα2 (CN ).
∗
Proof. By Lemma 2.4, we have Wϕ,ψ Kw = Wϕ,ψ
Kw for every w ∈ CN . Then
using Lemma 2.3, we get
ψ(z)Kw (ϕ(z)) = ψ(z)eαhϕ(z),wi = ψ(w)Kϕ(w) (z) = ψ(w)eαhz,ϕ(w)i
(1)
for all z, w ∈ CN .
Particularly, letting w = 0 in (1), we get ψ(z) = ψ(0)eαhz,ϕ(0)i for all z ∈ CN .
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) = aeαhz,bi .
Replacing ψ by ψ(z) = aeαhz,bi in (1), we have
eαhw,bi eαhz,ϕ(w)i = eαhz,bi+αhϕ(z),wi .
(2)
Since using a simple calculation implies that eαhw,bi = eαhw,bi , by (2) we obtain
eαhw,bi+αhz,ϕ(w)i = eαhz,bi+αhϕ(z),wi .
Then
hw, bi + hz, ϕ(w)i = hz, bi + hϕ(z), wi +
2πi
k(z, w), k(z, w) ∈ N.
α
(3)
From (3), we obtain
2πi
k(z, w), k(z, w) ∈ N.
α
Since b = ϕ(0), we can assume that ϕ(0) = 0. So, (4) becomes
hz, ϕ(w) − bi = hϕ(z) − b, wi +
(4)
2πi
k(z, w),
(5)
α
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 ∈ CN .
Let ej , j = 1, ..., N , denote the ordered N -tuple that has 1 in the jth
spot and 0 everywhere else. For fixed j, taking w = ej and letting ϕ(z) =
(ϕ1 (z), ..., ϕN (z)) in (5), we get
hz, ϕ(w)i = hϕ(z), wi +
ϕj (z) = z1 ϕ1 (ej ) + · · · + zN ϕN (ej ).
(6)
Once again setting z = ek in (6), we obtain that ϕj (ek ) = ϕj (ek ), which shows
that ϕj (ek ) 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).
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) = aeαhz,bi , then
∗
a straightforward calculation shows that Wϕ,ψ Kw = Wϕ,ψ
Kw for all w, by
N
2
Lemma 2.4, which means Wϕ,ψ is Hermitian on Fα (C ). As an application of Theorem 2.5, we have the following result.
Corollary 2.6. Let ϕ : CN → CN be an entire mapping. If the bounded
operator Cϕ : Fα2 (CN ) → Fα2 (CN ) 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
Cϕ : Fα2 (CN ) → Fα2 (CN ) 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 z0 ∈ CN such that Az0 + z0 = b.
If the bounded operator Wϕ,ψ : Fα2 (CN ) → Fα2 (CN ) 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 Wϕ,ψ : Fα2 (CN ) → Fα2 (CN ) is Hermitian
isometric.
Proof. If Wϕ,ψ is Hermitian isometric, then it follows that for all f ∈
Fα2 (CN ), (Wϕ,ψ )2 f = f . Then by Theorem 2.5, we have
ψ(z)ψ(ϕ(z))f (ϕ(ϕ(z))) = a2 eαhz+ϕ(z),bi f (ϕ(ϕ(z))) = f (z).
(7)
Taking f = 1 in (7), we get
a2 eαhz+ϕ(z),bi = 1.
α
(8)
2
Setting z = 0 in (8) shows that a = ±e− 2 |ϕ(0)| . Replacing a by this value in
(8), we see that
−α|ϕ(0)|2 + αhz + ϕ(z), bi = 2k(z)πi, k(z) ∈ N,
from which, b = ϕ(0) and ϕ(z) = Az + b, it follows that αhAz + z, bi = 0.
Noting that there is a point z0 in CN such that Az0 + z0 = b, we obtain
that b = ϕ(0) = 0. Hence ϕ(z) = Az, ψ(z) = a = ±1 and (7) becomes
f (A2 z) = f (z) for all f ∈ Fα2 (CN ). Since f (z) = zk ∈ Fα2 (CN ) for each
k = 1, ..., N , we conclude that A2 = I. From this and Theorem 2.5, it follows
that A is a real Hermitian unitary matrix.
Conversely, if weighted composition operators Wϕ,ψ satisfying ϕ(z) = Az
and ψ(z) ≡ a, where A is a real Hermitian unitary matrix and a = ±1, then
2
Wϕ,ψ is Hermitian and Wϕ,ψ
= I, which means Wϕ,ψ is Hermitian isometric on
2
N
Fα (C ). Acknowledgments. This work is supported by the National Natural Science Foundation of China (No.11201323), the Sichuan Province University Key
Hermitian weighted composition operators
3043
Laboratory of Bridge Non-destruction Detecting and Engineering Computing
(No.2013QZJ01) and the Introduction of Talent Project of SUSE (No.2014RC04).
References
[1] C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bergman space, Trans.
Amer. Math. Soc. 301 (1987), 813-829. http://dx.doi.org/10.1090/s0002-9947-19870882716-4
[2] A. Borichev, R. Dhuez and K. Kellay, Sampling and interpolation in
large Bergman and Fock spaces, J. Funct. Anal. 242 (2007), 563-606.
http://dx.doi.org/10.1016/j.jfa.2006.09.002
[3] B. J. Carswell, B. D. MacCluer and A. Schuster, Composition operators on the Fock
space, Acta Sci. Math. (Szeged) 69 (2003), 871-887.
[4] C. C. Cowen, An analytic Toeplitz operator that commutes with a compact operator and related class of Toeplitz operators, J. Funct. Anal. 36 (2) (1980), 169-184.
http://dx.doi.org/10.1016/0022-1236(80)90098-1
[5] C. C. Cowen, Linear fractional composition operarors on H 2 , Integral Equations Operator Theory. 11 (1988), 151-160. http://dx.doi.org/10.1007/bf01272115
[6] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
[7] D. Y. Du, Schatten class weighted composition operators on the Fock space Fα2 (CN ),
Int. Journal of Math. Analysis. 5(13)(2011), 625-630.
[8] F. Forelli, The isometries of H p , Canadian J. Math. 16 (1964), 721-728.
http://dx.doi.org/10.4153/cjm-1964-068-3
[9] S. M. Grudsky and N. L. Vasilevski, Toeplitz operators on the Fock space: radial component effects, Integral Equations Operator Theory. 44 (2002), 10-37.
http://dx.doi.org/10.1007/bf01197858
[10] Z. J. Jiang and S. Stević, Compact differences of weighted composition operators from
weighted Bergman spaces to weighted-type spaces, Applied Mathematics and Computation, 217(7)(2010), 3522-3530. http://dx.doi.org/10.1016/j.amc.2010.09.027
[11] Z. J. Jiang, Weighted composition operators from Bergman space to Bers-type space,
Acta Mathematica Chinese Series. 53(1)(2010), 67-74.
[12] Z. J. Jiang and H. B. Bai, Weighted composition operator on Hardy space H p (BN ),
Advances in Mathematics. 37(6)(2008), 749-754.
[13] N. Marco, X. Massaneda and J. Ortega-Cerdà, Interpolating and sampling sequences for
entire functions, J. Geom. Anal. 13 (2003), 862-914. http://dx.doi.org/10.1007/s00039003-0434-7
[14] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, 1980.
http://dx.doi.org/10.1007/978-1-4613-8098-6
[15] S. Stević and Z. J. Jiang, Differences of weighted composition operators
on the unit polydisk, Siberian Mathematical Journal. 52(2)(2011), 358-371.
http://dx.doi.org/10.1134/s0037446611020200
[16] K. Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39
(1992), 3-16. http://dx.doi.org/10.1307/mmj/1029004449
[17] S. Ueki, Weighted composition operator on the Fock space, Proc. Amer. Math. Sci. 135
(2007), 1405-1410. http://dx.doi.org/10.1090/s0002-9939-06-08605-9
[18] K. Zhu, Operator Theory in Function Space, Dekker, New York, 1990.
Received: July 1, 2014; Published: April 14, 2015