Spectral analysis of an operator associated with equations with time delay

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Head investigator and 8 investigators studied some properties of solutions in equations with time delay, and obtained many results on the subject. The contents of a part of results obtained are summarized in the following :First we analyzed some prperties of spectrum of the operator associated with functional difference equations and functional differential equations which are typical ones as equations with time delay. As applications of the result, we investigated the asymptotic equivalence of solutions and admissibility of some function spaces, and obtained some results on the subjects. These results are almost best possible ones in equations of finite dimension.Next we treated an abstract functional differential equation which is the one of infinite dimension and established a variation-of-constants formula which represents the segment of solutions in the phase space. This formula is crucial in the study of qualitative properties, because one can reduce the study of inifinite dimensional equations to the study of finite dimensional equations by using the formula. Indeed, by using the formula we established a result in admissibility theory for infinite dimensional equations.