On matching distance between eigenvalues of unbounded operators

Abstract

Let AA and ~AA~ be linear operators on a Banach space having compact resolvents, and let λk(A)λk(A) and λk(~A)(k=1,2,…)λk(A~)(k=1,2,…) be the eigenvalues taken with their algebraic multiplicities of AA and ~AA~, respectively. Under some conditions, we derive a bound for the quantity md(A,~A):=infπsupk=1,2,…∣∣λπ(k)(~A)−λk(A)∣∣,md⁡(A,A~):=infπsupk=1,2,…|λπ(k)(A~)−λk(A)|, where ππ is taken over all permutations of the set of all positive integers. That quantity is called the matching optimal distance between the eigenvalues of AA and ~AA~. Applications of the obtained bound to matrix differential operators are also discussed.