Journal article
Analyticity and spectral decompositions of L p for compact abelian groups
Pacific journal of mathematics, Vol.127(2), pp.247-260
04/01/1987
DOI: 10.2140/pjm.1987.127.247
Abstract
Let Γ be a dense subgroup of the real line R. Endow Γ with the discrete topology, and let K be the dual group of Γ. Helson’s classic theory uses the spectral representation in Stone’s Theorem for unitary groups to establish and implement a one-to-one correspondence Φ2 between the cocycles on K and the normalized simply invariant subspaces of L2(K). Using our recent extension of Stone’s Theorem to UMD spaces, we generalize Helson’s theory to LP(K), 1 < p < ∞, by producing spectral decompositions of LP(K) which provide a correspondence analogous to Φ2 In particular this approach shows that every normalized simply invariant subspace of LP(K) is the range of a bounded idempotent. However, unlike the situation in the Zasetting, our spectral decompositions do not stem from a projection-valued measure. Instead they owe their origins to the Hubert transform of LP(R). In the context of abstract UMD spaces, we develop the relationships between holomorphic semigroup extensions and the spectral decompositions of bounded one-parameter groups. The results are then applied to describe, in terms of generalized analyticity, the normalized simply invariant subspacesof LP(K). © 1987 by Pacific Journal of Mathematics.
Details
- Title: Subtitle
- Analyticity and spectral decompositions of L p for compact abelian groups
- Creators
- Earl BerksonThomas GillespiePaul Muhly
- Resource Type
- Journal article
- Publication Details
- Pacific journal of mathematics, Vol.127(2), pp.247-260
- DOI
- 10.2140/pjm.1987.127.247
- ISSN
- 0030-8730
- eISSN
- 1945-5844
- Language
- English
- Date published
- 04/01/1987
- Academic Unit
- Statistics and Actuarial Science; Mathematics
- Record Identifier
- 9984397937402771
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