Abstract:

Let $T$ and $S$ be densely defined closed linear operators and bounded linear operators in the Hilbert space $\mathcal{X}$, respectively. Denote by $\lambda S - T$ the operator pencils. This paper investigates the structure of the fine pseudo-spectra of $\lambda S - T$, including pseudo-point spectra, pseudo-continuous spectra, and pseudo-residual spectra. Furthermore, a necessary and sufficient condition is derived for the equality of the generalized pseudo-spectra of operator pencils to the minimal pseudo-spectra. Let $H$ denote Hamiltonian operators and $J$ denote symplectic identity operators; the symmetries of the pseudo-spectra and their subdivisions for the operator pencils $\lambda J - H$ are obtained. Additionally, it is proven that the pseudo-spectra of $\lambda J - H$ coincide with both the generalized pseudo-spectra and the minimal pseudo-spectra. Finally, an illustrative example is provided to substantiate the theoretical conclusions.

Key words: operator pencils, pseudo-spectra, operator matrices, Hamiltonian operators