Methods of Functional Analysis
and Topology

In this manuscript we define $A$-double density of a given point in $\mathbb{R}$ for double sequences. We state the relation between double density of a point and spliced double sequences and also examine limit points in view of density of a point.

In this paper, we prove that the space of strongly \((r, p, \sigma)\)-summable sequences is the topological dual of the space of \((p, \sigma)\)-weakly summable sequences. As an application, we provide a new proof that an operator \( T: E \to F \) is \((p, \sigma)\)-absolutely continuous if and only if its adjoint \( T^*: F^* \to E^* \) is strongly \((p^*, \sigma)\)-continuous.

We introduce a concept of antisymmetric Arens regularity of Banach spaces and prove that a Banach space is Arens regular if and only if it is both symmetrically Arens regular and antisymmetrically Arens regular. We show that if a Banach space is nonreflexive, then it is not antisymmetrically Arens regular and provide an example of a symmetrically Arens regular Banach space which is not antisymmetrically Arens regular.

In the paper, we introduce and analyze the notion of $\lambda$-slant Hankel operator and $(\lambda,\mu)$-slant Hankel operator on the Lebesgue space $L^2(\mathbb{T})$. Certain basic properties of these operators, connection between them and their co-relations with certain existing operators are also discussed.