## Abstract

Factorization theorems ate obtained for selfadjoint operator poly-nomials L(λ):= ε^{n} _{j=0} λ^{j} A_{j} where A_{0}, A_{1},..., A_{n} are selfadjoint bounded linear operators on a Hilbert space H{small}. 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 [1/2n] and [1/2(n + 1)] when the leading coefficient An is strictly positive and (b) of degree 1/2n (when n is even) when An is in-vertible and the spectrum of L (λ) is real. Consequences for the factorization of regular operator polynomials (when L(α) is invertible for some real α) are also discussed.

Original language | English |
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Pages (from-to) | 337-348 |

Number of pages | 12 |

Journal | Journal of Operator Theory |

Volume | 35 |

Issue number | 2 |

State | Published - 1 Dec 1996 |

## Keywords

- Companion operator
- Selfadjoint operator polynomial
- Spectral factorisation
- Supporting subspace

## ASJC Scopus subject areas

- Algebra and Number Theory