Methods of Functional Analysis
and Topology

This paper is devoted to an investigation of the relationship between the essential approximate point spectrum (respectively, the essential defect spectrum) of a sequ\-ence of closed linear operators $(T_n)_{n\in\mathbb{N}}$ on a Banach space $X$, and the essential approximate point spectrum (respectively, the essential defect spectrum) of a linear operator $T$ on $X$, where $(T_n)_{n\in\mathbb{N}}$ converges to $T$, in the case of convergence in generalized sense as well as in the case of the convergence compactly

We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums of Weyl-Nagy classes $W^r_{\beta,p}, 1\le p\le\infty,$ for rapidly growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform metric. We obtain similar estimates for approximations of the classes $W^r_{\beta,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.

We discuss the spectral problem for limit distributions of conflict dynamical systems on spaces subjected to fractal divisions. Conditions ensuring the existence of the point spectrum are established in two cases, the repulsive and the attractive interactions between the opponents. A key demand is the strategy of priority in a single region.

The (main) spectral problem for a star graph with three edges composed of Stieltjes strings is considered with the Dirichlet conditions at the pendant vertices. In addition we consider the Dirichlet-Neumann problem on the first edge (Problem 2) and the Dirichlet-Dirichlet problem on the union of the second and the third strings (Problem 3). It is shown that the spectrum of the main problem interlace (in a non-strict sense) with the union of spectra of Problems 2 and 3. The inverse problem lies in recovering the masses of the beads (point masses) and the lengths of the intervals between them using the spectra of the main problem and of Problems 2 and 3. Conditions on three sequences of numbers are proposed sufficient to be the spectra of the main problem and of Problems 2 and 3, respectively.