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Vol 213, No 8 (2022)
- Year: 2022
- Articles: 4
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7492
Representation of invariant subspaces of the Schwartz space
Abstract
A subspace $W$ of the Schwartz space $C^{\infty} (a,b)$ such that the restriction of the operator of differentiation to $W$ has a discrete spectrum is considered. Conditions for the representation of $W$ as a direct algebraic and topological sum of two subspaces, namely, the residual subspace and the subspace spanned by the exponential monomials from $W$, are investigated. One condition ensuring this representation turns out to be the existence of a functional annihilating $W$ such that the Fourier-Laplace transform of this functional is a slowly decreasing entire function. A new characteristic of complex sequences is introduced and investigated. Using this characteristic, the condition that an invariant subspace is equal to the direct sum of its residual and exponential subspaces can be put into a form that is similar to the previously discovered conditions for the possibility of weak spectral synthesis.Bibliography: 19 titles.
Matematicheskii Sbornik. 2022;213(8):3-25
3-25
Inner functions of matrix argument and conjugacy classes in unitary groups
Abstract
Let $\mathrm{B}_n$ denote the set of complex square matrices of order $n$ whose Euclidean operator norms are less than one. Its Shilov boundary is the set $\operatorname{U}(n)$ of all unitary matrices. A holomorphic map $\mathrm{B}_m\to\mathrm{B}_n$ is inner if it sends $\operatorname{U}(m)$ to $\operatorname{U}(n)$. On the other hand we consider the group $\operatorname{U}(n+mj)$ and its subgroup $\operatorname{U}(j)$ that is embedded in $\operatorname{U}(n+mj)$ in the block-diagonal way ($m$ blocks $\operatorname{U}(j)$ and a unit block of size $n$). To any conjugacy class of $\operatorname{U}(n+mj)$ with respect to $\operatorname{U}(j)$ we assign a ‘characteristic function’, which is a rational inner map $\mathrm{B}_m\to\mathrm{B}_n$. We show that the class of inner functions that can be obtained as ‘characteristic functions’ is closed with respect to such natural operations as pointwise direct sums, pointwise products, compositions, substitutions into finite-dimensional representations of general linear groups and so on. We also describe explicitly the corresponding operations on conjugacy classes. Bibliography: 24 titles.
Matematicheskii Sbornik. 2022;213(8):26-43
26-43
On multipliers for Fourier series in Sobolev orthogonal polynomials
Abstract
Multipliers are studied for Fourier series in polynomials orthogonal in continuous-discrete Sobolev spaces. For the multiplier operator, existence results and norm estimates are obtained. The proofs are based on a representation of the Fejer kernel, the construction of a ‘humpbacked majorant’ and estimates for the norms of maximal functions. Bibliography: 45 titles.
Matematicheskii Sbornik. 2022;213(8):44-82
44-82
Henselian division algebras and reduced unitary Whitehead groups for outer forms of anisotropic algebraic groups of the type $A_n$
Abstract
Some results on the structure of involutorial (that is, having an involution) Henselian tamely ramified division algebras are obtained. These results are then used to derive formulae for the computation of the reduced unitary Whitehead groups for outer forms of anisotropic algebraic groups of type $A_n$. Bibliography: 46 titles.
Matematicheskii Sbornik. 2022;213(8):83-148
83-148