  A classical result of Calkin [Ann. of Math. (2) 42 (1941), pp. 839-873] says
that an inner derivation $S\mapsto [T,S] = TS-ST$ maps the algebra of bounded
operators on a Hilbert space into the ideal of compact operators if and only if
$T$ is a compact perturbation of the multiplication by a scalar. In general, an
analogous statement fails for operators on Banach spaces. To complement
Calkin's result, we characterize Volterra-type inner derivations on Hardy
spaces using generalized area operators and compact intertwining relations for
Volterra and composition operators. Further, we characterize the compact
intertwining relations for multiplication and composition operators between
Hardy and Bergman spaces.
