# Vol. 20 (2014), no. 2

### Yury Stefanovich Samoilenko (to 70th birthday anniversary)

MFAT **20** (2014), no. 2, 101-102

101-102

### Parameter-elliptic problems and interpolation with a function parameter

Anna V. Anop, Aleksandr A. Murach

MFAT **20** (2014), no. 2, 103-116

103-116

Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to a Hilbert Sobolev scale. The latter are the Hörmander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary radial function RO-varying at $+\infty$. We prove that the operator corresponding to this problem sets isomorphisms between appropriate Hörmander spaces provided the absolute value of the parameter is large enough. For solutions to the problem, we establish two-sided estimates, in which the constants are independent of the parameter.

### A simplicity criterion for symmetric operator on a graph

E. N. Ashurova, A. N. Kandagura, I. I. Karpenko

MFAT **20** (2014), no. 2, 117-123

117-123

In the present paper we show that the topology of the underlying graph determines the domain and deficiency indices of a certain associated minimal symmetric operator. We obtaine a criterion of simplicity for the minimal operator associated with the graph.

### Continuity of operator-valued functions in the $*$-algebra of locally measurable operators

MFAT **20** (2014), no. 2, 124-133

124-133

In the present paper we establish sufficient conditions for a complex-valued function $f$ defined on $\mathbb{R}$ which guarantee continuity of an operator-function $T\mapsto f(T)$ w.r.t. the topology of local measure convergence in the $*$-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated to a von Neumann algebra $\mathcal{M}$.

### Trace formulae for Schrödinger operators on metric graphs with applications to recovering matching conditions

Yulia Ershova, Alexander V. Kiselev

MFAT **20** (2014), no. 2, 134-148

134-148

The paper is a continuation of the study started in [8]. Schrödinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of $\delta$ type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of $L_1$ and $C^M$ edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

### Some remarks on Hilbert representations of posets

MFAT **20** (2014), no. 2, 149–163

149–163

For a certain class of finite posets, we prove that all their irreducible orthoscalar representations are finite-dimensional and describe those, for which there exist essential (non-degenerate) irreducible orthoscalar representations.

### Spectral problem for a graph of symmetric Stieltjes strings

MFAT **20** (2014), no. 2, 164-174

164-174

A spectral problem generated by the Stieltjes string recurrence relations with a finite number of point masses on a connected graph is considered with Neumann conditions at pendant vertices and continuity and Kirchhoff conditions at interior vertices. The strings on the edges are supposed to be the same and symmetric with respect to the midpoint of the string. The characteristic function of such a problem is expressed via characteristic functions of two spectral problems on an edge: one with Dirichlet conditions at the both ends and the other one with the Neumann condition at one end and the Dirichlet condition at the other end. This permits to find values of the point masses and the lengths of the subintervals into which the masses divide the string from knowing the spectrum of the problem on the graph and the length of an edge. If the number of vertices is less than five then the spectrum uniquely determines the form of the graph.

### Arens algebras of measurable operators for Maharam traces

MFAT **20** (2014), no. 2, 175-185

175-185

We study order and topological properties of the non-commutative Arens algebra associated with arbitrary Maharam trace.

### An exponential representation for some integrals with respect to Lebesgue-Poisson measure

MFAT **20** (2014), no. 2, 186-192

186-192

We prove a theorem that allows to simplify some combinatorial calculations. An example of application of this theorem in statistical mechanics is given.