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St. Petersburg Mathematical Journal

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Spectral theory of operator measures in Hilbert space


Authors: M. M. Malamud and S. M. Malamud
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 15 (2003), nomer 3.
Journal: St. Petersburg Math. J. 15 (2004), 323-373
MSC (2000): Primary 47B15; Secondary 47A10
DOI: https://doi.org/10.1090/S1061-0022-04-00812-X
Published electronically: April 2, 2004
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Abstract: In §2 the spaces $ L^2(\Sigma,H) $ are described; this is a solution of a problem posed by M. G. Krein.

In §3 unitary dilations are used to illustrate the techniques of operator measures. In particular, a simple proof of the Naimark dilation theorem is presented, together with an explicit construction of a resolution of the identity. In §4, the multiplicity function $ N_{\Sigma} $ is introduced for an arbitrary (nonorthogonal) operator measure in $ H $. The description of $ L^2(\Sigma,H) $ is employed to show that this notion is well defined. As a supplement to the Naimark dilation theorem, a criterion is found for an orthogonal measure $ E $ to be unitarily equivalent to the minimal (orthogonal) dilation of the measure $ \Sigma $.

In §5 it is proved that the set $\Omega_{\Sigma}$ of all principal vectors of an arbitrary operator measure $ \Sigma $ in $ H $ is massive, i.e., it is a dense $ G_{\delta} $-set in $ H $. In particular, it is shown that the set of principal vectors of a selfadjoint operator is massive in any cyclic subspace.

In §6, the Hellinger types are introduced for an arbitrary operator measure; it is proved that subspaces realizing these types exist and form a massive set.

In §7, a model of a symmetric operator in the space $ L^2(\Sigma,H) $ is studied.


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Additional Information

M. M. Malamud
Affiliation: Department of Mathematics, Donetsk National University, Universitetskaya 24, Donetsk 83055, Ukraine
Email: mdmdc.donetsk.ua

S. M. Malamud
Affiliation: Department of Mathematics, Donetsk National University, Universitetskaya 24, Donetsk 83055, Ukraine
Email: mdmdc.donetsk.ua

DOI: https://doi.org/10.1090/S1061-0022-04-00812-X
Received by editor(s): June 19, 2002
Published electronically: April 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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