Some Properties of Essential Spectra of a Positive Operator

Some Properties of Essential Spectra of a Positive Operator Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ ew(T) of the operator T is a set $$\sigma_{\rm ew}(T) = {\mathop{\cap}\limits_{K \in {\mathcal {K}}(E)}} \sigma (T + K)$$ , where $${\mathcal K}(E)$$ is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum $$\sigma_{\rm ew}^+(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E)} \sigma (T + K), \ \ \ \ \ \sigma_{\rm ew}^-(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E) \le T} \sigma (T - K)$$ are introduced in the article. The conditions by which $$r(T) \not\in \sigma_{\rm ef}(T)$$ implies either $$r(T) \not\in \sigma_{\rm ew}^+(T)$$ or $$r(T) \not\in \sigma_{\rm ew}^-(T)$$ are investigated, where σ ef(T) is the Fredholm essential spectrum. By this reason, the relations between coefficients of the main part of the Laurent series of the resolvent R(., T) of a positive operator T around of the point λ  =  r(T) are studied. The example of a positive integral operator T : L 1→ L ∞ which doesn’t dominate a non-zero compact operator, is adduced. Applications of results which are obtained, to the spectral theory of band irreducible operators, are given. Namely, the criteria when the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T), are established, where T is a band irreducible abstract integral operator. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Some Properties of Essential Spectra of a Positive Operator

Positivity , Volume 11 (3) – Jan 1, 2007

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Publisher
Springer Journals
Copyright
Copyright © 2007 by Birkhäuser Verlag, Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2088-4
Publisher site
See Article on Publisher Site

Abstract

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ ew(T) of the operator T is a set $$\sigma_{\rm ew}(T) = {\mathop{\cap}\limits_{K \in {\mathcal {K}}(E)}} \sigma (T + K)$$ , where $${\mathcal K}(E)$$ is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum $$\sigma_{\rm ew}^+(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E)} \sigma (T + K), \ \ \ \ \ \sigma_{\rm ew}^-(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E) \le T} \sigma (T - K)$$ are introduced in the article. The conditions by which $$r(T) \not\in \sigma_{\rm ef}(T)$$ implies either $$r(T) \not\in \sigma_{\rm ew}^+(T)$$ or $$r(T) \not\in \sigma_{\rm ew}^-(T)$$ are investigated, where σ ef(T) is the Fredholm essential spectrum. By this reason, the relations between coefficients of the main part of the Laurent series of the resolvent R(., T) of a positive operator T around of the point λ  =  r(T) are studied. The example of a positive integral operator T : L 1→ L ∞ which doesn’t dominate a non-zero compact operator, is adduced. Applications of results which are obtained, to the spectral theory of band irreducible operators, are given. Namely, the criteria when the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T), are established, where T is a band irreducible abstract integral operator.

Journal

PositivitySpringer Journals

Published: Jan 1, 2007

References

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