Spectral Properties and Companion Forms of Operator- and Matrix Polynomials
Niels Hartanto · DepositOnce · 2011
In this thesis we study spectral properties of analytic $m$--monic operator- and matrix functions $F(\lambda)=\lambda^mI-A(\lambda)$. A major focus lies on $m$--monic operator- and matrix polynomials with coefficients that satisfy certain nonnegativity conditions. We first establish a general framework for the investigation of spectral properties of $m$--monic Banach algebra functions. A major role is played by a geometrically convex scalar function $\phi_A$ associated with the spectral radius of $A(\lambda)$. This function was introduced by K.-H. Förster and B. Nagy. Eigenvalues of $m$--monic operator functions which lie on some circles around zero are rotation invariant with respect to angles corresponding to certain roots of unity if the coefficients of the function are self-adjoint and satisfy some condition which is closely related to a condition on the function $\phi_A$ at some point $\rho>0$. This is also investigated in this thesis and extends results of H.K. Wimmer and J. Swoboda. Very crucial for the investigation of spectral properties of $m$--monic operator- and matrix polynomials is the canonical reduction from the $m$--monic to the $1$--monic case. It is a generalization of the linearization via the first companion form and allows the application of fixpoint iterations, which are an suitable tool for the study of factorizations of $1$--monic operator- and matrix polynomials. A one--to--one correspondence between factorizations of the $m$--monic polynomial and its canonical reduction guarantees that the original polynomial has a factorization if and only if its canonical reduction has one. Formulas which express one factorization in terms of the other one are developed. Furthermore, in the case of matrix polynomials, it is possible to calculate Jordan chains of the original polynomial from the Jordan chains of its canonical reduction and vice versa. Using fixpoint iterations, a condition for the existence of spectral factorizations of an $1$--monic matrix polynomial $P$ with nonnegative coefficients is given. Those factorizations separate the eigenvalues of $P$ with respect to certain circles. Via canonical reduction, this result can be transferred to the $m$--monic case. This is a special case of an abstract result in ordered Banach algebras given K.-H. Förster and B. Nagy and its proof uses only matrix theoretical concepts. $m$--monic matrix polynomials $P$ with entrywise nonnegative coefficients such that the sum of the coefficients is irreducible can have eigenvalues that have a certain rotation invariance. This symmetry is very similar to the rotation invariance of peripheral eigenvalues of entrywise nonnegative irreducible matrices which is a consequence of the well known Perron--Frobenius theory. The study of an infinite graph associated with $P$ is key for the analysis. A description of spectral factorizations, their existence and their eigenvalues in the irreducible case is given via the characterization of eight cases which can occur. These are extensions to results of K.-H. Förster and B. Nagy. Assumed that an $P$ satisfies the condition for the existence of a spectral factorization, a numerical algorithm for the calculation of the corresponding factors is given. Its core is constituted by a version of a cyclic reduction method suited for a certain type of Markov chains which is presented given by D.A. Bini, G. Latouche and B. Meini. We give a suitable transformation of the factorization problem to a Markov problem which allows the application of cyclic reduction.