Erratum to: A generalized Jentzsch theorem

Erratum to: A generalized Jentzsch theorem Positivity (2011) 15:537 DOI 10.1007/s11117-011-0125-9 Positivity ERRATUM A. K. Kitover Published online: 10 May 2011 © Springer Basel AG 2011 Erratum to: Positivity (2005) 9:501–509 DOI 10.1007/s11117-004-8291-7 There is an error in the proof of Lemma 10 in the above-mentioned paper. The impli- cation that operator T leaves invariant the principal band generated by the element z because it leaves invariant the principal ideal generated by z cannot be justified because we do not assume T to be σ -order continuous. I am grateful to Anton Schep who indicated to me the said error. The following changes have to be made. (1) In part (a) of Theorem 6 the operator R should be assumed order continuous instead of just σ -order continuous. (I do not know if the result remains true under the milder condition of σ -order continuity of R) (2) The proof of Lemma 10 then goes as follows: Let us prove first that T is σ -order continuous. Let x ↓ 0 then Rx ↓ 0 and n n w w Rx → 0 whence Sx ↓ 0 and Sx → 0. Assume contrary to what we claim that n n n Tx ≥ y ≥ 0 and y = 0. Then ST x ≥ Sy and ST x ≤ TSx → 0. Therefore n n n n Sy = 0. Let Z be the maximal by inclusion ideal in X such that S| Z = 0. Z obviously exists by Zorn’s lemma. Then Z is a band because S is order-continuous. We claim that TZ ⊆ Z . Indeed, otherwise there is a positive u ∈ Z such that Tu ∈ Z.But ST u ≤ TSu = 0 and we come to a contradiction with the maximality of Z.The remaining part of the proof does not have to be changed. The online version of the original article can be found under doi:10.1007/s11117-004-8291-7. A. K. Kitover ( ) Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden Street, Philadelphia, PA 19130, USA e-mail: akitover@ccp.edu http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Erratum to: A generalized Jentzsch theorem

Positivity , Volume 15 (3) – May 10, 2011
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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Econometrics; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0125-9
Publisher site
See Article on Publisher Site

Abstract

Positivity (2011) 15:537 DOI 10.1007/s11117-011-0125-9 Positivity ERRATUM A. K. Kitover Published online: 10 May 2011 © Springer Basel AG 2011 Erratum to: Positivity (2005) 9:501–509 DOI 10.1007/s11117-004-8291-7 There is an error in the proof of Lemma 10 in the above-mentioned paper. The impli- cation that operator T leaves invariant the principal band generated by the element z because it leaves invariant the principal ideal generated by z cannot be justified because we do not assume T to be σ -order continuous. I am grateful to Anton Schep who indicated to me the said error. The following changes have to be made. (1) In part (a) of Theorem 6 the operator R should be assumed order continuous instead of just σ -order continuous. (I do not know if the result remains true under the milder condition of σ -order continuity of R) (2) The proof of Lemma 10 then goes as follows: Let us prove first that T is σ -order continuous. Let x ↓ 0 then Rx ↓ 0 and n n w w Rx → 0 whence Sx ↓ 0 and Sx → 0. Assume contrary to what we claim that n n n Tx ≥ y ≥ 0 and y = 0. Then ST x ≥ Sy and ST x ≤ TSx → 0. Therefore n n n n Sy = 0. Let Z be the maximal by inclusion ideal in X such that S| Z = 0. Z obviously exists by Zorn’s lemma. Then Z is a band because S is order-continuous. We claim that TZ ⊆ Z . Indeed, otherwise there is a positive u ∈ Z such that Tu ∈ Z.But ST u ≤ TSu = 0 and we come to a contradiction with the maximality of Z.The remaining part of the proof does not have to be changed. The online version of the original article can be found under doi:10.1007/s11117-004-8291-7. A. K. Kitover ( ) Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden Street, Philadelphia, PA 19130, USA e-mail: akitover@ccp.edu

Journal

PositivitySpringer Journals

Published: May 10, 2011

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