On matching distance between eigenvalues of unbounded operators

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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