Integral Operators Between Fock Spaces

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

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 \(\mathop {\sup }\limits_{a \in \mathbb{C}} ||{S_\phi }{k_a}|{|_{p,\alpha }} < \infty ,\) $$\underset{a\in ℂ}{\sup }\mid \mid S_{\varphi }{k}_a\mid {\mid }_{p,\alpha }<\infty ,$$

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 \(\mathop {lim}\limits_{|a| \to \infty } ||{S_\phi }{k_a}|{|_{p,\alpha }} = 0.\) $$\underset{\mid a\mid \to \infty }{lim}\mid \mid S_{\varphi }{k}_a\mid {\mid }_{p,\alpha }=0.$$

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 \(\phi (z,\overline w ) = {e^{\alpha z\overline w }}\varphi (z - \overline w )\) $$\varphi (z,\overline{w})=e^{\alpha z\overline{w}}\phi (z-\overline{w})$$ 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 .