Toeplitz operators with non-trivial kernels and non-dense ranges on weak Hardy spaces

The well known Coburn lemma can be stated as follows: a nonzero Toeplitz operator $T(a)$ with symbol $a\in L^\infty(\mathbb{T})$ has a trivial kernel or a dense range on the Hardy space $H^p(\mathbb{T})$ with $p\in(1,\infty)$. We show that an analogue of this result does not hold for the Hardy-Marcinkiewicz (weak Hardy) spaces $H^{p,\infty}(\mathbb{T})$ with $p\in(1,\infty)$: there exist continuous nonzero functions $a:\mathbb{T}\to\mathbb{C}$ depending on $p$ such that $\operatorname{dim} \left(\operatorname{Ker} T(a)\right) = \infty$ and $\operatorname{dim} \left(H^{p,\infty}(\mathbb{T})/\operatorname{clos}_{H^{p,\infty}(\mathbb{T})}\big(\operatorname{Ran} T(a)\big)\right) = \infty$.