Methods of Functional Analysis
and Topology

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.

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.

Let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$ into itself. Given $A,B\in \mathcal{L}(H)$, the generalized derivation $\delta_{A,B}\in \mathcal{L}(\mathcal{L}(H))$ is defined by $\delta_{A,B}(X)=AX-XB$. For hyponormal operators $A$ and $B$, L.A. Fialkow gave necessary conditions for which the range of $\delta_{A,B}$ is norm closed in $\mathcal{L}(H)$ [13]. In this paper, we present necessary or sufficient conditions under which the generalized derivation $\delta_{A,B}$ has closed range, where $A$ and $B$ are $p$-hyponormal operators. We give a characterization for the range of $\delta_{A,B}$ to have closed range in the case when $A$ and $B$ are cyclic subnormal operators.

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.