Analytic Operator Functions with Selfadjoint Coefficients
K.-H. Förster, Niels Hartanto · PAMM · 2009
Abstract In this note, results of H. K. Wimmer [2] are extended to functions with values in the space of bounded linear operators $B(\cal{H})$ in the Hilbert space \cal{H}$, which are analytic on an annulus and have selfadjoint coefficients. The eigenvalues on the boundary of the spectrum of the functions considered in this note, corresponding to a fixed eigenvector possess a certain symmetry which is well known from the periphal spectrum of entrywise nonnegative matrices, due to the Perron‐Frobenius theorem. Similar results hold for polynomials of type (1) with entrywise nonnegative matrix coefficients Aj, see [1]. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)