Integral Operators Between Fock Spaces

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 ¹_α → F ^p_α 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 ²_α ; and, S_ϕ: F ¹_α → F ^p_α 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 ^q_α → F ^p_α .

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