Methods of Functional Analysis
and Topology

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.

In this paper, we introduce the notion of a generalized modified $F-$contraction via rectangular $G-\alpha-$admissible mapping and establish some new fixed point results under Wardowski's $F-$contractive condition $(F1)$ together with an auxiliary function $\phi$ in the setting of $G-$metric spaces. Our new fixed point results improve, generalize and unify several related result in the existing literature. Moreover, we give some non-trivial examples to verify our results. Finally, we derive the existence of a solution to an integral equation to support our main findings.

We prove sufficient conditions for the faithfulness of the family of cylinders generated by $\widetilde{Q}$ -expansions for the Hausdorff-Besicovitch dimension calculation. We also represent the conjecture about necessary and sufficient conditions for such a family of cylinders to be faithful. Based on these new results we study fine fractal properties of random variables with independent $\widetilde{Q}$-digits and prove exact formulae for the calculation of the Hausdorff dimension of the corresponding probability measure.

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.