Bounded and Compact Toeplitz Operators with Positive Measure Symbol on Fock-Type Spaces
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Bounded and Compact Toeplitz Operators with Positive Measure Symbol on Fock-Type Spaces Xiaofeng Wang1 · Zhihao Tu1
· Zhangjian Hu2
Received: 29 April 2019 © Mathematica Josephina, Inc. 2019
Abstract In this note, we discuss the Bergman projection P and Toeplitz operators Tμ with p q positive measure symbol μ between F (Cn ) and F (Cn ) for 1 ≤ p, q ≤ ∞. We first p p show that P is a bounded projection from L onto F when 1 ≤ p ≤ ∞, and then apply it to obtain results on the complex interpolation and the duality of the Focktype spaces. Furthermore, we obtain the equivalent conditions for the boundedness and compactness of Tμ in terms of the averaging function and the Berezin transform, which extend the main results about Toeplitz operators of Seip and Youssfi (J Geom Anal 23:170–201, 2013). Keywords Fock-type space · Toeplitz operator · Averaging function · Berezin transform Mathematics Subject Classification Primary 47B35; Secondary 32A37
1 Introduction Let : [0, +∞) → [0, +∞) be a C 3 -function satisfying (x) > 0, (x) ≥ 0, (x) ≥ 0,
B
(1.1)
Zhihao Tu [email protected] Xiaofeng Wang [email protected] Zhangjian Hu [email protected]
1
Key Laboratory of Mathematics and Interdisciplinary Sciences of the Guangdong Higher Education Institute, School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, Guangdong, China
2
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, Zhejiang, China
123
X. Wang et al.
for all x ∈ [0, +∞). We will refer to it as a logarithmic growth function. In fact, (1.1) says that should grow at least as a linear function. Let Cn be the n-dimensional complex space, H(Cn ) be the set of holomorphic functions on Cn , and dV be the Lebesgue measure on Cn . For 0 < p < ∞, the p 2 p p p Fock-type space F = H(Cn ) ∩ L , where L = L p (Cn , e− 2 (|z| ) dV (z)). For any p f ∈ F , we have f p =
Cn
1/ p p − 21 (|z|2 ) < ∞. f (z)e dV (z)
The Fock-type space F∞ is defined as the set of all holomorphic functions on Cn such that 1 2 f ∞ = supz∈Cn f (z)e− 2 (|z| ) < ∞. p
F is a Banach space with the norm · p as 1 ≤ p ≤ ∞. Particularly, F2 is a Hilbert space with the inner product
f , g =
f (z)g(z)e−(|z| ) dV (z). 2
Cn
p
p
Note that F coincides with the classic Fock space Fα when is a suitably normalized linear function. When n > 1, throughout this paper, we assume that the weight function has the additional property that there exists some η < 1/2 such that (x) = O(x −1/2 [ (x)]1+η ), x → ∞,
(1.2)
where (x) = x (x). As was observed in [12], F2 is a reproducing Hilbert space, with its reproducing kernel given by K (z, w) =
1 F (n−1) ( z, w ), (n − 1)!
where z, w = z 1 w 1 + · · · z n w n , F(ζ ) =
∞ ζk k=0
sk
, ζ ∈ C,
and sk = 0
123
+∞
x k e−(x) dx, k = 0, 1, 2, 3, . . . .
Bounded and Compact Toeplitz Operators with Positive Measure Symbol
Let P be the Bergman projection from L 2 onto F2 . Then for each f ∈ L 2 , we have P f (z) =
f (w)K (z, w)e−
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