Irreducible Semigroups of Positive Operators on Banach Lattices

2012/08/16 by Gao, Niushan, Troitsky, Vladimir G.
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1208.3498

Abstract

The classical Perron-Frobenius theory asserts that an irreducible matrix A has cyclic peripheral spectrum and its spectral radius r(A) is an eigenvalue corresponding to a positive eigenvector. In Radjavi (1999) and Radjavi and Rosenthal (2000), this was extended to semigroups of matrices and of compact operators on L_p-spaces. We extend this approach to operators on an arbitrary Banach lattice X. We prove, in particular, that if \iS is a commutative irreducible semigroup of positive operators on X containing a compact operator T then there exist positive disjoint vectors x₁,...,x_r in X such that every operator in \iS acts as a positive scalar multiple of a permutation on x₁,...,x_r. Compactness of T may be replaced with the assumption that T is peripherally Riesz, i.e., the peripheral spectrum of T is separated from the rest of the spectrum and the corresponding spectral subspace X₁ is finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator T, we show that T is a cyclic permutation on x₁,...,x_r, X₁=\Spanx₁,...,x_r, and if S=lim_j b_jT^n_j for some (b_j) in \mathbb R₊ and n_j→∞ then S=c(T_|X1)^k⊕ 0 for some c≥ 0 and 0≤ k

Related

• A note on the spectrum of irreducible operators and semigroups
• Compact right topological semigroups applied to operators
• Strong Monotonicity of Spectral Radius of Positive Operators
• Extensions of Perron-Frobenius Theory
• Growth rates and the peripheral spectrum of positive operators