Norms of partial sums operators for a basis with respect to a filter

Basis of a Banach space with respect to a filter F on N (F-basis for short) is a generalization of basis, where the ordinary convergence of series is substituted by convergence of partial sums with respect to the filter F. We study the behavior of the norms of partial sums operators for an F-basis, depending on the filter and on the space. One of the central results is: The following properties of a sequence $(a_n{)}_{n\in N}\subset (1,\mathrm{\infty })$ are equivalent: (i) ${\sum }_{n\in N}{a}_n^{-1}=\mathrm{\infty }$. (ii) There are a free filter F on N, an infinite-dimensional Banach space X and an F-basis $(u_k)$ of X such that the norms of the partial sums operators with respect of $(u_k)$ are equal to the corresponding $a_n$.