Let $K\in L^2([0,T]^2)$, and for each $t\in [0,T]$, let $\mathcal{T}_t $ be such that for all $f\in L^2([0,T])$, $\mathcal{T}_t f(s)=\int_0^T K(s,t)K(u,t)f(u)\, d u$ for all $s\in [0,T]$. One can show by the integrability of $K$ that $\mathcal{T}_t $ is a well-defined bounded linear operator acting on  $L^2([0,T];\mathbb{R})$ into itself (i.e., $\mathcal{T}_t\in \mathcal{L}(L^2([0,T]))$). Is it possible to prove that the map $[0,T]\ni t\mapsto \mathcal{T}_t\in \mathcal{L}(L^2([0,T]))$ is [strongly measurable](https://en.wikipedia.org/wiki/Bochner_measurable_function)?