Abstract
In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus \({\phi \mapsto \phi(N)}\) which is the proper analogue of \({\phi \mapsto \int \phi \, dE}\) in the Hilbert space situation. This paper is the first systematical study of definitizable normal operators on Krein spaces.
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Azizov, T.Ya., Strauss, V.A.: Spectral decompositions for special classes of self-adjoint and normal operators on Krein spaces. Spectral analysis and its applications, Theta Ser. Adv. Math., vol. 2, pp. 45–67. Theta, Bucharest (2003)
Kaltenbäck M., Pruckner R.: Functional calculus for definitizable selfadjoint linear relations on Krein spaces. Integr. Equ. Oper. Theory 83(4), 451–482 (2015)
Krein M.G., Smul’jan Ju.L.: J-polar representation of plus operators. Am. Math. Soc. Transl. (2) 85, 115–143 (1969)
Langer, H.: Spectral functions of definitizable operators in Krein spaces. Lecture Notes in Mathematics, vol. 948, pp. 1–46 (1982)
Langer, H., Szafraniec, F.H.: Bounded normal operators in Pontryagin spaces. Operator theory in Krein spaces and nonlinear eigenvalue problems, Oper. Theory Adv. Appl., vol. 162, pp. 231–251. Birkhäuser, Basel (2006)
Philipp F., Strauss V.A., Trunk C.: Local spectral theory for normal operators in Krein spaces. Math. Nachr. 286(1), 42–58 (2013)
Xiaoman C., Chaocheng H.: Normal operators on \({\Pi_\kappa}\) space. Northeast Math. J. 1(2), 247–252 (1985)
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This work was supported by a joint project of the Austrian Science Fund (FWF, I1536–N25) and the Russian Foundation for Basic Research (RFBR, 13-01-91002-ANF).
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Kaltenbäck, M. Spectral Theorem for Definitizable Normal Linear Operators on Krein Spaces. Integr. Equ. Oper. Theory 85, 221–243 (2016). https://doi.org/10.1007/s00020-016-2288-z
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DOI: https://doi.org/10.1007/s00020-016-2288-z