On matching distance between eigenvalues of unbounded operators

Year 2022, Volume: 5 Issue: 1, 46 - 53, 14.03.2022

Micheal Gil

https://doi.org/10.33205/cma.1060718

Abstract

Let $A$ and $\tilde{A}$ be linear operators on a Banach space having compact resolvents, and let ${\lambda }_k\left(A\right)$ and ${\lambda }_k\left(\tilde{A}\right)(k=1,2,\dots )$ be the eigenvalues taken with their algebraic multiplicities of $A$ and $\tilde{A}$, respectively. Under some conditions, we derive a bound for the quantity
$$\mathrm{md}(A,\tilde{A}):=\underset{\pi }{inf}\underset{k=1,2,\dots }{sup}\left|{\lambda }_{\pi \left(k\right)}\right(\tilde{A})-{\lambda }_k(A\left)\right|$$
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 $\tilde{A}$. 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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