Journal of Operator Theory

Volume 35, Issue 2, Spring 1996 pp. 337-348.

Factorization of selfadjoint operator polynomials

Authors: P. Lancaster (1), A. Markus (2) and V. Matsaev (3)
Author institution:(1) Dept. of Math. and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, CANADA
(2) Dept. of Math. and Computer Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, ISRAEL
(3) School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978, ISRAEL

Summary: Factorization theorems are obtained for selfadjoint operator polynomials $L(\lambda ){\rm{ : = }}\sum\limits_{j = 0}^{\rm{n}} {\lambda ^j A_j }$ where A_0, A_1,..., A_n are selfadjoint bounded linear operators on a Hilbert space $\mathcal H$. The essential hypotheses concern the real spectrum of $L(\lambda)$ and, in particular, ensure the existence of spectral sub-spaces associated with the real line for the (companion) linearization. Under suitable additional conditions, the main results assert the existence of polynomial factors (a) of degrees $[\frac{1}{2}n]$ and $[\frac{1}{2}(n+1)]$ when the leading coefficient A_n is strictly positive and (b) of degree $\frac{1}{2}n$ (when n is even) when A_n is invertible and the spectrum of $L(\lambda)$ is real. Consequences for the factorization of regular operator polynomials (when $L(\alpha)$ is invertible for some real $\alpha$) are also discussed.

Keywords: Selfadjoint operator polynomial, spectral factorization, companion operator, supporting subspace.

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