Irreducible Semigroups of Positive Operators on Banach Lattices

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 $x_1,...,x_r$ in $X$ such that every operator in $\iS$ acts as a positive scalar multiple of a permutation on $x_1,...,x_r$. 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 $X_1$ 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 $x_1,...,x_r$, $X_1=\Span{x_1,...,x_r}$, and if $S=\lim_j b_jT^{n_j}$ for some $(b_j)$ in $\mathbb R_+$ and $n_j\to\infty$ then $S=c(T_{|X_1})^k\oplus 0$ for some $c\ge 0$ and $0\le k<r$. We also extend results of Abramovich et al. (1992) and Grobler (1995) about peripheral spectra of irreducible operators.