Definitizable Operators and Quasihyperbolic Operator Polynomials

Abstract

The main concern of this paper is bounded operators A on a Hilbert space (with inner product (.,.)) which are selfadjoint in an indefinite scalar 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., which implies a strong stability property under perturbations. The central result states that the spectrum is of this type if and only if the operator in question is uniformly definitizable (i.e. Gp( A) ⪢ for some polynomial p). As a first application, characterizations 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 class of "quasihyperbolic polynomials" (QHP) with real and determinate spectrum is introduced. It is shown that QHP have nontrivial monic factors. The special cases of strongly hyperbolic and quadratic polynomials are also discussed. In particular, a factorization theorem is proved for a class of "gyroscopically stabilized" quadratic polynomials, which originate in recent investigations of problems in mechanics.