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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(λ):=nj=0λjAj where A_0, A_1,..., A_n are selfadjoint bounded linear operators on a Hilbert space H. The essential hypotheses concern the real spectrum of L(λ) 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 [12n] and [12(n+1)] when the leading coefficient A_n is strictly positive and b of degree 12n whenniseven when A_n is invertible and the spectrum of L(λ) is real. Consequences for the factorization of regular operator polynomials when$L(αisinvertibleforsomereal\alpha$) are also discussed.

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


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