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 _α ¹ → F _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 ¹ → F _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 ^q → F _α ^p.

In the particular case $\phi (z,\overline w ) = {e^{\alpha z\overline w }}\varphi (z - \overline w )$ with φ ∈ F _α ², for 1 ≤ q < p < ∞, they show that S _ϕ: F _α ^p → F _α ^q 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 _α ^p → F _α ^q.

Keywords

Fock spaces / Integral operators / Normalized reproducing kernel