Sectorial perturbations of self-adjoint matrices and operators

Abstract

This paper considers $N\times N$ matrices of the form $A_{\gamma }=A+\gamma B$, where $A$ is self-adjoint, $\gamma \in C$ and $B$ is a non-self-adjoint perturbation of $A$. We obtain some monodromy-type results relating the spectral behaviour of such matrices in the two asymptotic regimes $|\gamma |\to \infty$ and $|\gamma |\to 0$ under certain assumptions on $B$. We also explain some properties of the spectrum of $A_{\gamma }$ for intermediate sized $\gamma$ by considering the limit $N\to \infty$, concentrating on properties that have no self-adjoint analogue. A substantial number of the results extend to operators on infinite-dimensional Hilbert spaces.

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References in corpus (3)

• Rank one perturbations and singular integral operators
• Eigenvalues of rank one perturbations of unstructured matrices
• Rank one and finite rank perturbations - survey and open problems

Cited by in corpus (1)

• Spectral enclosure and superconvergence for eigenvalues in gaps