Addition theorems and related geometric problems of group representation theory

Ekaterina Shulman. "Addition theorems and related geometric problems of group representation theory." Banach Center Publications 99.1 (2013): 155-172. <http://eudml.org/doc/286113>.

@article{EkaterinaShulman2013,
abstract = {The Levi-Civita functional equation $f(gh) = ∑_\{k=1\}^\{n\} u_\{k\}(g)v_\{k\}(h)$ (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation $g ↦ T_\{g\}$ have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for which x, O(x) is contained in the sum of a fixed finite-dimensional subspace and a finite-dimensional invariant subspace? 3) what can be said about a pair L, M of finite-dimensional subspaces if $T_\{g\}L ∩ M ≠ \{0\}$ for all g ∈ G? 4) which finite-dimensional subspaces L have the property that for each g ∈ G there is 0 ≠ x ∈ L with $T_\{g\}x = x$? The problem 1) arises in the study of the Hyers-Ulam stability of the L-C equation. It leads to the theory of covariant widths - the analogues of Kolmogorov widths which measure the distances from a given set to n-dimensional invariant subspaces. The problem 2) is related to multivariable extensions of the L-C equation; the study of this problem is based on the theory of subadditive set-valued functions which was developed specially for this aim. To problems 3) and 4) one comes via the study of the equations $∑_\{i=1\}^\{m\} a_\{i\}(g)b_\{i\}(hg) = ∑_\{j=1\}^\{n\} u_\{j\}(g)v_\{j\}(h)$. We will finish by the consideration of “fractionally-linear version” of the L-C equation which is very important for the theory of integrable dynamical systems.},
author = {Ekaterina Shulman},
journal = {Banach Center Publications},
keywords = {functional equations on semigroups; addition theorems; representations of topological semigroups; stability in the Hyers-Ulam sense; subadditive set-valued functions on groups; elliptic functions},
language = {eng},
number = {1},
pages = {155-172},
title = {Addition theorems and related geometric problems of group representation theory},
url = {http://eudml.org/doc/286113},
volume = {99},
year = {2013},
}

TY - JOUR
AU - Ekaterina Shulman
TI - Addition theorems and related geometric problems of group representation theory
JO - Banach Center Publications
PY - 2013
VL - 99
IS - 1
SP - 155
EP - 172
AB - The Levi-Civita functional equation $f(gh) = ∑_{k=1}^{n} u_{k}(g)v_{k}(h)$ (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation $g ↦ T_{g}$ have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for which x, O(x) is contained in the sum of a fixed finite-dimensional subspace and a finite-dimensional invariant subspace? 3) what can be said about a pair L, M of finite-dimensional subspaces if $T_{g}L ∩ M ≠ {0}$ for all g ∈ G? 4) which finite-dimensional subspaces L have the property that for each g ∈ G there is 0 ≠ x ∈ L with $T_{g}x = x$? The problem 1) arises in the study of the Hyers-Ulam stability of the L-C equation. It leads to the theory of covariant widths - the analogues of Kolmogorov widths which measure the distances from a given set to n-dimensional invariant subspaces. The problem 2) is related to multivariable extensions of the L-C equation; the study of this problem is based on the theory of subadditive set-valued functions which was developed specially for this aim. To problems 3) and 4) one comes via the study of the equations $∑_{i=1}^{m} a_{i}(g)b_{i}(hg) = ∑_{j=1}^{n} u_{j}(g)v_{j}(h)$. We will finish by the consideration of “fractionally-linear version” of the L-C equation which is very important for the theory of integrable dynamical systems.
LA - eng
KW - functional equations on semigroups; addition theorems; representations of topological semigroups; stability in the Hyers-Ulam sense; subadditive set-valued functions on groups; elliptic functions
UR - http://eudml.org/doc/286113
ER -