Advertisement

Lithuanian Mathematical Journal

, Volume 58, Issue 1, pp 104–112 | Cite as

Subseries of \( \mathrm{\mathcal{I}}\)-convergent series

  • Jacek Tryba
Article

Abstract

We say that an ideal \( \mathrm{\mathcal{I}}\) has property (T) if for every \( \mathrm{\mathcal{I}}\)-convergent series \( {\sum}_{n=1}^{\infty }{x}_n \), there exists a set A\( \mathrm{\mathcal{I}}\) such that ∑n ∈ ℕ\Ax n converges in the usual sense. The main aim of this paper is to focus on several different classes of ideals, such as summable ideals, F σ ideals, and matrix summability ideals, and to show that they do not have the mentioned property.

Keywords

ideal filter P-ideal ideal convergence (T) property \( \mathrm{\mathcal{I}}\)-convergent series 

MSC

primary 40A35 secondary 40A05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, Generalized kinds of density and the associated ideals, Acta Math. Hung., 147(1):97–115, 2015, available from:  https://doi.org/10.1007/s10474.
  2. 2.
    R.C. Buck, The measure theoretic approach to density, Am. J. Math., 68:560–580, 1946.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Črveňanský, T. Šalát, and V. Toma, Remarks on statistical and I-convergence of series, Math. Bohem., 130(2):177–184, 2005.MathSciNetzbMATHGoogle Scholar
  4. 4.
    L. Drewnowski and P.J. Paúl, The Nikodým property for ideals of sets defined by matrix, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.), 94(4):485–503, 2000.MathSciNetzbMATHGoogle Scholar
  5. 5.
    I. Farah, Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers, Mem.Am.Math. Soc., Vol. 702, AMS, Providence, RI, 2000.Google Scholar
  6. 6.
    I. Farah, How many Boolean algebras \( \mathcal{P}\left(\mathrm{\mathbb{N}}\right)/\mathrm{\mathcal{I}}\) are there?, Ill. J. Math., 46(4):999–1033, 2002.Google Scholar
  7. 7.
    G. Grekos, L. Mišík, and M. Ziman, I-convergence and (T) property, manuscript, 2004.Google Scholar
  8. 8.
    A.R.D. Mathias, Solution of problems of Choquet and Puritz, in W. Hodhges (Ed.), Conference in Mathematical Logic – London ’70, Lect. Notes Math., Vol. 255, Springer, Berlin, Heidelberg, 1972, pp. 204–210.Google Scholar
  9. 9.
    K. Mazur, f σ-ideals and \( {\omega}_1{\omega}_1^{\ast } \)-gaps in the Boolean algebras P(ω)/I, Fundam. Math., 138(2):103–111, 1991.CrossRefGoogle Scholar
  10. 10.
    L. Mišík, (I)-convergence of series and Šalát’s conjecture, manuscript.Google Scholar
  11. 11.
    O. Toeplitz, Über allgemeine lineare Mittelbildungen, Prace Mat.-Fiz., 22(1):113–119, 1911, available from: http://eudml.org/doc/215310.
  12. 12.
    B.C. Tripathy, On statistically convergent series, J. Math., Punjab Univ., 32:1–7, 1999.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of GdańskGdańskPoland

Personalised recommendations