On positive invertibility of operators and their decompositions

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)