General methods of convergence and summability

Abstract

AbstractThis paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is a space of convergence associated with any free ultrafilter of $\mathbb{N} $ N ; and that if X is not complete, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is never the space of convergence associated with any free filter of $\mathbb{N} $ 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 $\ell _{\infty }(X)$ ℓ ∞ ( X ) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$ c ( X ) is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$ 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 $\mathcal{HB}(\lim ):= \{T\in \mathcal{B} (\ell _{\infty }(X),X): T|_{c(X)}= \lim \text{ and }\|T\|=1\}$ HB ( lim ) : = { T ∈ B ( ℓ ∞ ( X ) , X ) : T | c ( X ) = lim  and  ∥ T ∥ = 1 } and prove that $\mathcal{HB}(\lim )$ HB ( lim ) is a face of $\mathsf{B} _{\mathcal{L}_{X}^{0}}$ B L X 0 if X has the Bade property, where $\mathcal{L}_{X}^{0}:= \{ T\in \mathcal{B} (\ell _{\infty }(X),X): c_{0}(X) \subseteq \ker (T) \} $ L X 0 : = { T ∈ B ( ℓ ∞ ( X ) , X ) : c 0 ( X ) ⊆ ker ( T ) } . Finally, we study the multipliers associated with series for the above methods of convergence.