Javier Garcia-Pacheco, FranciscoKama, Ramazandel Carmen Listan-Garcia, Maria2024-12-242024-12-2420211029-242Xhttps://doi.org/10.1186/s13660-021-02587-xhttps://hdl.handle.net/20.500.12604/7301This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of N and study the space of convergence associated with the filter. We notice that c(X) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then l(infinity)(X) is a space of convergence associated with any free ultrafilter of N; and that if X is not complete, then l(infinity)(X) is never the space of convergence associated with any free filter of N. Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that l(infinity)(X) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then c(X) is a space of convergence through a certain class of such operators; and that if X is not complete, then c(X) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set HB(lim):={T is an element of B(l(infinity)(X),X):T|(c(X))=lim and parallel to T parallel to=1} and prove that HB(lim) is a face of B-LX(0) if X has the Bade property, where L-X(0):={T is an element of B(l(infinity)(X),X):c(0)(X)subset of ker(T)}. Finally, we study the multipliers associated with series for the above methods of convergence.eninfo:eu-repo/semantics/openAccessMethodsConvergenceSummability47A05Article20211Q1WOS:000636464600001Q12-s2.0-8510358731410.1186/s13660-021-02587-x