Operator Theoretic Aspects of Ergodic Theory by Tanja Eisner Bálint Farkas Markus Haase & Rainer Nagel

Author:Tanja Eisner, Bálint Farkas, Markus Haase & Rainer Nagel
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

for each f ∈ C(G) and u ∈ E, and we conclude the proof by an appeal to Lemma 15.5 above. □

Remarks 15.15.

1)It follows from Theorem 15.14 that each infinite-dimensional (continuous) representation of a compact group G has a nontrivial finite-dimensional invariant subspace. In particular, each irreducible unitary representation of G is finite-dimensional.

2)Two unitary representations (π, H) and (ρ, K) are called (unitarily) equivalent if there is a unitary operator U: H → K intertwining the representations. The Peter–Weyl theorem states that decomposes orthogonally

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