On Hereditariness for Real and Complex Interpolation

Every isometric property of Banach spaces preserved by real or complex interpolation is subspace-hereditary, and every isomorphic property of separable Banach spaces so preserved is quotient-hereditary. Introduction Many properties of Banach spaces are known to pass to interpolation spaces obtained from the real k-method of interpolation or from the complex method, completeness itself being the most obvious example. Other properties that are known to pass to interpolation spaces include reflexivity and uniform convexity [1]. In general, however, the problem of identifying properties that do not pass to interpolation spaces in every case yielded relatively few results (see e.g. [3]) before a very general result of D. J. H. Garling and S. J. Montgomery-Smith identified several such properties, among them the Radon-Nikodým property, having non-trivial cotype, and having the unconditional martingale difference sequence property [4]. The main result of this paper identifies a large class of Banach space properties that do not necessarily pass to interpolation spaces constructed by the real k-method or by the complex method: those properties that are not subspace-hereditary, e.g. the property of being isomorphic to a separable dual space or the property of being weakly compactly generated. Combining our result with the main result of [4], it follows that any isomorphic property of separable Banach spaces which is preserved by the real or the complex methods must also be quotient-hereditary. Consequently, the only isomorphic property of Banach spaces that is enjoyed by `1 and preserved by real or complex interpolation is separability. Real Interpolation We begin with a brief review of the real k-method of interpolation. We refer the reader to [2] for a detailed treatment. Let (X1, ‖ · ‖1) and (X2, ‖ · ‖2) be Banach spaces which each embed continuously into a common Hausdorff space H. We call such a pair of spaces 1991 Mathematics Subject Classification. Primary 46B70. Secondary 46B20; 46B04.