Compact and strictly singular operators on certain function spaces (Q793964)

Suppose X is a quasi-Banach space and \(T:L_ p(X)\to Y\) (where \(1\leq p<\infty)\), is an operator which is strictly singular on every Hilbertian subspace of \(L_ p(X)\); we show that T factors through the containing Banach space of \(L_ p(X)\). Using this and a similar result for compact operators we study operators on symmetric function space on [0,1], and characterize those non-locally convex symmetric function spaces which have a Schauder basis.