Functional Analysis and Its Applications最新文献

In this paper, a parametric Korteweg–de Vries hierarchy is defined that depends on an infinite set of graded parameters (a = (a_4,a_6,dots)). It is shown that, for any genus (g), the Klein hyperelliptic function (wp_{1,1}(t,lambda)) defined on the basis of the multidimensional sigma function (sigma(t, lambda)), where (t = (t_1, t_3,dots, t_{2g-1})) and (lambda = (lambda_4, lambda_6,dots, lambda_{4 g + 2})), specifies a solution to this hierarchy in which the parameters (a) are given as polynomials in the parameters (lambda) of the sigma function. The proof uses results concerning the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of (g) third-order differential operators in (g) variables. Such families are defined for all (g geqslant 1), the operators in each of them pairwise commute with each other and also commute with the Schrödinger operator. In this paper a relationship between these families and the Korteweg–de Vries parametric hierarchy is described. A similar infinite family of third-order operators on an infinite set of variables is constructed. The results obtained are extended to the case of such a family.

Given disjoint countable dense subsets (C) and (D) of the half-line ((1,+infty)), there exists a flow (T_t) preserving a sigma-finite measure and such that all automorphisms (T_1otimes T_{c}) with (cin C) have simple singular spectrum and all automorphisms (T_1otimes T_{d}) with (din D) have Lebesgue spectrum of countable multiplicity.

In (L_2(mathbb R^d;mathbb C^n)) we consider a self-adjoint elliptic second-order differential operator (A_varepsilon). It is assumed that the coefficients of (A_varepsilon) are periodic and depend on (mathbf x/varepsilon), where (varepsilon>0) is a small parameter. We study the behavior of the operator exponential (e^{-iA_varepsilontau}) for small (varepsilon) and (tauinmathbb R). The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation (ipartial_tau mathbf{u}_varepsilon(mathbf x,tau) = - (A_varepsilon{mathbf u}_varepsilon)(mathbf x,tau)) with initial data in a special class. For fixed (tau) and (varepsilonto 0), the solution ({mathbf u}_varepsilon(,boldsymbolcdot,,tau)) converges in (L_2(mathbb R^d;mathbb C^n)) to the solution of the homogenized problem; the error is of order (O(varepsilon)). We obtain approximations for the solution ({mathbf u}_varepsilon(,boldsymbolcdot,,tau)) in (L_2(mathbb R^d;mathbb C^n)) with error (O(varepsilon^2)) and in (H^1(mathbb R^d;mathbb C^n)) with error (O(varepsilon)). These approximations involve appropriate correctors. The dependence of errors on (tau) is traced.

In this paper we generalize the notion of the Taylor spectrum to modules over an arbitrary Lie algebra and study it for finite-dimensional modules. We show that the spectrum can be described as the set of simple submodules in the case of nilpotent and semisimple Lie algebras. We also show that this result does not hold for solvable Lie algebras and obtain a precise description of the spectrum in the case of Borel subalgebras of semisimple Lie algebras.

An analytic semigroup of operators on a Banach space is approximated by a sequence of positive integer powers of a linear-fractional operator function. It is proved that the order of the approximation error in the domain of the generating operator equals (O(n^{-2}ln(n))). For a self-adjoint positive definite operator (A) decomposed into a sum of self-adjoint positive definite operators, an approximation of the semigroup (exp(-tA)) ((tgeq0)) by weighted averages is also considered. It is proved that the order of the approximation error in the operator norm equals (O(n^{-1/2}ln(n))).

Let (E) be a unital (f)-module over an (f)-algebra (A). With the help of Arens extension theory, a ((A^{sim})_{n}^{sim}) module structure on (E^{sim}) can be defined. The paper deals mainly with properties of the Arens homomorphism (etacolon(A^{sim})_{n}^{sim}to operatorname {Orth}(E^{sim})), which is defined by the ((A^{sim})_{n}^{sim}) module structure on (E^{sim}). Necessary and sufficient conditions for an (A) submodule of (E) to be an order ideal are obtained.

We characterize Newton polytopes of nondegenerate quadratic forms and Newton polyhedra of Morse singularities.

Let (alpha) be a complex scalar, and let (A) be a bounded linear operator on a Hilbert space (H). We say that (alpha) is an extended eigenvalue of (A) if there exists a nonzero bounded linear operator (X) such that (AX=alpha XA). In weighted Hardy spaces invariant under automorphisms, we completely compute the extended eigenvalues of composition operators induced by linear fractional self-mappings of the unit disk (mathbb{D}) with one fixed point in (mathbb{D}) and one outside (overline{mathbb{D}}). Such classes of transformations include elliptic and loxodromic mappings as well as a hyperbolic nonautomorphic mapping.

We study the densities of measures that are polynomial images of the standard Gaussian measure on (mathbb{R}^n). We assume that the degree of a polynomial is fixed and each variable appears in the monomials of the polynomial to powers bounded by another fixed number.

According to a known characterization, a function (f) belongs to the Sobolev space (W^{p,1}(mathbb{R}^n)) of functions contained in (L^p(mathbb{R}^n)) along with their generalized first-order derivatives precisely when there is a function (gin L^p(mathbb{R}^n)) such that