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On positive invertibility of operators and their decompositions

✍ Scribed by M. R. Weber

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
122 KB
Volume
282
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

In Banach spaces ordered by a normal cone that contains interior points the positive invertibility of operators is studied. If there exists a uniformly positive functional then any positively invertible operator A possesses a B ‐decomposition, i.e., a positive decomposition A = U – V with the properties: U^–1^ exists, VU^–1^ ≥ 0, Ax ≥ 0, U x ≥ 0 imply x ≥ 0 and r (VU^–1^) < 1. Earlier it was shown that the existence of a B ‐decomposition with r (VU^–1^) < 1 is sufficient for the positive invertibility of the operator A. Peris' result is obtained as a special case of the main theorem. The decomposition is demonstrated for a positively invertible operator in a Banach space ordered by an ice cream cone (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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