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We present Korovkin approximation theorems that incorporate summability methods. These result allows us to obtain a unified treatment of several previous results, focusing on the underlying structure and the properties that a summability method should satisfy in order to establish a Korovkin-type approximation result. As a by-product we obtain new Korovkin-type results incorporating summability methods, for example for regular matrix convergence methods, convergence through ideals of natural numbers, etc, and we provide further directions and a preparation for questions which remain open., Comment: arXiv admin note: substantial text overlap with arXiv:2307.02237

We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods. Sometimes very sharp conditions are obtained, giving a focused view of the subject and from which new problems emerge.

We explore some convergence notions for set-convergence coming from modern summability methods. Specifically we will see the connections between Wijsman $f$-statistical convergence and Wijsman $f$-strong Ces\`aro convergence, when $f$ is a modulus function that induces a density in $\mathbb{N}$.

Here we fully complete the studies initiated by Vinod K. Bhardwaj and Shweta Dhawan in \cite{hindawi} which relate different convergence methods which involves the classical statistical and the classical strong Ces\`aro convergences by means of lacunary sequences and measures of density in $\mathbb{N}$ modulated by a modulus function $f$.

A continuous linear operator L defined on the space of entire functions H(C) is said to be an extended $lambda$-eigenoperator of the differentiation operator D provided DL = $lambda$LD. Here we fully characterize when an extended $lambda$-eigenoperator of D is supercyclic, it has a hypercyclic subspace or it has a supercyclic subspace.

A closed subspace $S$ of $\ell_\infty$ is said to be a \emph{$\ell_\infty$-Grothendieck subspace} if $c_0\subset S$ (hence $\ell_\infty\subset S^{**}$) and every $\sigma(S^*,S)$-convergent sequence in $S^*$ is $\sigma(S^*,\ell_\infty)$-convergent. Here we give examples of closed subspaces of $\ell_\infty$ containing $c_0$ which are or fail to be $\ell_\infty$-Grothendieck.

An algebra of bounded linear operators on a Banach space is said to be {\em strongly compact} if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be {\em strongly compact} if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fern\'andez-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator., Comment: 7 pages, to appear in Proc. Amer. Math. Soc