On the bounded approximation property on subspaces of ℓp when 0 < p < 1 and related issues

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator {\ell_{p}\to X} has the BAP when X has it and {0<p\leq 1}, which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

On a Generalization of the Hilbert Frame Generated by the Bilinear Mapping

The concept ofb-frame which is a generalization of the frame in Hilbert spaces generated by the bilinear mapping is considered.b-frame operator is defined; analogues of some well-known results of frame theory are obtained in Hilbert spaces. Conditions for the existence ofb-frame in Hilbert spaces are obtained; the relationship between the definite bounded surjective operator andb-frame is also studied. The concept ofb-orthonormalb-basis is introduced.

Perturbations of normally solvable nonlinear operators, I

LetXandYbe Banach spaces and letℱand𝒢be Gateaux differentiable mappings fromXtoYIn this note we study when the operatorℱ+𝒢is surjective for sufficiently small perturbations𝒢of a surjective operatorℱThe methods extend previous results in the area of normal solvability for nonlinear operators.