Real and complex interpolation and extrapolation of compact operators (Q1187741)

The author settles a long standing important problem of real interpolation theory. If \(T\) is a linear operator mapping the Banach interpolation couple \((A_ 0,A_ 1)\) to the couple \((B_ 0,B_ 1)\) so that \(T: A_ 0\to B_ 0\) is continuous and \(T: A_ 1\to B_ 1\) is compact, then for \(0<\vartheta<1\) and \(q\geq 1\), \(T: (A_ 0,A_ 1)_{\vartheta,q}\to (B_ 0,B_ 1)_{\vartheta,q}\) compactly. The result for two-sided compactness, i.e. under the assumption that both \(T: A_ 0\to B_ 0\) and \(T: A_ 1\to B_ 1\) are compact goes back to Hayakawa, 1969. Some progress on the related, important problem in the context of complex interpolation is also described.