A quantitative approach to disjointly non-singular operators

We introduce and study some operational quantities which characterize the disjointly non-singular operators from a Banach lattice E to a Banach space Y when E is order continuous, and some other quantities which characterize the disjointly strictly singular operators for arbitrary E.

Genericity and Universality for Operator Ideals

A bounded linear operator $U$ between Banach spaces is universal for the complement of some operator ideal $\mathfrak{J}$ if it is a member of the complement and it factors through every element of the complement of $\mathfrak{J}$. In the first part of this paper, we produce new universal operators for the complements of several ideals, and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use descriptive set theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal $\mathfrak{J}$ generic if whenever an operator $A$ has the property that every operator in $\mathfrak{J}$ factors through a restriction of $A$, then every operator between separable Banach spaces factors through a restriction of $A$. We prove that many classical operator ideals (such as strictly singular, weakly compact, Banach–Saks) are generic and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W. B. Johnson, which states that there is no universal operator for the complement of the ideal of completely continuous operators.

On Twisted Sums of Sequence Spaces

We prove the existence of non trivial twisted sums involving the pth James space Jp(1 < p < 1), the Johnson-Lindenstrauss space JL; the James tree space JT , the Tsirelson’s space T and the Argyros and Deliyanni space AD . We also present non trivial twisted sums involving their duals and biduals. We show that there are strictly singular quasi-linear maps from the spaces T ;T* ; AD and JT into C[0; 1]. We discuss the Pelczynski’s property(u) for the twisted sums involving these spaces which extends a pthJames-Schreier spaces Vp(1 < p <1) or J2.