### On some sublattices of regular operators on Banach lattices

Methods Funct. Anal. Topology 14 (2008), no. 4, 297-301

We give some sufficient conditions under which the linear span of positive compact (resp. Dunford-Pettis, weakly compact, AM-compact) operators cannot be a vector lattice without being a sublattice of the order complete vector lattice of all regular operators. Also, some interesting consequences are obtained.

### On certain resolvent convergence of one non-local problem to a problem with spectral parameter in boundary condition

E. V. Cheremnikh

Methods Funct. Anal. Topology 14 (2008), no. 4, 302-313

A family of non-local problems with the same finite point spectrum is given. The resolvent convergence on a dense linear subspace which gives a problem with spectral parameter in the boundary condition is considered. The spectral eigenvalue decomposition of the last problem on the half line for Sturm-Liouville operator with trivial potential is given.

Methods Funct. Anal. Topology 14 (2008), no. 4, 314-322

In the present paper we describe semiadditive functionals and establish that the construction generated by semiadditive functionals forms a covariant functor. We show that the functor of semiadditive functionals is a normal functor acting in category of compact sets.

### On solvability of a partial integral equation in the space ${L_2(\Omega \times\Omega)}$

Yu. Kh. Eshkabilov

Methods Funct. Anal. Topology 14 (2008), no. 4, 323-329

In this paper we investigate solvability of a partial integral equation in the space $L_2(\Omega\times\Omega),$ where $\Omega=[a,b]^ u.$ We define a determinant for the partial integral equation as a continuous function on $\Omega$ and for a continuous kernels of the partial integral equation we give explicit description of the solution.

### One remark about the unconditional exponential bases and cosine bases, connected with them

Methods Funct. Anal. Topology 14 (2008), no. 4, 330-333

In the paper we consider examples of basis families $\{\cos \lambda_k t\}^\infty_1$, $\lambda_k>0$, in the space $L_2(0,\sigma)$, such that systems $\{e^{i\lambda_kt},e^{-i\lambda_kt}\}^\infty_1$ don't form an unconditional basis in space $L_2(-\sigma,\sigma)$.

### Generalized stochastic derivatives on parametrized spaces of regular generalized functions of Meixner white noise

N. A. Kachanovsky

Methods Funct. Anal. Topology 14 (2008), no. 4, 334-350

We introduce and study Hida-type stochastic derivatives and stochastic differential operators on the parametrized Kondratiev-type spaces of regular generalized functions of Meixner white noise. In particular, we study the interconnection between the stochastic integration and differentiation. Our researches are based on the general approach that covers the Gaussian, Poissonian, Gamma, Pascal and Meixner cases.

### Vanishing of the first $(\sigma, \tau)$-cohomology group of triangular Banach algebras

Methods Funct. Anal. Topology 14 (2008), no. 4, 351-360

In this paper, we define the first topological $(\sigma,\tau)$-cohomology group and examine vanishing of the first $(\sigma,\tau)$-cohomology groups of certain triangular Banach algebras. We apply our results to study the $(\sigma,\tau)$-weak amenability and $(\sigma,\tau)$-amenability of triangular Banach algebras.

### Representation of commutants for composition operators induced by a hyperbolic linear fractional automorphisms of the unit disk

Yu. S. Linchuk

Methods Funct. Anal. Topology 14 (2008), no. 4, 361-371

We describe the commutant of the composition operator induced by a hyperbolic linear fractional transformation of the unit disk onto itself in the class of linear continuous operators which act on the space of analytic functions. Two general classes of linear continuous operators which commute with such composition operators are constructed.

### The criteria of maximal dissipativity and self-adjointness for a class of differential-boundary operators with bounded operator coefficients

Methods Funct. Anal. Topology 14 (2008), no. 4, 372-379

A class of the second order differential-boundary operators acting in the Hilbert space of infinite-dimensional vector-functions is investigated. The domains of considered operators are defined by nonstandard (e.g., multipoint-integral) boundary conditions. The criteria of maximal dissipativity and the criteria of self-adjointness for investigated operators are established.

### On unitary operators in weighted spaces $A^2_\omega(\mathbb{C})$ of entire functions

Methods Funct. Anal. Topology 14 (2008), no. 4, 380-385

The paper gives a complete characterization of all unitary operators acting in some wide Hilbert spaces $A^2_\omega(\mathbb{C})$ of entire functions possessing weighted square integrable modulus over the whole finite complex plane, which exhaust the set of all entire functions.

### Sufficient conditions for superstability of many-body interactions

M. V. Tertychnyi

Methods Funct. Anal. Topology 14 (2008), no. 4, 386-396

A detailed analysis of sufficient conditions on a family of many-body potentials, which ensure stability, superstability or strong superstability of a statistical system is given in present work.There has been given also an example of superstable many-body interaction.