Universal Non-completely-continuous Operators

A bounded linear operator between Banach spaces is called completely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from L1 into an arbitrary Banach space, namely, the operator from L1 into `∞ defined by T0(f) = Z rnf dμ n≥0 , where rn is the nth Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach space. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space. Suppose that C is a class of (always bounded, linear, between Banach spaces) operators so that an operator S is in C whenever the domain of S is the domain of some operator in C and there exist operators A, B so that BSA is in C; the natural examples of such classes are all the operators that do not belong to a given operator ideal. A subset S of such a class C is said to be universal for C provided for each U in C, some member of S factors through U ; that is, there exist operators A and B so that BUA is in S. In case S is singleton; say, S = {S}; we say that S is universal for C. In order to study a class C of operators, it is natural to try to find a universal subclass of C consisting of specific, simple operators. For certain classes, such a subclass is known to exist. For example, Lindenstrauss and Pe lczyński, who introduced the concept of universal operator, proved [LP] that the “summing operator” from `1 to `∞, defined by {an}n=1 7→ { ∑n k=1 ak}n=1, is universal for the class of non-weakly-compact operators; while in [J] it was pointed out that the formal identity from `1 to `∞ is universal for the class of non-compact operators. 1991 Mathematics Subject Classification. 47B99, 47A68, 47D50, 47B38, 46B20, 46B28, 46B07, 46B22. *Supported in part by NSF grants DMS-9306460 and DMS-9003550. ‡Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University.