$M$-operators on partially ordered Banach spaces

Document Type: Original Article

Authors

1 Institute for Analysis, Department of Mathematics, Technical University of Dresden, D - 01062 Dresden, Germany.

2 Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India.

3 IIT Madras, Chennai

Abstract

‎For a matrix $A \in \mathbb{R}^{n \times n}$ whose off-diagonal entries are nonpositive‎, ‎there are several well-known properties that are equivalent to $A$ being an invertible $M$-matrix‎. ‎One of them is the positive stability of $A$‎. ‎A generalization of this characterization to partially ordered Banach spaces is considered in this article‎. ‎Relationships with certain other equivalent conditions are derived‎. ‎An important result on singular irreducible $M$-matrices is generalized using the concept of $M$-operators and irreducibility‎. ‎Certain other invertibility conditions of $M$-operators are also investigated‎.

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