For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently, a sequence of Toeplitz operators $(T_{\phi_n})$ converges to a Toeplitz operator $T_\phi$ in the operator norm iff $\phi_n$ converges to $\phi$ in $L^{\infty}$.
Is there something that can be said about convergence of the symbols in some sense when $(T_{\phi_n})$ converges to a Toeplitz operator $T_\phi$ in the strong operator topology and the weak operator topology? References will be appreciated.