Johannes M. Schumacher

arXiv2025

Keldysh's theorem revisited

In a variety of applications, the problem comes up of describing the principal part of the inverse of a holomorphic operator at an eigenvalue in terms of left and right root functions associated to the eigenvalue. Such a description was given by Keldysh in 1951. His theorem, the proof of which was published only in 1971, is a fundamental result in the local spectral theory of operator-valued functions. Here we present a streamlined derivation in the matrix case, and we extend Keldysh's theorem by means of a new principal part formula. Special attention is given to the semisimple case (first-order poles).

math.FA↗

arXiv2017

A Multi-Objective Interpretation of Optimal Transport

This paper connects discrete optimal transport to a certain class of multi-objective optimization problems. In both settings, the decision variables can be organized into a matrix. In the multi-objective problem, the notion of Pareto efficiency is defined in terms of the objectives together with non-negativity constraints and with equality constraints that are specified in terms of column sums. A second set of equality constraints, defined in terms of row sums, is used to single out particular points in the Pareto efficient set which are referred to as "balanced solutions". Examples from several fields are shown in which this solution concept appears naturally. Balanced solutions are shown to be in one-to-one correspondence with solutions of optimal transport problems. As an example of the use of alternative interpretations, the computation of solutions via regularization is discussed.

math.OC↗