Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that makes it easier). I have a few questions regarding their interpolation theory:

- Is $C^1(M)$ an interpolation space for the pair $(C^0(M),C^2(M))$? According to Bergh-Lofström, this means that any linear map $T:C^0(M)\rightarrow C^0(M)$ which leaves $C^2(M)$ invariant, also leaves $C^1(M)$ invariant. I don't see how one would prove this. The reason I am wondering is that this would be a sufficient (but not a necessary) condition for the association $(C^0,C^2)\mapsto C^1$ to extend to an interpolation functor on Banach spaces (Aronszajn-Gagliardo Theorem).
- Can we identify the interpolation spaces $[C^k, C^l]_\theta$ or $[C^k,C^l]_{\theta,p}$ (where the brackets stand for complex and real interpolation respectively)? I only find results of this kind for Hölder-Zygmund spaces $C_*^k$, which differ from $C^k$ for integer values of $k$. Maybe one can even identify $C^k$ as member of some larger scale of spaces (Besov, Triebel, etc.)?