# Vol. 23 (2017), no. 1

### Myroslav Lvovych Gorbachuk (Obituary)

Editorial Board

Methods Funct. Anal. Topology 23 (2017), no. 1, 1-2

### The Liouville property for harmonic functions on groups and hypergroups

Herbert Heyer

Methods Funct. Anal. Topology 23 (2017), no. 1, 3-25

A survey is given on the Liouville property of harmonic functions on groups and hypergroups. The discussion of a characterization of that property in terms of the underlying algebraic structures yields interesting open problems.

### Infinitesimal generators of invertible evolution families

Yoritaka Iwata

Methods Funct. Anal. Topology 23 (2017), no. 1, 26-36

A logarithm representation of operators is introduced as well as a concept of pre-infinitesimal generator. Generators of invertible evolution families are represented by the logarithm representation, and a set of operators represented by the logarithm is shown to be associated with analytic semigroups. Consequently generally-unbounded infinitesimal generators of invertible evolution families are characterized by a convergent power series representation.

### On Fourier algebra of a hypergroup constructed from a conditional expectation on a locally compact group

Methods Funct. Anal. Topology 23 (2017), no. 1, 37-50

We prove that the Fourier space of a hypergroup constructed from a conditional expectation on a locally compact group has a Banach algebra structure.

### Sturm-Liouville operators with matrix distributional coefficients

Methods Funct. Anal. Topology 23 (2017), no. 1, 51-59

The paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ with the matrix-valued coefficients $p,q$ such that $$q=Q', \quad p^{-1},\, p^{-1}Q, \,\, Qp^{-1}, \,\, Qp^{-1}Q \in L_1,$$ where the derivative of the function $Q$ is understood in the sense of distributions. Due to a suitable regularization, the corresponding operators are correctly defined as quasi-differentials. Their resolvent convergence is investigated and all self-adjoint, maximal dissipative, and maximal accumulative extensions are described in terms of homogeneous boundary conditions of the canonical form.

### On certain spectral features inherent to scalar type spectral operators

Marat V. Markin

Methods Funct. Anal. Topology 23 (2017), no. 1, 60-65

Important spectral features such as the emptiness of the residual spectrum, countability of the point spectrum, provided the space is separable, and a characterization of spectral gap at 0, known to hold for bounded scalar type spectral operators, are shown to naturally transfer to the unbounded case.

### On new inverse spectral problems for weighted graphs

Methods Funct. Anal. Topology 23 (2017), no. 1, 66-75

In this paper, we consider various new inverse spectral problems (ISP) for metric graphs, using maximal eigen values of the adjacency matrix of the graph and its subgraphs as well as the corresponding eigen vectors or some of their components as spectral data. We give examples of spectral data that uniquely determine the metric on the graph. Effective algorithms for solving the considered ISP are given.

### Tannaka-Krein reconstruction for coactions of finite quantum groupoids

Methods Funct. Anal. Topology 23 (2017), no. 1, 76-107

We study coactions of finite quantum groupoids on unital $C^*$-algebras and obtain a Tannaka-Krein reconstruction theorem for them.