DEFINITIZABLE OPERATORS AND QUASIHYPERBOLIC OPERATOR POLYNOMIALS

The main concern of this paper is bounded operators A on a Hilbert spa ce (with inner product (.,.)) which are selfadjoint in an indefinite s calar product (say [x, y] = (Gx, y)) and have entirely real spectrum. In addition, all points of spectrum are required to have ''determinate type''; a notion refining earlier ideas of Krein, Langer, et al., whi ch implies a strong stability property under perturbations. The centra l result states that the spectrum is of this type if and only if the o perator in question is uniformly definitizable (i.e. Gp(A) much greate r than 0 for some polynomial p). As a first application, characterizat ions of compact uniformly definitizable operators on Pontrjagin spaces are obtained. Then the basic ideas are extended to selfadjoint monic operator polynomials via their linearizations. In particular, a new cl ass of ''quasihyperbolic polynomials'' (QHP) with real and determinate spectrum is introduced. It is shown that QHP have nontrivial monic fa ctors. The special cases of strongly hyperbolic and quadratic polynomi als are also discussed. In particular, a factorization theorem is prov ed for a class of ''gyroscopically stabilized'' quadratic polynomials, which originate in recent investigations of problems in mechanics. (C ) 1995 Academic Press, Inc.