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On positive invertibility of operators and their decompositions. (English) Zbl 1190.47039

A real \(n \times n\) matrix \(M\) is called monotone (or a matrix of “monotone kind”) if \(Mx \geq 0 ~\Rightarrow x \geq 0\). Here, \(y \geq 0\) for \((y_1,y_2,\dots,y_n)=y \in \mathbb R^n\) means that \(y_i \geq 0\) for all \(i=1,2,\dots,n\). This notion was introduced by L.Collatz [Arch.Math., 3, 366–376 (1952; Zbl 0048.09802)], where he showed that \(M\) is monotone if and only if \(M^{-1}\) exists and \(M^{-1} \geq 0\), where the latter denotes that all the entries of \(M^{-1}\) are nonnegative. The book by L.Collatz [“Functional analysis and numerical mathematics” (German) (1964; Zbl 0139.09802), English translation (1966; Zbl 0148.39002)] has details of how monotone matrices arise naturally in the study of finite difference approximation methods for certain elliptic partial differential equations.
The problem of characterizing monotone (also referred to as inverse-positive) matrices has been extensively dealt with in the literature. Much effort has also been devoted on characterizing inverse-positive matrices in terms of the so-called splittings of the matrix concerned. The book by A.Berman and R.J.Plemmons [“Nonnegative matrices in the mathematical sciences” (1979; Zbl 0484.15016)] gives an excellent account of many of these characterizations and certain extensions to generalized inverses. Generalizations of inverse-positivity to operators over infinite-dimensional spaces were studied by J.Schroeder, M.A.Krasnoselskij (and his collaborators) and, much more recently, by M.I.Gill (citations are given in the paper under review).
In regard to connections between splittings and inverse-positivity, J.Peris [Linear Algebra Appl.154–156, 45–58 (1991; Zbl 0734.15008)] came up with the following characterization: An invertible matrix \(M\) has a non-negative inverse if and only if for any decomposition \(M= U - V\) as a difference of two non-negative matrices \(U\) and \(V\), there exist a vector \(x > 0\) (meaning that all the components are positive) and a scalar \(\mu \in [0, 1)\) such that \(Vx= \mu Ux\). The author of the present paper in [M.Weber, Math.Nachr.163, 145–149 (1993), erratum ibid.171, 325–326 (1995; Zbl 0834.47030)] extended this particular result to ordered normed spaces where the positive cone allows plastering. The reviewer (in a collaborative work) subsequently generalized the results of Weber to the case of nonnegative Moore-Penrose inverses for Hilbert space operators [T.Kurmayya and K.C.Sivakumar, Positivity 12, No.3, 475–481 (2008; Zbl 1169.47003)].
Returning to the work of Peris, in the present context, mention must be made of a characterization of inverse-positivity in terms of another type of nonnegative splittings. In what follows, we say that \(M=U-V\) is a \(B\)-splitting of \(M\) if (1) \(U\) is invertible, (2) \(Ux \geq 0 \Rightarrow Mx \geq 0\) for all \(x \in\mathbb R^n\), and (3) \(Ux \geq 0\Rightarrow Mx \geq 0\), \(x \geq 0\) for all \(x \in\mathbb R^n\). Peris demonstrated that \(M\) is inverse-positive if and only if \(M\) has a \(B\)-splitting \(M=U-V\) such that \(VU^{-1}\) has spectral radius strictly less than one.
In the paper under review, the author, by adapting the proof of Peris, nicely extends this result to Banach spaces ordered by a normal cone (with nonempty interior such that the cone admits a uniformly positive linear functional). The main result (Theorem 3.7) of the author is the following: Let \(X\) be an ordered Banach space with a closed normal cone with nonempty interior, allowing plastering. If \(M\) is a bounded linear operator on \(X\) which is positively invertible, then \(M\) possesses a decomposition \(M=U-V\) such that (a) \(U \geq 0\), \(V \geq 0\), (b) \(U^{-1}\) exists (as a bounded linear operator on \(X\)), (c) \(VU^{-1} \geq 0\), (d) \(Mx \geq 0\), \(Ux \geq 0\) imply \(x \geq 0\) and (e) the spectral radius of \(VU^{-1}\) is strictly less than one. An illustrative example demonstrates the main result for a positively invertible operator on a Banach space ordered by the ice-cream cone.

MSC:

47B60 Linear operators on ordered spaces
46A40 Ordered topological linear spaces, vector lattices
15B48 Positive matrices and their generalizations; cones of matrices
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