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The projection spectral theorem and Jacobi fields. (English) Zbl 1340.47061

The projection spectral theorem for a family, possibly uncountable, of commuting selfadjoint operators was proved by Yu. M. Berezanskiĭ [Russ. Math. Surv. 39, No. 4, 1–62 (1984; Zbl 0577.47019)]. The author gives a survey of its applications to Jacobi fields, the families \(J(\varphi)\) where the index \(\varphi\) is typically an element of a nuclear space. The operators \(J(\varphi)\) are defined on a Hilbert space with an orthogonal decomposition and have a kind of tridiagonal structure. Applying the spectral theorem, one obtains important classes of measures on infinite-dimensional spaces, such as Lévy white noise measures. Non-commutative versions of the theory are discussed.

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
60H40 White noise theory
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Citations:

Zbl 0577.47019
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