I am studying the compactness of some convolution operators. Let the convolution $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a $C_0$-semigroup on some Banach space $X$, $B(s)\in\mathcal{L}(X)$ is a compact operator for all $s\in [0,t]$ and $t\ge 0$.
$\star$ If $B(s)=B$ are independent of $s$, it is not difficult to prove the compactness of $\Gamma$ ( I did it using Riemann summation)
$\star$ I have a positive answer if $s\mapsto B(s)$ is immediately continuous ( continuous in the norm of $\mathcal{L}(X)$).
${\color{blue}\star}$ Now if $s\mapsto B(s)$ is strongly continuous what can we say about the compactness in this case?
${\color{red}\star}$ Another problem which I am facing right now is if we consider $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$ This time $B(s)$ arrives in the Favard space $\mathbb{F}_1$ associated with the $C_0-$semigroup (so that the convolution makes sense). $T_{-1}$ is the extrapolated semigroup. Do we still recover compactness?
Many thanks.
