Methods of Functional Analysis
and Topology

Any abstract Gram operator is consistent with some graph. For an arbitrary operator $B_\Gamma$ that is consistent with a graph $\Gamma$, the question arises as to when it is an abstract Gram operator, i.e., whether it is nonnegative. We study this question for certain types of graphs. The simplest case is a star graph. Next, we use the results obtained for star graphs to explore a more general case, where a graph $\Gamma$ can be treated as a collection of rooted trees, with their roots connected by additional edges into a connected subgraph $\Gamma_0$. The work shows that the question about the nonnegativity of an operator $B_\Gamma$ for such a graph can be reduced to the corresponding question for some operator that is consistent with the subgraph $\Gamma_0$.

The paper is devoted to the study of conditions for the Hausdorff-Besicovitch faithfulness of the family of cylinders generated by Cantor series expansions. We show that there exist subgeometric Cantor series expansions for which the corresponding families of cylinders are not faithful for the Hausdorff-Besicovitch dimension on the unit interval. On the other hand we found a rather wide subfamily of subgeometric Cantor series expansions generating faithful families of cylinders.

We also study conditions for the Hausdorff-Besicovitch dimension preservation on [0;1] by probability distribution functions of random variables with independent symbols of arithmetic Cantor series expansions.

This paper is devoted to the study of hyperstability phenomena in the context of convex modular spaces. In particular, we investigate the hyperstability of three fundamental functional equations: the quadratic equation \begin{equation} \varphi(x+y)+\varphi(x-y)=2\varphi(x)+2\varphi(y) \end{equation} the general linear equation \begin{equation} \varphi(ax+by)=A\varphi(x)+B\varphi(y)+C \end{equation} and the $n$-dimensional quadratic equation \begin{equation} f\left(\sum_{i=1}^{m}x_{i}\right)+\sum_{1\leq i < j\leq m}f\big(x_{i}-x_{j}\big)=m\sum_{i=1}^{m}f(x_{i}). \end{equation} Using the direct method, we establish sufficient conditions under which every approximate solution of these equations in modular spaces coincides exactly with an exact solution. Our results extend earlier contributions obtained in Banach spaces via fixed point techniques, and provide new insights into the stability of functional equations in the broader context of modular spaces.

In this paper, we construct, and study a certain type of definite, or indefinite inner product spaces over the real field $\mathbb{R}$, induced by the scaled hypercomplex numbers $\mathbb{H}_{t}$ for a fixed scale $t\in\mathbb{R}$, and some bounded operators acting on such vector spaces. In particular, we are interested in the vector spaces $\mathbb{H}_{t}^{N}$ consisting of all $N$-tuples of scaled hypercomplex numbers of $\mathbb{H}_{t}$, and the $\left(N\times N\right)$-matrices acting on $\mathbb{H}_{t}^{N}$ whose entries are from $\mathbb{H}_{t}$, i.e., $\mathbb{H}_{t}$-matrices, for all $N\in\mathbb{N}$. For an arbitrarily fixed $N\in\mathbb{N}$, we define $\mathbb{H}_{t}^{N}$ as a subspace of a certain functional vector space $\mathbf{H}_{t:2}$ equipped with a well-defined definite (if $t<0$), or indefinite (if $t\geq0$) inner product introduced in [6, 7, 8]. So, one can check immediately that our subspace $\mathbb{H}_{t}^{N}$ becomes a restricted definite, or indefinite inner product Banach space. Operator-theoretic, operator-algebraic and free-probabilistic properties of $\mathbb{H}_{t}$-matrices are considered and characterized on $\mathbb{H}_{t}^{N}$.