Methods of Functional Analysis
and Topology

We study the emergence problem of new points in the discrete spectrum under singular perturbations of a positive operator. We start with the sequential approach to construction of additional eigenvalues for perturbed operators, which was produced by V. Koshmanenko on the base of rigged Hilbert spaces methods. Two new observations are established. We show that one can construct a point of the discrete spectrum of any finite multiplicity in a single step. And that the method of rigged Hilbert spaces admits an application to the modified construction of a new point of the discrete spectrum under super-singular perturbations.

We study relations between spectra of two operators that are connected to each other through some intertwining conditions. As an application, we obtain new results on the spectra of multiplication operators on $B(\mathcal H)$ relating it to the spectra of the restriction of the operators to the ideal $\mathcal C_2$ of Hilbert-Schmidt operators. We also solve one of the problems, posed in [6], about the positivity of the spectrum of multiplication operators with positive operator coefficients when the coefficients on one side commute. Using the Wiener-Pitt phenomena we show that the spectrum of a multiplication operator with normal coefficients satisfying the Haagerup condition might be strictly larger than the spectrum of its restriction to $\mathcal C_2$.

In this paper we study the effect of subordination to the solution of a model of spatial ecology in terms of the evolution density. The asymptotic behavior of the subordinated solution for different rates of spatial propagation is studied. The difference between subordinated solutions to non-linear equations with classical time derivative and solutions to non-linear equation with fractional time derivative is discussed.