On matching distance between eigenvalues of unbounded operators

Abstract

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