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
Free
1 page

Loading next page...
1 Page
 
/lp/springer_journal/erratum-to-a-generalized-jentzsch-theorem-cxDCNfxqHt
Publisher
Springer Journals
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

There are no references for this article.

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off