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Eventual Cone Invariance EVENTUAL CONE INVARIANCE By MICHAEL KASIGWA A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Mathematics JULY 2017 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of MICHAEL KASIGWA find it satisfactory and recommend that it be accepted. Michael Tsatsomeros, Ph.D., Chair Judith McDonald, Ph.D. Nikos Voulgarakis, Ph.D. ii Acknowledgments Thanks to all in the Ndoleriire family for your steadfast love and support, to my adviser for your invaluable guidance and to all the committee members for your support and dedication. iii EVENTUAL CONE INVARIANCE Abstract by Michael Kasigwa, Ph.D. Washington State University July 2017 Chair: Michael Tsatsomeros n If K is a proper cone in R some results in the theory of eventually (entrywise) nonneg- ative matrices have equivalent analogues in eventual K-invariance. We develop these analogues using the classical Perron-Frobenius theory for cone preserving maps. Using an ice-cream cone we n×n demonstrate that unlike the entrywise nonnegative case of a matrix A 2 R , eventual cone invari- ance and eventual exponential cone invariance are not equivalent. The notions of inverse positivity of M-matrices, M-type and Mv-matrices are extended to Mv;K -matrices (operators), that have the form A = sI − B, where B is eventually K-nonnegative, that is, BmK ⊆ K for all sufficiently large m. Eventual positivity of semigroups of linear operators on Banach spaces ordered by a cone can be investigated using resolvents or resolvent type operators constructed from the generators. iv Contents Acknowledgments ......................................... iii Abstract ............................................... iv Contents ............................................... v Symbols ............................................... vii 1 Introduction .......................................... 1 1.1 Introduction . 1 1.2 Nonnegative Matrices . 3 1.3 Exponential Nonnegativity . 5 1.4 Organization of the Dissertation . 6 2 Cones and Perron-Frobenius Theory ........................... 7 2.1 Introduction and Definitions . 7 2.2 Perron-Frobenius Theory . 10 3 Matrices and Cone Invariance ............................... 13 3.1 Introduction . 13 v 3.2 Results from the Theory of Invariant Cones . 14 3.3 Eventually (exponentially) K-Positive Matrices . 15 3.4 Eventually (exponentially) K-Nonnegative Matrices . 19 3.5 Points of K-Potential . 23 3.6 Generalizing M-Matrices based on Eventual K-Nonnegativity . 26 4 Semigroups of Linear Operators and Invariance .................... 33 4.1 Introduction . 33 4.2 Operator Semigroups on Ordered Spaces . 34 4.3 Semigroup Generation . 37 4.4 Functional Calculus for Bounded Linear Operators . 39 4.5 Positive Semigroups . 42 Bibliography ............................................ 50 vi Symbols R Set of real numbers. R+ Nonnegative real numbers. n R+ Nonnegative orthant. R(λ) Real part of λ. C Complex numbers. n C n-dimensional complex space. Mn(R or C) A set of all n × n matrices over R or C. (T (t))t≥0 A one-parameter semigroup over R (or C). ρ(A) Resolvent set of operator A. σ(A) Spectrum of operator A. σb(A) Boundary spectrum of A. Rλ or R(λ, A) The resolvent operator. L(X) Bounded linear operators on a normed linear space X. r(A) Spectral radius of A. S(A) Spectral bound, i.e. SupfR(λ): λ 2 σ(A)g 2 [−∞; 1]. +@U Positive oriented boundary of an open set U. C(Ω) Continuous functions on a compact Hausdorff space Ω. vii C0(X) Space of continuous functions that vanish at infinity. f ≥ 0 A function f is positive, pointwise. f 0 A function f is strongly positive. T ≥ 0 (T 0) Operator T is positive (strongly positive). E+ ff 2 E : f ≥ 0g, a positive cone in a Banach space E. int K Interior of a cone K. π(K) The set of square matrices that leave a cone K-invariant. XA(K) Points of K-potential. Mv;K -matrix Matrix of form A = sI − B, B eventually K-nonnegative, s ≥ r(B) ≥ 0. indexλ(B) The index of the eigenvalue λ of B. viii Chapter 1 Introduction 1.1 Introduction In this dissertation we start with introductory notions on the theory of nonnegative matrices in n×n R and basics of the Perron-Frobenius theory. This will be followed by the generalizations of n this theory to cones in R and a description of the extension to semigroups of linear operators on Banach lattices. This approach also reflects the trend of research in this field where results on the n n spectrum of positive operators on R ordered by the cone R+, first analyzed by Perron (1907) and extended to nonnegative matrices by Frobenius (1909), were further extended to general spaces and cones by Krein and Rutman in the 1950's. Quite often in the literature on the theory of linear operators one observes that, whenever possible, some ideas developed in infinite dimension are tested or exemplified in finite dimension and conversely ideas developed in finite dimension are extended to infinite dimension. 1 A systematic theory has been developed on eventually positive semigroups of linear operators on some Banach lattices. This development has advanced the notion of positivity (nonnegativity) of matrices to operators in general and applications. One such work examined is the reachability and holdability of nonnegative states for linear differential systems utilizing the notion of Perron- Frobenius type properties [25]. Nonnegative matrices are used in different fields such as Biology, Economics, Dynamical Systems, Numerical analysis, Probability theory and others. Several scholars have developed characteri- zations of inverse positive matrices and have made extensions from Z-(negated-Metzler) and M- (Minkowski) matrices to generalized matrices and operators. Le and McDonald [19, 2006] char- acterized inverses of M-type matrices (created from eventually nonnegative matrices) and Olesky, Tsatsomeros and Van Den Driessche [27, 2009] extended the theory to Mv-matrices (a generalization of M-matrices based on eventually nonnegative matrices). We have some results on the extension of the above theory to eventual cone invariance for general n cones in R and as far as we know this has not been studied before using our particular perspective. The Perron-Frobenius theorem and its generalization has indeed been studied by several authors in the context of both finite and infinite dimensions. The present work seeks to generalize some key results from nonnegative matrix theory to cones and examine the corresponding Perron-Frobenius type properties. Some new theorems are formulated that correspond to cones. Recently, Daners et al. [7, 2016] investigated strongly continuous semigroups with eventual positiv- ity properties on C(Ω), the space of complex-valued continuous functions on a compact non-empty Hausdorff space Ω. They looked at positive initial conditions and the corresponding eventually positive solutions to the Cauchy problem for large enough time. 2 We examine further extensions to the work in [7] by using eventual cone invariance and in particular to Banach spaces ordered by a cone, in order to describe a new analogue of what is known as an Mv;K -type matrix (operator). We make use of the notions of eventually positive matrices (operators) and look at resolvent positive (matrices) operators and generators of positive linear operator semigroups. In this way, we link the notions in M-matrix and inverse M-matrix theory in [4], [19], [27] (among others) to those in [1], [7]. We make some observations and provide some answers to questions on the characterization of eventual cone positivity for linear operators in a different vein from what has been done to date. 1.2 Nonnegative Matrices n×n Definition 1.1. Let [aij] = A 2 R , with aij 2 R. (i) A is nonnegative (positive) if aij ≥ 0 (> 0) for all i; j 2 f1; 2; :::; ng and is denoted by A ≥ 0 (A > 0). (ii) A is eventually nonnegative (positive) if there exists a k0 2 N, such that for all k ≥ k0; Ak ≥ 0 (Ak > 0). n×n −1 (iii) Given A 2 C ; the set σ(A) = fλ 2 C :(λI − A) is undefinedg is called the spectrum of A and is denoted by σ(A). In finite dimension this is simply the set of all eigenvalues of A or the point spectrum. (iv) The spectral radius of A, denoted by r(A), is the maxfjλj : λ 2 σ(A)g: (v) A is essentially nonnegative (positive), also known as Metzler, if there exists an α ≥ 0 such that A + αI ≥ 0 (> 0); (−A is called a Z-matrix). 3 (vi) When a Z-matrix is invertible and the inverse is nonnegative then it is called an invertible n×n M-matrix. Another definition of an M matrix A is A = sI − B, where B 2 R is (entrywise) nonnegative and s > r(B), (s > r(B) implies A is an invertible M-matrix while s = r(B) implies A is a singular M matrix). (vii) Pseudo M-matrices are of the form A = sI −B where s ≥ r(B) > 0 such that r(B) 2 σ(B) and r(B) ≥ jλj, λ 2 σ(B) n fr(B)g with the corresponding left (right) eigenvectors strictly positive. (viii) Matrices of the form A = sI − B, where s ≥ r(B) > 0 and B is eventually nonnegative are called EM-matrices. (ix) M-type matrices are of the form A = sI − B , where B is eventually nonnegative, s > r(B), such that the possible index of 0 as an eigenvalue of B is at most 1, and there exists β > r(B) such that A = sI − B has a positive inverse for all s 2 (r(B); β), [19]. (x) Olesky, Tsatsomeros and van den Driessche generalized M-matrices to Mv-matrices. An Mv matrix has the form A = sI − B, where s ≥ r(B) ≥ 0 and B is eventually nonnegative (notice that Mv ⊇ EM-matrices), [27].
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