Let ${\mathbb T}$ denote the ordinary $\Lambda$-adic Hecke algebra of say tame level $N$. If I specialize ${\mathbb T}$ to a classical weight $k \geq 2$, then it is proven by Hida that the result is the Hecke algebra of classical $p$-ordinary forms of weight $k$ and level $pN$. (Here by specialize I mean mod out by the prime ideal $p_k \subseteq \Lambda = {\mathbb Z}_p[[\Gamma]]$ generated by $[\gamma] - \gamma^{k-2}$ where $\gamma$ is a topogical generator of $\Gamma$.) I guess this is Hida's control theorem.

My question is the following: if one takes $\kappa$ to be some $p$-adic weight (but not necessarily classical), how can one describe ${\mathbb T}/ p_{\kappa} {\mathbb T}$?

I assume the answer is that the result is the Hecke algebra of the space of overconvergent modular forms of weight $\kappa$ and slope 0 -- but I don't have a proof of this nor a reference. Either would be greatly appreciated!