# An F. and M. Riesz theorem for bounded symmetric domains

• Volume: 37, Issue: 2, page 139-150
• ISSN: 0373-0956

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## Abstract

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We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.The proof uses a criterion for absolute continuity involving ${L}^{p}$ spaces with $p<1$: A measure $\mu$ on a compact metrisable group $K$ is absolutely continuous with respect to Haar measure $dk$ on $K$ if for some $p<1$ a certain subspace of ${L}^{p}\left(K,dk\right)$ which is related to $\mu$ has sufficiently many continuous linear functionals to separate its points. For abelian $K$ this criterion is due to J.H. Shapiro.

## How to cite

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Brummelhuis, R. G. M.. "An F. and M. Riesz theorem for bounded symmetric domains." Annales de l'institut Fourier 37.2 (1987): 139-150. <http://eudml.org/doc/74749>.

@article{Brummelhuis1987,
abstract = {We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.The proof uses a criterion for absolute continuity involving $L^p$ spaces with $p&lt; 1$: A measure $\mu$ on a compact metrisable group $K$ is absolutely continuous with respect to Haar measure $dk$ on $K$ if for some $p&lt; 1$ a certain subspace of $L^p(K,dk)$ which is related to $\mu$ has sufficiently many continuous linear functionals to separate its points. For abelian $K$ this criterion is due to J.H. Shapiro.},
author = {Brummelhuis, R. G. M.},
journal = {Annales de l'institut Fourier},
keywords = {Riesz theorem; metrizable compact groups; bounded symmetric domains; absolute continuity; spaces},
language = {eng},
number = {2},
pages = {139-150},
publisher = {Association des Annales de l'Institut Fourier},
title = {An F. and M. Riesz theorem for bounded symmetric domains},
url = {http://eudml.org/doc/74749},
volume = {37},
year = {1987},
}

TY - JOUR
AU - Brummelhuis, R. G. M.
TI - An F. and M. Riesz theorem for bounded symmetric domains
JO - Annales de l'institut Fourier
PY - 1987
PB - Association des Annales de l'Institut Fourier
VL - 37
IS - 2
SP - 139
EP - 150
AB - We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.The proof uses a criterion for absolute continuity involving $L^p$ spaces with $p&lt; 1$: A measure $\mu$ on a compact metrisable group $K$ is absolutely continuous with respect to Haar measure $dk$ on $K$ if for some $p&lt; 1$ a certain subspace of $L^p(K,dk)$ which is related to $\mu$ has sufficiently many continuous linear functionals to separate its points. For abelian $K$ this criterion is due to J.H. Shapiro.
LA - eng
KW - Riesz theorem; metrizable compact groups; bounded symmetric domains; absolute continuity; spaces
UR - http://eudml.org/doc/74749
ER -

## References

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11. [11] J. H. SHAPIRO, Subspaces of Lp(G) spanned by characters, 0 &lt; p &lt; 1, Israel J. Math., 29, Nos 2-3 (1978), 248-264. Zbl0382.46015MR57 #17123
12. [12] W. SCHMID, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Raumen, Invent. Math., 9 (1969), 61-80. Zbl0219.32013MR41 #3806
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