Eigenmeasures and stochastic diagonalization of bilinear maps

Abstract

A new stochastic approach is presented to understand general spectral type problems for (not necessarily linear) functions between topological spaces. In order to show its potential applications, we construct the theory for the case of bilinear forms acting in couples of a Banach space and its dual. Our method consists of using integral representations of bilinear maps that satisfy particular domination properties, which is shown to be equivalent to having a certain spectral structure. Thus, we develop a measure‐based technique for the characterization of bilinear operators having a spectral representation, introducing the notion of eigenmeasure, which will become the central tool of our formalism. Specific applications are provided for operators between finite and infinite dimensional linear spaces.

Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

“…The aim of this section is to explain the motivation of our investigation and to provide the first general results. Let us start by presenting how we can rewrite the well-known context of diagonalization or linear operators in terms of integral expressions, under the formalism provided by the recently introduced eigenmeasures [26]. We expect to find the best diagonal approximation to a given (linear and continuous) operator T : E → E. Following the mentioned paper, we introduce first a measure on B E * .…”

“…Let us fix now the measure structure on the measurable space (B E , B(B E )) that is needed to complete the integral diagonal representation of the approximation to the operator T. Let us recall the definition of weak eigenmeasure (Definition 2 in [26]), adapted for the case of linear maps. Definition 1.…”

“…In this paper, we will use a stronger version of this notion, which has been also introduced in [26]. We say that a probability measure satisfying the requirement above is an eigenmeasure if λ N (x) does not depend on N, that is, it has a constant value once x is fixed.…”

“…Thus, let us describe now how to get an approximation based on eigenmeasures in this case. In [26], it is shown, for the case of bilinear maps, how the calculus of eigenmeasures can be used to deal with non-linear diagonalization. Here, we assume that such an eigenmeasure µ is given for a certain non-linear map f : E → E, where E is a (finite dimensional) Euclidean space, and we show how we can use it to obtain an approximation to f .…”

Extension procedures for lattice Lipschitz operators on Euclidean spaces

“…Nonlinear spectral theory is already a classical subject and has a long development, both from a theoretical and practical point of view ([2, 5, 10]). For example, for the case of multilinear mappings [6,7,8,16], polynomials [14] or Lipschitz maps [9]. In particular, we have studied suitable versions of these notions for Lipschitz maps in [3,9].…”