More about the kernel convergence and the ideal convergence

Abstract

Let I ⊂ 2∕ be an ideal and let $X_\mathcal{I} = \overline {span} \{ \chi _I :I \in \mathcal{I}\} $ , and let p I be the quotient norm of l ∞/X I . In this paper, we show first that for each proper ideal I ⊂ 2ℕ, the ideal convergence deduced by I is equivalent to p I -kernel convergence. In addition, let $\mathcal{K} = \{ x^* \circ \chi _{( \cdot )} :x^* \in \partial p(e)\} $ , where $p(x) = \lim \sup _{n \to \infty } \tfrac{1} {n}\sum\nolimits_{k = 1}^n {\left| {x(k)} \right|} $ , and let $\mathcal{I}_\mu = \{ A \subset \mathbb{N}:\mu (A) = 0\} $ for all $\mu = x^* \circ \chi _{( \cdot )} \in \mathcal{K}$ . Then $\mathcal{I}_\mu $ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal $\mathcal{I}_\mu $ , the p-kernel convergence and the statistical convergence are also equivalent.