# Methods of Functional Analysis

and Topology

Editors-in-Chief: Yu. M. Berezansky,
Yu. G. Kondratiev

ISSN: 1029-3531 (Print) 2415-7503 (Online)

Methods of Functional Analysis and Topology (MFAT), founded in 1995, is a peer-reviewed arXiv overlay journal publishing original articles and surveys on general methods and techniques of functional analysis and topology with a special emphasis on applications to modern mathematical physics.

MFAT is an open access journal, free for authors and free for readers.

MFAT is indexed in: MathSciNet, zbMATH, Scopus, Web of Science, DOAJ, Google Scholar

## Latest Articles (June, 2019)

### Abstract formulation of the Cole-Hopf transform

Methods Funct. Anal. Topology **25** (2019), no. 2, 142-151

Operator representation of Cole-Hopf transform is obtained based on the logarithmic representation of infinitesimal generators. For this purpose the relativistic formulation of abstract evolution equation is introduced. Even independent of the spatial dimension, the Cole-Hopf transform is generalized to a transform between linear and nonlinear equations defined in Banach spaces. In conclusion a role of transform between the evolution operator and its infinitesimal generator is understood in the context of generating nonlinear semigroup.

### Operators preserving orthogonality on Hilbert $\it{K}(H)$-modules

R. G. Sanati, E. Ansari-piri, M. Kardel

Methods Funct. Anal. Topology **25** (2019), no. 2, 189-194

In this paper, we study the class of orthogonality preserving operators on a Hilbert $\it{K(H)}$-module $W$ and show that an operator $T$ on $W$ is orthogonality preserving if and only if it is orthogonality preserving on a special dense submodule of $W$. Then we apply this fact to show that an orthogonality preserving operator $T$ is normal if and only if $T^*$ is orthogonality preserving.

### Weak-coupling limit for ergodic environments

Martin Friesen, Yuri Kondratiev

Methods Funct. Anal. Topology **25** (2019), no. 2, 118-133

The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [27]. Our approach is based on the study of forward Kolmogorov equations (a.k.a. Fokker-Planck equations). We describe a system evolving as a Markov process on the space of finite configurations, whereas its rates depend on the actual state of another (equilibrium) process on the space of locally finite configurations. We will show that ergodicity of the environment process implies the averaging principle for the solutions of the coupled Fokker-Planck equations.

### On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph

Olga Boiko, Olga Martynyuk, Vyacheslav Pivovarchik

Methods Funct. Anal. Topology **25** (2019), no. 2, 104-117

Recurrence relations of the second order on the edges of a metric connected graph together with boundary and matching conditions at the vertices generate a spectral problem for a self-adjoint finite-dimensional operator. This spectral problem describes small transverse vibrations of a graph of Stieltjes strings. It is shown that if the graph is cyclically connected and the number of masses on each edge is not less than 3 then the maximal multiplicity of an eigenvalue is $\mu+1$ where $\mu$ is the cyclomatic number of the graph. If the graph is not cyclically connected and each edge of it bears at least one point mass then the maximal multiplicity of an eigenvalue is expressed via $\mu$, the number of edges and the number of interior vertices in the tree obtained by contracting all the cycles of the graph into vertices.