Methods of Functional Analysis
and Topology

The Perron--Frobenius theorem states that a linear stochastic operator associated with a positive square stochastic matrix has a unique fixed point in the simplex and it is strongly ergodic to that fixed point. However, in general, the similar result for quadratic stochastic operators associated with positive cubic stochastic matrices does not hold true. Namely, it may have more than one fixed point in the simplex. Moreover, the uniqueness of fixed points does not imply the strong ergodicity of quadratic stochastic operators. In this paper, for some classes of positive cubic stochastic matrices, we provide a uniqueness criterion for fixed points of quadratic stochastic operators acting on a 2D simplex. Some supporting examples are also presented.

The article is devoted to extensions of linear functionals, generated by scalar products, in a scale of Hilbert spaces. Such extensions are used to consider non-symmetrically singular rank one perturbations of ${\mathcal H}_{-2}$-class. For comparison, we give main definitions and descriptions of singular non-symmetric perturbations of ${\mathcal H}_{-1}$ and ${\mathcal H}_{-2}$-classes.

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 $\mathcal C_2$.