Authors Index, Vol. 14, 2008

Boundary problems for fully nonlinear parabolic equations with Lévy Laplacian

Methods Funct. Anal. Topology 14 (2008), no. 1, 1-9

We suggest a method to solve boundary and initial-boundary value problems for a class of nonlinear parabolic equations with the infinite dimensional L'evy Laplacian $\Delta _L$ $$f\Bigl(U(t,x),\frac{\partial U(t,x)}{\partial t},\Delta_LU(t,x)\Bigl)=0$$ in fundamental domains of a Hilbert space.

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.

About my first meeting with S. I. Zuchovitsky

Yu. M. Berezansky

Methods Funct. Anal. Topology 14 (2008), no. 3, 207-208

Recursion relation for orthogonal polynomials on the complex plane

Methods Funct. Anal. Topology 14 (2008), no. 2, 108-116

The article deals with orthogonal polynomials on compact infinite subsets of the complex plane. Orthogonal polynomials are treated as coordinates of generalized eigenvector of a normal operator $A$. It is shown that there exists a recursion that gives the possibility to reconstruct these polynomials. This recursion arises from generalized eigenvalue problem and, actually, this means that every gene alized eigenvector of $A$ is also a generalized eigenvector of $A^*$ with the complex conjugated eigenvalue. If the subset is actually the unit circle, it is shown that the presented algorithm is a generalization of the well-known Szego recursion from OPUC theory.

Inverse problem for Stieltjes string damped at one end

Methods Funct. Anal. Topology 14 (2008), no. 1, 10-19

Small transversal vibrations of the Stieltjes string, i.e., an elastic thread bearing point masses is considered for the case of one end being fixed and the other end moving with viscous friction in the direction orthogonal to the equilibrium position of the string. The inverse problem of recovering the masses, the lengths of subintervals and the coefficient of damping by the spectrum of vibrations of such a string and its total length is solved.

Inverse spectral problem for a star graph of Stieltjes strings

Methods Funct. Anal. Topology 14 (2008), no. 2, 159-167

We solve the inverse spectral problem for a star graph of Stieltjes strings (these are threads bearing a finite number of point masses) with the pendant ends fixed, i.e., we recover the masses and lengths of the intervals between them from the spectra of small transverse vibrations of the graph together with the spectra of the Dirichlet problems on the edges and the total lengths of the edges.

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.

On rank one perturbation of continuous spectrum which generates given finite point spectrum

Methods Funct. Anal. Topology 14 (2008), no. 1, 20-31

The perturbations of Nevanlinna type functions which preserve the set of zeros of this function or add to this set new points are discussed.

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.

An exact inner structure of the block Jacobi-type unitary matrices connected with the corresponding direct and inverse spectral problems

Mykola E. Dudkin

Methods Funct. Anal. Topology 14 (2008), no. 2, 168-176

We discuss a problem posed by M. J. Cantero, L. Moral, and L.~Vel\'azquez about representing an arbitrary unitary operator with a CMV-matrix. We consider this problem from the point of view of a one-to-one correspondence between a non-finite unitary operator and an infinite (five-diagonal) block three-diagonal Jacobi-type matrix in the form of the corresponding direct and inverse spectral problems for the trigonometric moment problem. Since the earlier obtained block three-diagonal Jacobi-type unitary matrix has not been fully described, we continue this investigations in the present article. In particular, we show that this exact inner structure coincides with an earlier obtained CMV-matrix.

Semion Israilevich Zuchovitsky (to the centenary of his birth)

Editorial Board

Methods Funct. Anal. Topology 14 (2008), no. 3, 201-205

S. I. Zuchovitsky’s recollection on M. F. Kravchuk

Editorial Board

Methods Funct. Anal. Topology 14 (2008), no. 3, 206

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.

$g$-frames and stability of $g$-frames in Hilbert spaces

Methods Funct. Anal. Topology 14 (2008), no. 3, 271-286

Wenchang Sun in his paper [Wenchang Sun, $G$-frames and $g$-Riesz bases, J. Math. Anal. Appl. 322 (2006), 437--452] has introduced $g$-frames which are generalized frames and include ordinary frames and many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. In this paper we develop the $g$-frame theory for separable Hilbert spaces and give characterizations of $g$-frames and we show that $g$-frames share many useful properties with frames. We present a version of the Paley-Wiener Theorem for $g$-frames which is in spirit close to results for frames, due to Ole Christensen.

Quasilinear parabolic equations with a Lévy Laplacian for functions of infinite number of variables

Methods Funct. Anal. Topology 14 (2008), no. 2, 117-123

We construct solutions to initial, boundary and initial-boundary value problems for quasilinear parabolic equations with an infinite dimensional Lévy Laplacian $\Delta _L$, $$\frac{\partial U(t,x)}{\partial t}=\Delta_LU(t,x)+f_0(U(t,x)),$$ in fundamental domains of a Hilbert space. The solution is defined in the functional class where a solution of the corresponding problem for the heat equation $\frac {\partial U(t,x)}{\partial t}=\Delta_LU(t,x)$ exists.

On two-component contact model in continuum with one independent component

Methods Funct. Anal. Topology 14 (2008), no. 3, 209-228

Properties of a contact process in continuum for a system of particles of two types, one which is independent of the other, are considered. We study dynamics of the first and the second order correlation functions, their asymptotics, and the dependence on parameters of the~system.

On the approximation to solutions of operator equations by the least squares method

Methods Funct. Anal. Topology 14 (2008), no. 3, 229-241

We consider the equation $Au = f$, where $A$ is a linear operator with compact inverse in a Hilbert space. For the approximate solution $u_n$ of this equation by the least squares method in a coordinate system that is an orthonormal basis of eigenvectors of a self-adjoint operator $B$ similar to $A \ ({\mathcal{D}} (A) = {\mathcal{D}} (B))$, we give a priori estimates for the asymptotic behavior of the expression $R_n = \|Au_n - f\|$ as $n \to \infty$. A relationship between the order of smallness of this expression and the degree of smoothness of the solution $u$ with respect to the operator $B$ (direct and converse theorems) is established.

On solutions of parabolic and elliptic type differential equations on $(-\infty, \infty)$ in a Banach space

Volodymyr M. Gorbachuk

Methods Funct. Anal. Topology 14 (2008), no. 2, 177-183

We show that every classical solution of a parabolic or elliptic type homogeneous differential equation on $(-\infty, \infty)$ in a Banach space may be extended to an entire vector-valued function. The description of all the solutions is given, and necessary and sufficient conditions for a solution to be continued to a finite order and finite type entire vector-valued function are presented.

About nilpotent $C_0$-semigroups of operators in the Hilbert spaces and criteria for similarity to the integration operator

Methods Funct. Anal. Topology 14 (2008), no. 1, 60-66

In the paper, we describe a class of operators $A$ that have empty spectrum and satisfy the nilpotency property of the generated $C_0$-semigroup $U(t)=\exp\{-iAt\},\, t\geqslant 0$, and such that the operator$A^{-1}$ is similar to the integration operator on the corresponding space $L_2(0,a)$.

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 $L_2(-\sigma,\sigma)$.

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.

Generalized stochastic derivatives on a space of regular generalized functions of Meixner white noise

N. A. Kachanovsky

Methods Funct. Anal. Topology 14 (2008), no. 1, 32-53

We introduce and study generalized stochastic derivatives on a Kondratiev-type space of regular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to the properties of the stochastic derivatives in the Gaussian analysis. As an example we calculate the generalized stochastic derivative of the solution of some stochastic equation with a Wick-type nonlinearity.

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.

The involutive automorphisms of $\tau$-compact operators affiliated with a type I von Neuman algebra

Methods Funct. Anal. Topology 14 (2008), no. 1, 54-59

Let $M$ be a type I von Neumann algebra with a center $Z,$ and a faithful normal semi-finite trace $\tau.$ Consider the algebra $L(M, \tau)$ of all $\tau$-measurable operators with respect to $M$ and let $S_0(M, \tau)$ be the subalgebra of $\tau$-compact operators in $L(M, \tau).$ We prove that any $Z$-linear involutive automorphisms of $S_0(M, \tau)$ is inner.

On some approximations and main topological descriptions for special classes of Banach spaces with integrable derivatives

Methods Funct. Anal. Topology 14 (2008), no. 3, 255-270

We consider some classes of Banach spaces with integrable derivatives. An important compactness lemma for nonreflexive spaces is obtained. However some main topological properties for the given spaces are obtained.

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.

A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes

Methods Funct. Anal. Topology 14 (2008), no. 1, 67-80

We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space $X$, for which certain fermion point processes are invariant. The Glauber dynamics is a birth-and-death process in $X$, while in the case of the Kawasaki dynamics interacting particles randomly hop over $X$. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.

One-dimensional Schrödinger operators with singular periodic potentials

Methods Funct. Anal. Topology 14 (2008), no. 2, 184-200

We study the one-dimensional Schrödinger operators $$S(q)u:=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q) \right),$$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the Hilbert space $L_{2}\left(\mathbb{R} \right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described.

Interpolation with a function parameter and refined scale of spaces

Methods Funct. Anal. Topology 14 (2008), no. 1, 81-100

The interpolation of couples of separable Hilbert spaces with a function parameter is studied. The main properties of the classical interpolation are proved. Some applications to the interpolation of isotropic Hörmander spaces over a closed manifold are given.

Douglis-Nirenberg elliptic systems in the refined scale of spaces on a closed manifold

Aleksandr A. Murach

Methods Funct. Anal. Topology 14 (2008), no. 2, 142-158

Douglis-Nirenberg elliptic systems of linear pseudodifferential equations are studied on a smooth closed manifold. We prove that the operator generated by the system is a Fredholm one on the refined two-sided scale of the functional Hilbert spaces. Elements of this scale are the special isotropic spaces of H\"{o}rmander--Volevich--Paneah. The refined smoothness of a solution of the system is studied. The elliptic systems with a parameter are investigated as well.

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.

Finding generalized Walras-Wald equilibrium

Roman A. Polyak

Methods Funct. Anal. Topology 14 (2008), no. 3, 242-254

The Generalized Walras-Wald Equilibrium (GE) was introduced by S. I. Zuchovitsky et al. in 1973 (see \cite{17}) as an alternative to Linear Programming (LP) approach for optimal resources allocation. There are two fundamental differences between the GE and LP approach for the best resources allocation. First, the prices for goods (products) are not fixed as it is in LP; they are functions of the production output. Second, the factors (resources) used in the production process are not fixed either; they are functions of the prices for resources. In this paper we show that under natural economic assumptions on both price and factor functions the GE exists and is unique. Finding GE is equivalent to solving a variational inequality with a strongly monotone operator. For solving the variational inequality we introduce projected pseudo-gradient method. We prove that under the same assumptions on price and factor functions the projected pseudo-gradient method converges globally with $Q$-linear rate. It allows estimating its computational complexity and finding parameters critical for the complexity bound. The method can be viewed as a natural pricing mechanism for establishing economics equilibrium.

On stability, superstability and strong superstability of classical systems of statistical mechanics

Methods Funct. Anal. Topology 14 (2008), no. 3, 287-296

A detailed analysis of conditions on 2-body interaction potential, which ensure stability, superstability or strong superstability of statistical systems is given. We give a connection between conditions of superstability (strong superstability) and the problem of minimization of Riesz energy in bounded volumes.

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.

A stochastic integral of operator-valued functions

Volodymyr Tesko

Methods Funct. Anal. Topology 14 (2008), no. 2, 132-141

In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes with respect to normal martingales and the Ito integral in a Fock space.

The portrait of a young man (from memory)

Leonid I. Vainerman

Methods Funct. Anal. Topology 14 (2008), no. 2, 102-107

The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices

S. M. Zagorodnyuk

Methods Funct. Anal. Topology 14 (2008), no. 2, 124-131

We consider a difference equation associated with a semi-infinite complex $(2N+1)$-diagonal transposition-antisymmetric matrix $J=(g_{k,l})_{k,l=0}^\infty$ with $g_{k,k+N} \not=0$, $k=0,1,2,\ldots ,$ ($g_{k,l}=-g_{l,k}$): $\sum_{j=-N}^N g_{k,k+j} y_{k+j} = \lambda^N y_k,\ k=0,1,2,\ldots ,$ where $y=(y_0,y_1,y_2,\ldots )$ is an unknown vector, $\lambda$ is a complex parameter, $g_{k,l}$ and $y_l$ with negative indices are equal to zero, $N\in\mathbb N$. We introduce a notion of the spectral function for this difference equation. We state and solve the direct and inverse problems for this equation.