Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

2-convexity and 2-concavity in Schatten ideals

G. J. O. Jamesona1

a1 Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, xs2016 xs2016E) is 2-convex, then there is another Banach sequence space (F, xs2016 xs2016F) such that xs2016x;xs2016 S0305004100001651inline1 = xs2016x2xs2016F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

(Received August 07 1995)