Let
μ
\mu
be a positive Borel measure on the positive real axis. We study the integral operator
H
μ
(
f
)
(
z
)
=
∫
(
0
,
∞
)
1
t
f
(
z
t
)
d
μ
(
t
)
,
z
∈
C
,
\begin{equation*} \mathcal {H}_{\mu }(f)(z)=\int _{(0,\infty )}\frac {1}{t}f\left (\frac {z}{t}\right ) d\mu (t),\quad z\in \mathbb {C}, \end{equation*}
acting on the Fock spaces
F
α
p
F^{p}_{\alpha }
,
p
∈
[
1
,
∞
]
,
α
>
0
p\in [1,\infty ],\alpha >0
. Its action is easily seen to be a coefficient multiplication operator by the moment sequence
μ
n
=
∫
[
1
,
∞
)
1
t
n
+
1
d
μ
(
t
)
.
\begin{equation*} \mu _n= \int _{[1,\infty )}\frac {1}{t^{n+1}} d\mu (t). \end{equation*}
We prove that
‖
H
μ
‖
F
α
p
→
F
α
p
=
∫
[
1
,
∞
)
1
t
d
μ
(
t
)
,
1
≤
p
≤
∞
.
\begin{equation*} \|\mathcal {H}_{\mu }\|_{F^{p}_{\alpha }\to F^{p}_{\alpha }}=\int _{[1,\infty )}\frac {1}{t} d\mu (t),\quad 1\leq p\leq \infty . \end{equation*}
It turns out that
H
μ
\mathcal {H}_{\mu }
is compact on
F
α
p
,
p
∈
(
1
,
∞
)
F^{p}_{\alpha },p\in (1,\infty )
if and only if
μ
(
{
1
}
)
=
0
\mu (\{1\})=0
. In addition, we completely characterize the Schatten class membership of
H
μ
\mathcal {H}_{\mu }
.