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Abstract

The Levi-Civita functional equation $f(gh) = \sum_{k=1}^n u_k(g)v_k(h)$ $(g,h\in G)$, 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 $g\mapsto T_g$ 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 $T_gL\cap M \neq \{0\}$ for all $g\in G$? 4) which finite-dimensional subspaces $L$ have the property that for each $g\in G$ there is $0\neq x\in L$ with $T_gx = x$? 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 $\sum_{i=1}^m a_i(g)b_i(hg) = \sum_{j=1}^n u_j(g)v_j(h)$. 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.