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).