Spectral inclusion for unbounded diagonally dominant n × n operator matrices

Rasulov, Tulkin; Tretter, Christiane (2018). Spectral inclusion for unbounded diagonally dominant n × n operator matrices. Rocky Mountain journal of mathematics, 48(1), pp. 279-324. Rocky Mountain Mathematics Consortium 10.1216/RMJ-2018-48-1-279

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In this paper, we establish an analytic enclosure for the spectrum of unbounded linear operators ~A admitting an n×n matrix representation in a Hilbert space H=H₁⊕⋯⊕Hn. For diagonally dominant operator matrices of order 0, we show that this new enclosing set, the block numerical range Wⁿ(A), contains the eigenvalues of A and that the approximate point spectrum of A is contained in its closure Wⁿ(A). Since the block numerical range turns out to be a subset of the usual numerical range, Wⁿ(A)⊂W(A), it may give a tighter enclosure of the spectrum. Moreover, we prove Gershgorin theorems for diagonally dominant n×n operator matrices and compare our results to both Gershgorin bounds and classical perturbation theory. Our results are illustrated by deriving new lower bounds for 3×3 self-adjoint operator matrices and applying the latter to three-channel Hamiltonians in quantum~mechanics.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Rasulov, Tulkin and Tretter, Christiane

Subjects:

500 Science > 510 Mathematics

ISSN:

0035-7596

Publisher:

Rocky Mountain Mathematics Consortium

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

22 May 2019 14:31

Last Modified:

22 May 2019 14:31

Publisher DOI:

10.1216/RMJ-2018-48-1-279

URI:

https://boris.unibe.ch/id/eprint/125531

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