Spektrale Invarianz bei verallgemeinerten Eigenfunktionsentwicklungen

Let T be a selfadjoint operator in a Gelfand triplet H. Examples show that there can occur generalized eigenvalues of T which are not in the Hilbert space spectrum (T) of T. Moreover, the fuction d, which assigns to each real number s the dimension of the generalized eigenspace corresponding to s, can be essentially greater then the von Neumann multiplicity function of T. We therefore construct a new triplet H, closely related to the given Gelfand triplet, according to which the set of generalized eigenvalues of T is contained in (T), and the function d essentially equals the von Neumann multiplicity function of T. Then, in particular, the closure of the set of generalized eigenvalues equals (T). The expansion theorems in H are transferred to H.

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Grundlage dieser Arbeit ist ein Teil meiner Dissertation. Ich danke Herrn Prof. Dr. H.G. Tillmann für die Anregung hierzu und für viele wertvolle Hinweise.