Abstract
In this chapter we specialize some of the considerations of Chapter 5 to the case of unitary C 0-Groups in a Hubert space H. The theory of unitary representations \(W(x) = {e^{iA - x}}\) of ℝn is a very well understood classical subject and will not be presented here. However we mention that a n-dimensional version of Stone’s theorem states that there is a unique spectral measure E on ℝn such that \( W(x) = \int_{{^n}} {{e^{ix \cdot y}}E(dy)} \) and this allows one to extend the functional calculus which we already have for functions in \(C_{pol}^\infty ({^n})\) to all Borel functions \(\varphi :{^n} \to \). The natural definition of φ(A) for a Borel function φ is \( \varphi :(A) = \int_{{^n}} {\varphi (y)E(dy)} \), and one may check that for \( \varphi \in C_{pol}^\infty ({^n})\) the two definitions lead to the same operator φ(A).
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© 1996 Springer Basel AG
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Amrein, W.O., de Monvel, A.B., Georgescu, V. (1996). Unitary Representations and Regularity for Self-adjoint Operators . In: C 0-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians. Progress in Mathematics, vol 135. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7762-6_6
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DOI: https://doi.org/10.1007/978-3-0348-7762-6_6
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7764-0
Online ISBN: 978-3-0348-7762-6
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