On matching distance between eigenvalues of unbounded operators

Authors : Micheal GİL

Pages : 46-53

Doi:10.33205/cma.1060718

View : 40 | Download : 19

Publication Date : 2022-03-14

Article Type : Research Paper

Abstract :Let A A and ~ A A~ be linear operators on a Banach space having compact resolvents, and let λ k insert ignore into journalissuearticles values( A ); λkinsert ignore into journalissuearticles values(A); and λ k insert ignore into journalissuearticles values( ~ A ); insert ignore into journalissuearticles values( k = 1 , 2 , … ); λkinsert ignore into journalissuearticles values(A~);insert ignore into journalissuearticles values(k=1,2,…); be the eigenvalues taken with their algebraic multiplicities of A A and ~ A A~ , respectively. Under some conditions, we derive a bound for the quantity md insert ignore into journalissuearticles values( A , ~ A ); : = inf π sup k = 1 , 2 , … ∣ ∣ λ π insert ignore into journalissuearticles values( k ); insert ignore into journalissuearticles values( ~ A ); − λ k insert ignore into journalissuearticles values( A ); ∣ ∣, md⁡insert ignore into journalissuearticles values(A,A~);:=infπsupk=1,2,…|λπinsert ignore into journalissuearticles values(k);insert ignore into journalissuearticles values(A~);−λkinsert ignore into journalissuearticles values(A);|, where π π 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 A and ~ A A~ . Applications of the obtained bound to matrix differential operators are also discussed.
Keywords : Banach space, perturbations of eigenvalues, matching distance, differential operator, tensor product of Hilbert spaces

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