Functional Analysis and Its Applications最新文献

We give a description of finite-zone (mathcal{PT})-potentials in terms of explicit theta-functional formulas.

Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if (X) is a countably compact space and (C_p(X)) is a space of continuous functions on (X) in the topology of pointwise convergence, then any countably compact subspace of the space (C_p(X)) is precompact, that is, it has a compact closure. The paper provides an overview of the results on this topic. It is proved that if a pseudocompact (X) contains a dense Lindelöf (Sigma)-space, then pseudocompact subspaces of the space (C_p(X)) are precompact. If (X) is the product Čech complete spaces, then bounded subsets of the space (C_p(X)) are precompact. Results on the continuity of separately continuous functions are also obtained.

The Poincare construction in CR geometry allows us to estimate the dimension of the stabilizer in the Lie algebra of infinitesimal holomorphic automorphisms of the germ of a CR manifold by the dimension of the stabilizer in the corresponding algebra of the model surface of this germ. We give a negative answer to the following natural question: is there an algebraic Poincare construction, i.e., is it true that the stabilizer in the Lie algebra of automorphisms of the germ of a CR manifold is isomorphic to a Lie subalgebra of the stabilizer in the algebra of its model surface? We also give a negative answer to the corresponding question for the whole automorphisms algebra.

In this paper, we demonstrate that for a locally compact Hausdorff space (S) and a decomposable Borel measure (mu), metric projectivity, injectivity, or flatness of the (C_0(S))-module (L_p(S,mu)) implies that (mu) is purely atomic with at most one atom.

The diffusion mechanism in Hamiltonian systems, close to completely integrable, is usually connected with the existence of the so-called “transition chains”. In this case slow diffusion occurs in a neighborhood of intersecting separatrices of hyperbolic periodic solutions (or, more generally, lower-dimensional invariant tori) of the perturbed system. In this note we discuss another diffusion mechanism that uses destruction of invariant tori of the unperturbed system with an almost resonant set of frequencies. We demonstrate this mechanism on a particular isoenergetically nondegenerate Hamiltonian system with three degrees of freedom. The same phenomenon also occurs for general higher-dimensional Hamiltonian systems. Drift of slow variables is shown using analysis of integrals of quasi-periodic functions of the time variable (possibly unbounded) with zero mean value. In addition, the proof uses the conditions of topological transitivity for cylindrical cascades.

We describe subgroups of elements of uniformly bounded degrees in Cremona groups of arbitrary rank. These subgroups appear naturally in (mathrm{CR})-geometry as holomorphic automorphism groups of nondegenerate homogeneous model surfaces.

We suggest a general construction of continuous Banach bundles of holomorphic function algebras on subvarieties of the closed noncommutative ball. These algebras are of the form (mathcal{A}_d/overline{I_x}), where (mathcal{A}_d) is the noncommutative disc algebra defined by G. Popescu, and (overline{I_x}) is the closure in (mathcal{A}_d) of a graded ideal (I_x) in the algebra of noncommutative polynomials, depending continuously on a point (x) of a topological space (X). Moreover, we construct bundles of Fréchet algebras (mathcal{F}_d/overline{I_x}) of holomorphic functions on subvarieties of the open noncommutative ball. The algebra (mathcal{F}_d) of free holomorphic functions on the unit ball was also introduced by G. Popescu, and (overline{I_x}) stands for the closure in (mathcal{F}_d) of a graded ideal (I_x) in the algebra of noncommutative polynomials, depending continuously on a point (xin X).

Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.

It is shown how certain recent results in the theory of determinants and traces can be applied to obtain new theorems on the distribution of eigenvalues of nuclear operators on Banach spaces and to prove the equality of the spectral and nuclear traces of such operators. As an example, we consider a new class of operators: the class of generalized Lapresté nuclear operators.