6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Ekaterina Shulman

Approximation of invariant subsets by invariant subspaces with application to Levi-Civita functional equations
Let T be a continuous representation of an amenable group G on a linear topological space X. Let Y ⊂ X be a closed subspace of X invariant for T and supplied with a norm ∥·∥Y such that the restriction of T to Y is an isometric representation.
For ξ ∈ X and any subspace X0⊂X, we define the Y-distance from ξ to X0 by

dY(ξ,X0) = inf{∥y∥: y ∈ Y, ξ -y ∈ X0}.

Statement: For each ϵ>0 and n∈N there is a δ>0 with the following property:
if L is a finite-dimensional subspace of X with dim(L) ≤ n and

dY(Tgξ,L) < δ   for all  g ∈ G

then there is a T-invariant subspace M⊂X with dim(M) ≤ 3n and

dY(ξ,M) < ϵ.

We apply the result to the study of stability of the Levi-Civita functional equation

f(gh)=∑ j=1N uj(g)vj(h), g, h ∈ G

in the class of measurable functions on an amenable group G.
Stability in the Ulam-Hyers sense means that each function that “almost” satisfies the equation is “close” to a proper solution.
For bounded functions the study of stability of ([1]) leads to the following geometric problem:

given a representation g ↦ Tg of a group G on a Banach space X, for any invariant subset K⊂ X to estimate its distance to invariant subspaces of X via the distances to arbitrary n-dimensional subspaces.

In this setting the problem was studied in [1] using a specially developed techniques of covariant width, and the estimate is more strict: dim(M) ≤ dim(L).

  1. E. Shulman,: Group representations and stability of functional equations, J. London Math. Soc., 54 (1996), 111-120.
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