On the pseudo-spectra and the related properties of infinite-dimensional Hamiltonian operators

Abstract

Given a linear operator T on a Hilbert space, the pseudo-point spectrum, the pseudo-residual spectrum, and the pseudo-continuous spectrum of T are defined, and some of their properties are studied. Furthermore, the structure of the pseudo-spectra of infinite-dimensional Hamiltonian operators is discussed, thus revealing their symmetry about the imaginary axis. Along the way, some examples are constructed to show the validity of our results.

Keywords:

• Pseudo-spectra
• pseudo-point spectrum
• pseudo-residual spectrum
• pseudo-continuous spectrum
• Hamiltonian operators

Acknowledgments

The authors are grateful to the reviewer for his/her kind comments and valuable suggestions, especially for English polishing, which greatly improved the readability and quality of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11861048, 11761029], the Program for Young Talents of Science and Technology in Universities of Inner Mongolia [grant number NJYT-15-B03], the NSF of Inner Mongolia [grant number 2021MS01004] and the YTTP of Prairie Excellence Program of Inner Mongolia Autonomous Region [grant number 1300020608].