We give necessary and sufficient conditions under which a C 0 -semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded. r 2004 Elsevier Inc. All rights reserved. MSC: primary 47D60; 47B50; secondary 47A20; 47A15
The article describes a small and portable apparatus designed to measure the dynamic stiffness of roads and soils under sinusoidal loading, at frequencies between 5 and lo00 Hz. An electromagnetic vibrator applies forces of up to rf: 14 lb (about its dead weight of 45 lb) to the road surface, the force itself and the resulting displacement of the road surface being measured by means of electrical transducers feeding a phase-sensitive voltmeter. Values of the elastic stiffness of the construction and values of the damping are obtained. Check tests of the behaviour of the apparatus are described, and some typical results on various types of road pavement are given.
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.
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