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.

Let us consider the scattering theory of one-dimensional Schrödinger equation defined by piecewise constant potentials with compact support. We construct the asymptotics of reflection coefficient by iterating the transition matrices. The jump points of the step potential appear in the asymptotics of reflection coefficient. Accordingly, we investigate the modulus of reflection coefficient. The modulus is connected to the zero set of reflection coefficient via Nevanlinna-Levin type of representation theorem. The growth theory of zero set of the reflection coefficient plays a role.

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 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.