# Vol. 15 (2009), no. 2

### The integration of double-infinite Toda lattice by means of inverse spectral problem and related questions

MFAT **15** (2009), no. 2, 101-136

101-136

The solution of the Cauchy problem for differential-difference double-infinite Toda lattice by means of inverse spectral problem for semi-infinite block Jacobi matrix is given. Namely, we construct a simple linear system of three differential equations of first order whose solution gives the spectral matrix measure of the aforementioned Jacobi matrix. The solution of the Cauchy problem for the Toda lattice is given by the procedure of orthogonalization w.r.t. this spectral measure, i.e. by the solution of the inverse spectral problem for this Jacobi matrix.

### Expansion in eigenfunctions of relations generated by pair of operator differential expressions

MFAT **15** (2009), no. 2, 137-151

137-151

For relations generated by a pair of operator symmetric differential expressions, a class of generalized resolvents is found. These resolvents are integro-differential operators. The expansion in eigenfunctions of these relations is obtained.

### Extension of some Lions-Magenes theorems

MFAT **15** (2009), no. 2, 152-167

152-167

A general form of the Lions-Magenes theorems on solvability of an elliptic boundary-value problem in the spaces of nonregular distributions is proved. We find a general condition on the space of right-hand sides of the elliptic equation under which the operator of the problem is bounded and has a finite index on the corresponding couple of Hilbert spaces. Extensive classes of the spaces satisfying this condition are constructed. They contain the spaces used by Lions and Magenes and many others spaces.

### About $\ast$-representations of polynomial semilinear relations

MFAT **15** (2009), no. 2, 168-176

168-176

In the present paper we study $\ast$-representations of semilinear relations with polynomial characteristic functions. For any finite simple non-oriented graph $\Gamma$ we construct a polynomial characteristic function such that $\Gamma$ is its graph. Full description of graphs which satisfy polynomial (degree one and two) semilinear relations is obtained. We introduce the $G$-orthoscalarity condition and prove that any semili ear relation with quadratic characteristic function and condition of $G$-orthoscalarity is $\ast$-tame. This class of relations contains, in particular, $\ast$-representations of $U_{q}(so(3)).$

### Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms

S. Albeverio, Sh. A. Ayupov, A. A. Zaitov, J. E. Ruziev

MFAT **15** (2009), no. 2, 177-187

177-187

In the present paper derivations and $*$-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all $L^0$-linear derivations and $L^{0}$-linear $*$-automorphisms are inner. Moreover, it is proved that each $L^0$-linear automorphism of the algebra of all linear operators on a $bo$-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.

### A class of distal functions on semitopological semigroups

MFAT **15** (2009), no. 2, 188-194

188-194

The norm closure of the algebra generated by the set $\{n\mapsto {\lambda}^{n^k}:$ $\lambda\in{\mathbb {T}}$ and $k\in{\mathbb{N}}\}$ of functions on $({\mathbb {Z}}, +)$ was studied in \cite{S} (and was named as the Weyl algebra). In this paper, by a fruitful result of Namioka, this algebra is generalized for a general semitopological semigroup and, among other things, it is shown that the elements of the involved algebra are distal. In particular, we examine this algebra for $({\mathbb {Z}}, +)$ and (more generally) for the discrete (additive) group of any countable ring. Finally, our results are treated for a bicyclic semigroup.

### On the group of foliation isometries

A. Ya. Narmanov, A. S. Sharipov

MFAT **15** (2009), no. 2, 195-200

195-200

The purpose of our paper is to introduce some topology on the group $G_F^{r}(M)$ of all $C^{r}$-isometries of foliated manifold $(M,F)$, which depends on a foliation $F$ and coincides with compact-open topology when $F$ is an $n$-dimensional foliation. If the codimension of $F$ is equal to $n$, convergence in our topology coincides with pointwise convergence, where $ n=\operatorname{dim}M.$ It is proved that the group $G_F^{r}(M)$ is a topological group with compact-open topology, where $r\geq{0}.$ In addition it is showed some properties of F-compact-open topology.