Integral Operators Between Fock Spaces

Yongqing Liu , Shengzhao Hou

Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (2) : 265 -278.

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Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (2) : 265 -278. DOI: 10.1007/s11401-024-0016-6
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Integral Operators Between Fock Spaces

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Abstract

In this paper, the authors study the integral operator ${S_\phi }f(z) = \int_{\mathbb{C}} \phi (z,\overline w )f(w){\rm{d}}{\lambda _\alpha }(w)$

induced by a kernel function ϕ(z,·) ∈ F α between Fock spaces. For 1 ≤ p ≤ ∞, they prove that S ϕ: F α 1F α p is bounded if and only if $\mathop {\sup }\limits_{a \in \mathbb{C}} ||{S_\phi }{k_a}|{|_{p,\alpha }} < \infty ,$

where k a is the normalized reproducing kernel of F α 2; and, S ϕ: F α 1F α p is compact if and only if $\mathop {lim}\limits_{|a| \to \infty } ||{S_\phi }{k_a}|{|_{p,\alpha }} = 0.$

When 1 < q ≤ ∞, it is also proved that the condition (†) is not sufficient for boundedness of S ϕ: F α qF α p.

In the particular case $\phi (z,\overline w ) = {e^{\alpha z\overline w }}\varphi (z - \overline w )$ with φF α 2, for 1 ≤ q < p < ∞, they show that S ϕ: F α pF α q is bounded if and only if φ = 0; for 1 < pq < ∞, they give sufficient conditions for the boundedness or compactness of the operator S ϕ: F α pF α q.

Keywords

Fock spaces / Integral operators / Normalized reproducing kernel

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Yongqing Liu, Shengzhao Hou. Integral Operators Between Fock Spaces. Chinese Annals of Mathematics, Series B, 2024, 45(2): 265-278 DOI:10.1007/s11401-024-0016-6

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