Advances in Operator Theory

$M$-operators on partially ordered Banach spaces

A. ‎Kalauch, S. ‎Lavanya, and K. C. ‎Sivakumar

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‎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.

Article information

Adv. Oper. Theory, Volume 4, Number 2 (2019), 481-496.

Received: 13 June 2018
Accepted: 27 October 2018
First available in Project Euclid: 1 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B60: Operators on ordered spaces
Secondary: 15B48‎ ‎46B40‎ ‎47B65

$M$-operator‎ ‎positive stability ‎ ‎‎‎invertibility‎ ‎ ‎irreducibility‎


‎Kalauch, A.; ‎Lavanya, S.; ‎Sivakumar, K. C. $M$-operators on partially ordered Banach spaces. Adv. Oper. Theory 4 (2019), no. 2, 481--496. doi:10.15352/aot.1806-1383.

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