Example of an integral operator satisfying Holmgren's condition but not compact

Question

Let $F$ be an integral operator with the kernel $K(s,t)$. This means that $F(f)(t)=\int K(s,t)f(s)ds$ for square integrable functions $f(s)$.

Here $K$ satisfies the condition $[sup_{s} \int \mid K(s,t) \mid dt] \cdot [sup_{t} \int \mid K(s,t) \mid ds] < \infty$.

Is there an example of $K(s,t)$ such that the operator $F$ is not compact as a mapping from some $L^2$ space to another $L^2$ space? I cannot find one myself.

Answer 1

$K(s,t)=\frac{1}{s}$, if $0<t<s$ and $0$ otherwise.

This will give a non-compact operator

Answer 2

The identity operator on $\ell^2$ is not compact, but is an integral operator whose kernel has the property in the question.