Translator Disclaimer
2018 Spectral inclusion for unbounded diagonally dominant $n\times n$ operator matrices
Tulkin H. Rasulov, Christiane Tretter
Rocky Mountain J. Math. 48(1): 279-324 (2018). DOI: 10.1216/RMJ-2018-48-1-279

Abstract

In this paper, we establish an analytic enclosure for the spectrum of unbounded linear operators~$\mathcal{A} $ admitting an $n \times n$ matrix representation in a Hilbert space $\mathcal{H} =\mathcal{H} _1\oplus \cdots \oplus \mathcal{H} _n$. For diagonally dominant operator matrices of order 0, we show that this new enclosing set, the block numerical range $W^n(\mathcal{A} )$, contains the eigenvalues of $\mathcal{A} $ and that the approximate point spectrum of $\mathcal{A} $ is contained in its closure $\overline {W^n(\mathcal{A} )}$. Since the block numerical range turns out to be a subset of the usual numerical range, $W^n(\mathcal{A} )\subset W(\mathcal{A} )$, it may give a tighter enclosure of the spectrum. Moreover, we prove Gershgorin theorems for diagonally dominant $n \times n$ operator matrices and compare our results to both Gershgorin bounds and classical perturbation theory. Our results are illustrated by deriving new lower bounds for $3\times 3$ self-adjoint operator matrices and applying the latter to three-channel Hamiltonians in quantum~mechanics.

Citation

Download Citation

Tulkin H. Rasulov. Christiane Tretter. "Spectral inclusion for unbounded diagonally dominant $n\times n$ operator matrices." Rocky Mountain J. Math. 48 (1) 279 - 324, 2018. https://doi.org/10.1216/RMJ-2018-48-1-279

Information

Published: 2018
First available in Project Euclid: 28 April 2018

zbMATH: 06866711
MathSciNet: MR3795744
Digital Object Identifier: 10.1216/RMJ-2018-48-1-279

Subjects:
Primary: 47A10, 47A12
Secondary: 47B25, 47B50

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

JOURNAL ARTICLE
46 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.48 • No. 1 • 2018
Back to Top