## Advances in Operator Theory

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

### $M$-operators on partially ordered Banach spaces

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

#### 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.

#### Article information

**Source**

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

**Dates**

Received: 13 June 2018

Accepted: 27 October 2018

First available in Project Euclid: 1 December 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1543633239

**Digital Object Identifier**

doi:10.15352/aot.1806-1383

**Mathematical Reviews number (MathSciNet)**

MR3883148

**Zentralblatt MATH identifier**

07009321

**Subjects**

Primary: 47B60: Operators on ordered spaces

Secondary: 15B48 46B40 47B65

**Keywords**

$M$-operator positive stability invertibility irreducibility

#### Citation

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. https://projecteuclid.org/euclid.aot/1543633239