On nonnegative operators and fully cyclic peripheral spectrum

Abstract

In this note the properties of the peripheral spectrumof a nonnegative linear operator A (for which the spectral radius is a pole of its resolvent) in a complex Banach lattice are studied. It is shown, e.g., that the peripheral spectrum of a natural quotient operator is always fully cyclic. We describe when the nonnegative eigenvectors corresponding to the spectral radius r span the kernel N(r A). Finally, we apply our results to the case of a nonnegative matrix, and show that they sharpen earlier results by B.-S. Tam [Tamkang J. Math. 21:65{70, 1990] on such matrices and full cyclicity of the peripheral spectrum. AMS subject classi cations. 47B65, 47A10, 15A48