Let $B_1$ be the unit ball in $\mathbb{R}^d$ and $(f_n)$ be a sequence in $L^2(B_1)$ such that $f_n$ converges weakly to some $f_0 \in L^2(B_1)$. Consider the sequence $T_n$ of operators defined by $$(T_nu)(x)= \int_{B(0,|x|)}f_n(y)u(y)dy,~~~x \in B_1,~ u\in L^2(B_1),~ n \in \mathbb{N}.$$
I've managed to show that every $T_n: L^2(B_1) \longrightarrow L^2(B_1)$ is a bounded linear operator and that $T_nu \rightarrow Tu$ for every $u \in L^2(B_1)$ pointwise. Now, I'm asking myself if is it true that $||T_n-T||_{\mathscr{L}(L^2(B_1))} \rightarrow 0$. I guess the answer is no, but I can't figure out a counterexample. Also, is there any hypothesis that I could add in order to make $(T_n)$ convergent in $\mathscr{L}(L^2(B_1))$?
Any suggestion or comment is appreciated, thank you.