Abstract
We solve the problem of intrinsic description of the space \(L^2 (\sum ,H)\), posed by M. G. Krein. It is proved that the set of principal vectors of a nonorthogonal operator measure Σ in H is an everywhere dense set of type G δ. A theory of Hellinger types for a measure Σ is also constructed.
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References
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications, Inc., New York, 1993.
Yu. M. Berezanskii, Dokl. Acad. Nauk SSSR, 108, No.3, 379–382 (1956).
Yu. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monographs, Vol. 17, Amer. Math. Soc., Providence, R.I., 1968.
M. Sh. Birman and M. Z. Solomyak, (RS-STPT) TI: Spectral Theory of Selfadjoint Operators in Hilbert Space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987.
M. Sh. Birman and C. B. Entina, Izv. AN SSSR, 31, 401–430 (1967).
I. M. Gelfand and A. G. Kostyuchenko, Dokl. Acad. Nauk SSSR, 103, No.3, 349–352 (1955).
N. Dunford and J. T. Schwartz, Linear operators. Part II: Spectral Theory. Selfadjoint operators in Hilbert space, John Wiley & Sons, New York–London, 1963.
I. S. Kac, Zap. Khark. Mat. Obshch., (4), 22, 95–113 (1950).
M. G. Krein, Trudy. Mat. Inst. Acad. Nauk UkrSSR, 10, 83–106 (1948).
V. Paulsen, J. Funct. Anal., 109, 113–129 (1992).
A. I. Plesner, Spectral Yheory of Linear Operators. Vols. I, II, Frederick Ungar Publishing Co., New York, 1969.
G. Pisier, Similarity Problems and Completely Bounded Maps, Springer-Verlag, 1996.
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Malamud, M.M., Malamud, S.M. On the Spectral Theory of Operator Measures. Functional Analysis and Its Applications 36, 154–158 (2002). https://doi.org/10.1023/A:1015630909658
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DOI: https://doi.org/10.1023/A:1015630909658