Translator Disclaimer
2014 On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges
Ivan S. Feshchenko
Banach J. Math. Anal. 8(1): 55-63 (2014). DOI: 10.15352/bjma/1381782087

Abstract

Let $\mathcal{H}$ be a complex Hilbert space, and $A_1,\ldots,A_N$ be bounded self-adjoint operators in $\mathcal{H}$ such that $A_i A_j$ is compact for any $i\neq j$. It is well-known that $\sigma_e(\sum_{i=1}^N A_i)\setminus\{0\}=(\cup_{i=1}^N\sigma_e(A_i))\setminus\{0\}$, where $\sigma_e(B)$ stands for the essential spectrum of a bounded self-adjoint operator $B$. In this paper we get necessary and sufficient conditions for $0\in\sigma_e(\sum_{i=1}^N A_i)$. This conditions are formulated in terms of the projection valued spectral measures of $A_i$, $i=1,\ldots,N$. Using this result, we obtain necessary and sufficient conditions for the sum of ranges of $A_i$, $i=1,\ldots,N$ to be closed.

Citation

Download Citation

Ivan S. Feshchenko. "On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges." Banach J. Math. Anal. 8 (1) 55 - 63, 2014. https://doi.org/10.15352/bjma/1381782087

Information

Published: 2014
First available in Project Euclid: 14 October 2013

zbMATH: 1310.47003
MathSciNet: MR3161682
Digital Object Identifier: 10.15352/bjma/1381782087

Subjects:
Primary: 47B15
Secondary: 46C07

Rights: Copyright © 2014 Tusi Mathematical Research Group

JOURNAL ARTICLE
9 PAGES


SHARE
Vol.8 • No. 1 • 2014
Back to Top