## Berezin transform on real bounded symmetric domains

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- by Genkai Zhang PDF
- Trans. Amer. Math. Soc.
**353**(2001), 3769-3787 Request permission

## Abstract:

Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$ with a real form $V$ and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space $V$. We construct the Berezin kernel and consider the Berezin transform on the $L^2$-space on $D$. The corresponding representation of $G$ is then unitarily equivalent to the restriction to $G$ of a scalar holomorphic discrete series of holomorphic functions on $\mathbb D$ and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the $L^2$-space.## References

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## Additional Information

**Genkai Zhang**- Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
- Email: genkai@math.chalmers.se
- Received by editor(s): January 16, 2000
- Received by editor(s) in revised form: October 10, 2000
- Published electronically: May 4, 2001
- Additional Notes: Research supported by the Swedish Natural Sciences Research Council (NFR)
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**353**(2001), 3769-3787 - MSC (2000): Primary 22E46, 43A85, 32M15, 53C35
- DOI: https://doi.org/10.1090/S0002-9947-01-02832-X
- MathSciNet review: 1837258