M. Langer and M. Strauss,
Spectral properties of unbounded J-self-adjoint block operator matrices,
J. Spectr. Theory (2017), 137–190

Abstract:
We study the spectrum of unbounded J-self-adjoint block operator matrices. In particular, we prove enclosures for the spectrum, provide a sufficient condition for the spectrum being real and derive variational principles for certain real eigenvalues even in the presence of non-real spectrum. The latter lead to lower and upper bounds and asymptotic estimates for eigenvalues.

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