# 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, Web of Science, DOAJ, Google Scholar.

## Latest Articles (March, 2018)

### Adjunction formula, Poincaré residue and holomorphic differentials on Riemann surfaces

Methods Funct. Anal. Topology **24** (2018), no. 1, 41-52

There is still a big gap between knowing that a Riemann surface of genus $g$ has $g$ holomorphic differential forms and being able to find them explicitly. The aim of this paper is to show how to construct holomorphic differential forms on compact Riemann surfaces. As known, the dimension of the space $H^1(\mathcal{D}, \mathbb{C})$ of holomorphic differentials of a compact Riemann surface $\mathcal{D}$ is equal to its genus, $\dim H^1(\mathcal{D}, \mathbb{C})=g(\mathcal{D})=g$. When the Riemann surface is concretely described, we show that one can usually present a basis of holomorphic differentials explicitly. We apply the method to the case of relatively complicated Riemann surfaces.

### On similarity of unbounded perturbations of selfadjoint operators

Methods Funct. Anal. Topology **24** (2018), no. 1, 27-33

We consider a linear unbounded operator $A$ in a separable Hilbert space with the following property: there is an invertible selfadjoint operator $S$ with a discrete spectrum such that $\|(A-S)S^{-\nu}\|<\infty$ for a $\nu\in [0,1]$. Besides, all eigenvalues of $S$ are assumed to be different. Under certain assumptions it is shown that $A$ is similar to a normal operator and a sharp bound for the condition number is suggested. Applications of that bound to spectrum perturbations and operator functions are also discussed. As an illustrative example we consider a non-selfadjoint differential operator.

### Parametrization of scale-invariant self-adjoint extensions of scale-invariant symmetric operators

Miron B. Bekker, Martin J. Bohner, Alexander P. Ugolʹnikov, Hristo Voulov

Methods Funct. Anal. Topology **24** (2018), no. 1, 1-15

On a Hilbert space $\frak H$, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scale-invariant self-adjoint extension in $ \frak H$. We prove that there is a one-to-one correspondence between (generalized) resolvents of scale-invariant extensions and solutions of some functional equation. Two examples of Dirac-type operators are considered.

### Categories of unbounded operators

Methods Funct. Anal. Topology **24** (2018), no. 1, 71-81

In this article we introduce the concept of an $LK^\ast$-algebroid, which is defined axiomatically. The main example of an $LK^\ast$-algebroid is the category of all subspaces of a Hilbert space and closed (not necessarily bounded) linear operators. We prove that for any $LK^\ast$-algebroid there is a faithful functor that respects its structure and maps it into this main example.