On matching distance between eigenvalues of unbounded operators

Let A and A similar to be linear operators on a Banach space having compact resolvents, and let lambda k(A) and lambda k( A similar to) (k = 1, 2, ...) be the eigenvalues taken with their algebraic multiplicities of A and A similar to, respectively. Under some conditions, we derive a bound for the quantitymd(A, A similar to) := inf pi sup k=1,2,... |lambda pi(k)( A similar to) - lambda k(A)|,where pi 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 similar to. Applications of the obtained bound to matrix differential operators are also discussed.