Abstract
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.
Highlights
A quasi-Banach space X is said to be a twisted sum of two Banach spaces Y and U if it contains a subspace A isomorphic to Y and the quotient X/A is isomorphic to U
Two exact sequences 0 −→ Y −→ X1 −→ U −→ 0 and 0 −→ Y −→ X2 −→ U −→ 0 are equivalent if there is a bounded linear operator T making the diagram
An exact sequence 0 −→ Y −→ X −→ U −→ 0 is said to split and X is said to be trivial if it is equivalent to the trivial sequence 0 −→ Y −→ Y ⊕ U −→ U −→ 0
Summary
A quasi-Banach space X is said to be a twisted sum of two Banach spaces Y and U if it contains a subspace A isomorphic to Y and the quotient X/A is isomorphic to U . We denote by Ext (U , Y ) the space of all equivalence classes of locally convex twisted sums of Y and U . Given a family E of finite dimensional Banach spaces, a Banach space X is said to contain E uniformly complemented if there exists a constant c such that for every E ∈ E, there is a c-complemented subspace A of X which is cisomorphic to E.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
