Gaussian linear operators. (Opérateurs linéaires gaussiens.) (French) Zbl 0808.46059
Summary: We extend operators from the Cameron-Martin space to Gaussian Lusinian locally convex space. We then are allowed to give sense to the Mehler formula for every such bounded operator. An application is made to Hilbert-Schmidt operators. Next, we show that capacities associated to second quantizations of operators are tight on compact sets, and this is a general result even if the underlying space is not a Banach space.
MSC:
| 46G12 | Measures and integration on abstract linear spaces |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
| 28A12 | Contents, measures, outer measures, capacities |
| 31C15 | Potentials and capacities on other spaces |
| 47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 47B38 | Linear operators on function spaces (general) |
| 60B11 | Probability theory on linear topological spaces |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
Keywords:
Gaussian measures; Fock space; linear measurable operators; Mehler formula; semigroups; capacities; Sobolev spaces; tightness of capacities; Cameron-Martin space; Gaussian Lusinian locally convex space; Hilbert- Schmidt operators; second quantizationsReferences:
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