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

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

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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).