Some Spectral Properties of Contractive and Expansive Operators in Indefinite Inner Product Spaces

Abstract

The linear operator Tin an inner product space ( X , [ . , a ] ) is called contractive (expansive, XI, resp.) for all x E X . Eigenvalues, in particular those in the unit disc, and the signatures of the corresponding eigenspaces were studied e.g. in [IKL], [AI], [B], where also references to earlier papers can be found. It is the aim of this note to prove results of this type under fairly general assumptions, to improve earlier results, e.g. Lemma 11.8 of [IKL] about an expansive but not doubly expansive operator (in [IKL] because of the different sign of the inner product these operators are contractive), and to show that e.g. in a Pontrjagin space the inner product on an eigenspace of a contraction Tat z, IzI = 1, is very similar to the inner product on an eigenspace at z of a unitary operator. Most of the statements below have analogues for operators which are dissipative with respect to an indefinite inner product, and, in fact, some of them were earlier proved in this context (see [AI]). The formulation of these analogues is left to the reader.