Factoring absolutely summing operators through Hilbert-Schmidt operators

Source abstract

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K . It is well-known that every 1-summing operator S:C(K)→l 2 is also nuclear, and therefore factors S = S 1 S 2 , with S 1 :l 2 →l 2 a Hilbert–Schmidt operator and S 1 :C(K)→l 2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X ** , does. This led A. Pełczyński [ 15 ] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K) -space if X has the property that every 1-summing operator X-→l 2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.