Methods of Functional Analysis
and Topology

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.

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.

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.