Abstract
A generalization of the notion of a Julia operator is used to produce models to give a straightforward construction of invariant, and in fact reducing, subspaces for a class of operators on Kreĭn spaces known as weakly definitizable operators. This includes, among others, the definitizable selfadjoint and definitizable unitary operators. A simple proof of the existence of an orthogonal pair of maximal definite invariant subspaces for positive operators on a Kreĭn space is found, and applied to show the existence of such invariant subspaces for definitizable selfadjoint operators. A new proof of the Spectral Theorem for continuous definitizable selfadjoint operators is also given, along with a functional calculus which includes functions that are Borel measurable and bounded after division by the definitizing polynomial on a suitably large subset of the real line.
The author was supported by the National Science Foundation.
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Dritschel, M.A. (1993). A Method for Constructing Invariant Subspaces for Some Operators on Kreĭn Spaces. In: Gheondea, A., Timotin, D., Vasilescu, FH. (eds) Operator Extensions, Interpolation of Functions and Related Topics. Operator Theory: Advances and Applications, vol 61. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8575-1_5
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