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Том 51, № 2 (2017)
- Год: 2017
- Статей: 11
- URL: https://journals.rcsi.science/0016-2663/issue/view/14569
Article
Lyudvig Dmitrievich Faddeev
81-83
Mikhail Semenovich Agranovich
84-85
Moduli algebras of some non-semiquasihomogeneous singularities
Аннотация
Under some additional restrictions we find dimensions and bases of moduli algebras of isolated singularities of polynomials in n variables that are sums of n monomials of equal weighted degrees and one monomial of lower degree.
86-97
Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument
Аннотация
We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of order α+∞+···+∞ (m times) and its subgroup K ≅ U(∞), which consists of block diagonal unitary matrices with the identity block of order α and a matrix u ∈ U(∞) repeated m times. It turns out that there is a natural multiplication on the space G//K of conjugacy classes. We construct “spectral data” of conjugacy classes, which visualize the multiplication and are sufficient for reconstructing a conjugacy class.
98-111
On operators with orbits dense relative to nontrivial subspaces
Аннотация
In the present paper we consider bounded linear operators which have orbits dense relative to nontrivial subspaces. We give nontrivial examples of such operators and establish many of their basic properties. An example of an operator which has an orbit dense relative to a certain subspace but is not subspace-hypercyclic for this subspace is given. This, in turn, provides a new answer to a question posed in [18]. Other hypercyclic-like properties of such operators are also considered.
112-122
Cohomology of the rational “electric” tetrahedron relation
Аннотация
A tetrahedral cochain complex is generalized to the case of the functional “electric” solution of the tetrahedron equation, which is expressed in terms of rational functions. The nontrivial part of the 3-cohomology group for this solution is calculated. It turns out to be the free Abelian group with one generator; the generator is explicitly specified.
123-134
Spectra of 3 × 3 upper triangular operator matrices
Аннотация
Let H1, H2, and H3 be complex separable Hilbert spaces. Given A ∈ B(H1), B ∈ B(H2), and C ∈ B(H3), write \({M_{D,E,F}} = \left( {\begin{array}{*{20}{c}} A&D&E \\ 0&B&F \\ 0&0&C \end{array}} \right)\), where D ∈ B(H2,H1), E ∈ B(H3,H1), and F ∈ B(H3,H2) are unknown operators. This paper gives a complete description of the intersection ∩D,E,Fσ(MD,E,F), where D, E, and F range over the respective sets of bounded linear operators. Further, we show that σ(A) ∪ σ(B) ∪ σ(C) = σ(MD,E,F) ∪ W, where W is the union of certain gaps in σ(MD,E,F), which are subsets of (σ(A) ∩ σ(B)) ∪ (σ(B) ∩ σ(C)) ∪ (σ(A) ∩ σ(C)). Finally, we obtain a necessary and sufficient condition for the relation σ(MD,E,F) = σ(A)∪σ(B)∪σ(C) to hold for any D, E, and F.
135-143
Brief Communications
144-147
A short and simple proof of the Jurkat–Waterman theorem on conjugate functions
Аннотация
It is well known that certain properties of continuous functions on the circle T related to the Fourier expansion can be improved by a change of variable, i.e., by a homeomorphism of the circle onto itself. One of the results in this area is the Jurkat–Waterman theorem on conjugate functions, which improves the classical Bohr–Pál theorem. In the present work we propose a short and technically very simple proof of the Jurkat–Waterman theorem. Our approach yields a stronger result.
148-151
On homogenization for non-self-adjoint locally periodic elliptic operators
Аннотация
In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on Rd of the form Aε = −divA(x, x/ε)∇. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (Aε − μ)−1, including one with a corrector, and for (−Δ)s/2(Aε − μ)−1 in the operator norm on L2(Rd)n. For s ≠ 0, we also give estimates of the rates of approximation.
152-156
Fourier transform and quasi-analytic classes of functions of bounded type on tubular domains
Аннотация
A condition for a function of bounded type to belong to the Hardy class H1 in terms of the Fourier transform of the boundary values of this function on Rn is found. Applications of the obtained result to the theories of Hardy classes and of quasi-analytic classes of functions are given.
157-160