Banach Lattices and Positive Operators by Dr. rer. nat. Helmut H. Schaefer (auth.)

By Dr. rer. nat. Helmut H. Schaefer (auth.)

Show description

Read Online or Download Banach Lattices and Positive Operators PDF

Best abstract books

Intégration: Chapitres 7 et 8

Intégration, Chapitres 7 et 8Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce quantity du Livre d’Intégration, sixième Livre du traité, traite de l’intégration sur les groupes localement compacts et de ses functions.

Extra info for Banach Lattices and Positive Operators

Sample text

C) If A is power positive and s, t denote the smallest and largest row sums of A, respectively, then s ~ Ao ~ t. 16. Let A be real and suppose r( IA I) = 1. (a) If t; is a unimodular eigenvalue of A, then t;2k+ 1 is an eigenvalue of A and t;2k is an eigenvalue of IAI, for each kE7L. (b) Show that t;2n=1 for each unimodular eigenvalue of A. Deduce from this that if B is a real n x n-matrix with a unimodular eigenvalue not a 2n-th root of unity, then r(IBI»1. 45 § 10. Bounds for Eigenvalues (c) If A possesses no non-trivial invariant ideals in JR" and if <: is any unimodular eigenvalue of A, then <:2 E Gm and the unimodular spectrum of A is the coset <: Gm , where Gm is the group of all m-th roots of unity for some m, 1 ~ m ~ n.

This implies Bz=Az=Az and CZ= -Az= -Az. Moreover, rCy=(M -r)y and rBy=(r-m)y. 3) Cor. that M - rand r - m are the spectral radii of C and B, respectively. This proves (3). We show now that (3) implies (4), supposing that m>O. We distinguish the cases M -r~r-m and M -r

Then for each i, 1 ;£i;£n, But for any fixed i we cannot have aij=O for all ji=i, since otherwise the ideal r>a ii for all i. Further, ).. 2), hence a simple root of 11()") =0. Finally, if z>O satisfies pz=Az then pi=r would imply (z,y) =0 for some y=r-lCA)y~O (cf. 2)(c)), which is impossible. 2) it follows that z ~ O. 0 J={X:~i=O} were A-invariant. Thus r~i>aii~i hence, Corollary. If A is irreducible and n ~2, then r>O and r- 1A is similar toa stochastic matrix. Proof. Let rx=Ax where X=(~i)~O, and denote by Dx the diagonal matrix with diagonal entries ~i' If S=D;1(r- 1A)Dx , then rx=Ax shows that e=Se so S is stochastic (§ 4).

Download PDF sample

Rated 4.78 of 5 – based on 39 votes