Vol. 26 (2020), no. 2

We consider an elliptic problem with unknowns on the boundary of the domain of the elliptic equation and suppose that the right-hand side of this equation is square integrable and that the boundary data are arbitrary (specifically, irregular) distributions. We investigate local (up to the boundary) properties of generalized solutions to the problem in Hilbert distribution spaces that belong to the refined Sobolev scale. These spaces are parametrized with a real number and a function that varies slowly at infinity. The function parameter refines the number order of the space. We prove theorems on local regularity and a local a priori estimate of generalized solutions to the problem under investigation. These theorems are new for Sobolev spaces as well.

The paper investigates spectral properties of multi-interval Sturm-Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions and also all generalized resolvents in terms of boundary conditions are given.

We obtain a new sufficient condition under which generalized solutions to a parabolic initial boundary-value problem for a Petrovskii system and the homogeneous Cauchy data are classical. The condition is formulated in terms of the belonging of the right-hand sides of the problem to some anisotropic Hörmander spaces.

We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable functions $y:[a,b]\to\mathbb{C}^{m}$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)})^{m}$ to $\mathbb{C}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)})^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.

We consider when the universal $C^*$-algebras associated to separated graphs are exact. Specifically, for finite separated graphs we show that the universal $C^*$-algebra is exact if and only if the $C^*$-algebra is isomorphic to a graph $C^*$-algebra which occurs precisely when the universal and reduced $C^*$-algebras of the separated graph are isomorphic.

In this paper, we introduce a generalization of quasi-Fredholm operators [7] to $k$-quasi-Fredholm operators on Hilbert spaces for nonnegative integer $k$. The case when $k = 0,$ represents the set of quasi-Fredholm operators and the meeting of the classes of $k$-quasi-Fredholm operators is called the class of pseudo-quasi-Fredholm operators. We present some fundamental properties of the operators belonging to these classes and, as applications, we prove some spectral theorem and finite-dimensional perturbations results for these classes. Also, the notion of new index of a pseudo-quasi-Fredholm operator called $pq$-index is introduced and the stability of this index by finite-dimensional perturbations is proved. This paper extends some results proved in [5] to closed unbounded operators.

An extension of the first eigenvalue-type Ambarzumyan's theorem are provided for arbitrary self-adjoint Sturm-Liouville differential operators.

Let $G$ be a locally compact group. In this note, we characterise non-degenerate $*$-representations of $A_\Phi(G)$ and $B_\Phi(G).$ We also study spectral subspaces associated to a non-degenerate Banach space representation of $A_\Phi(G).$

We study an analogue of the logarithmic $(u,v)$-derivative. The last one has many interesting properties and good ways to calculate it. To show how it can be used we apply it to a model class of nowhere monotone functions that are composition of Salem function and nowhere differentiable functions.