Multiplier spaces and the summing operator for series

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C*-Algebras and Factorization Through Diagonal Operators

Let A be a C*-algebra and E be a Banach space with the Radon-Nikodym property. We prove that if j is an embedding of E into an injective Banach space then for every absolutely summing operator T : A → E, the composition j ○ T factors through a diagonal operator from l2 into l1. In particular, T factors through a Banach space with the Schur property. Similarly, we prove that for 2 < p < ∞, any absolutely summing operator from A into E factors through a diagonal operator from lp into l2.

(r,p)-absolutely summing operators on the spaceC (T,X)and applications

We give necessary and sufficient conditions for an operator on the spaceC (T,X)to be(r,p)-absolutely summing. Also we prove that the injective tensor product of an integral operator and an(r,p)-absolutely summing operator is an(r,p)-absolutely summing operator.

Factoring absolutely summing operators through Hilbert-Schmidt operators

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)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 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-→l2 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.

A non-absolutely summing operator

In the case when 0 < p < 1 it is proved, using a method of Macphail that the identity map i: lp → lp is not (r, s)-absolutely summing for any r, s.