I am studying the compactness of some convolution operators. Let the convolution with extrapolation $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a $C_0$-semigroup on some Banach space $X$.
${\bf B(s)\in \mathcal{L}(X,\mathbb{F}_{-1})}$ is a compact operator for all $s\in[0,t]$ and $t \ge 0$ and ${\bf B(s)x\in L^1(0,t; \mathbb{F}_{-1})}$ (so that the convolution makes sense),
$T_{-1}$ is the extrapolated semigroup and $\mathbb{F}_{-1}$ is the associated Favard space.
Do we still recover the compactness (or weak-compactness) of $\Gamma$?
(see Compact convolution for the case without extrapolation)
Many thanks.
