Could we interpolate the compactness of compact operators?

Question

Classical theorems of Marcinkiewicz and Riesz and their extensions to general Banach spaces by Calderón, Lions, Peetre, et al. allow us to interpolate the continuity of two operators, viz., the suitable "intermediate operators" of the two operators inherit the boundedness of the latter. Do these operators inherit other properties of the endpoint operators, such as compactness? A quick google search turns up a 2006 survey which states that the problem of interpolating compactness is open, and I would like to know what is known about this and other related problems.

Edit:The linked survey is from 2006, not 1991. Here's also a 2008 paper by the first author of the survey, which seems to indicate that the problem of interpolating compactness is still open. I'd still like to know if interpolating any other functional-analytic properties of operators has been considered (or is of interest, even).

Answer 1

For $L^p$ spaces over countably generated $\sigma$-finite measure spaces (with $1 \leq p \leq \infty$), the answer is yes - compactness at one endpoint propagates to the midpoints. My reference for this is Davies' 'Heat Kernels and Spectral Theory', Theorem 1.6.1, but apparently this goes back to Persson 'Compact linear mappings between interpolation spaces' (1964) (I don't have access to the reference at the moment).

More precisely, when we have an operator $A$ which is compact on $L^{p_0}$ and $L^{p_1}$ ($1 \leq p_0 < p_1 \leq \infty$), we can extend $A$ to a compact operator on $L^p$ ($p_0 \leq p < p_1$). There's nothing special about compactness being on the lower endpoint, except that we would then have to take $p_1 < \infty$.

The key idea of the proof is that we can strongly approximate the identity operator on $L^q$ (for all $1 \leq q < \infty$) by a sequence $(P_{k})_{k=1}^\infty$ of finite rank projections $P$, of the form $$ Pf = \sum_{r=1}^n \chi_{E_r} |E_r|^{-1} \int_{E_r} f(x) \; dx$$ where ${E_n}$ is a sequence of disjoint subsets of finite positive measure. These projections are contractions on $L^q$ for all $1 \leq q \leq \infty$.

The rest of the proof is a nice interpolation argument. Since $A$ is compact we have that $\lim_{k \to \infty} P_k A = A$ (as bounded operators on $L^{p_0}$), and since each $P_k$ is a contraction on $L^{p_1}$ we have $$\lim_{k \to \infty}{||A - P_k A||}_{p_1} \leq 2||A||_{p_1}.$$ Interpolating between $p_0$ and $p_1$ for each $k$ then shows that $P_k A$ converges to $A$ as bounded operators on $L^p$.

I have no idea what the situation is for more general Banach spaces, and this is certainly the most simple setting to work in (and probably not what you were looking for), but I thought this argument was too nice not to post!