This is an announcement for the paper "On convergence with respect to an ideal and a family of matrices" by Jan-David Hardtke.

Abstract: Recently P. Das, S. Dutta and E. Savas introduced and studied the notions of strong $A^I$-summability with respect to an Orlicz function $F$ and $A^I$-statistical convergence, where $A$ is a non-negative regular matrix and $I$ is an ideal on the set of natural numbers. In this note, we will generalise these notions by replacing $A$ with a family of matrices and $F$ with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' $\sup$-$\limsup$-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal $I$ has a countable base), continuing the author's previous work.

Archive classification: math.FA

Mathematics Subject Classification: 40C05, 40C99, 46B20

Remarks: 32 pages

Submitted from: hardtke@math.fu-berlin.de

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1210.1350

or

http://arXiv.org/abs/1210.1350