## Abstract

Let D be a Cartan domain in ℂ^{d} and let G = Aut(D) be the group of all biholomorphic automorphisms of G. Consider the projective representation of G on spaces of holomorphic functions on D (U_{v}(g)f)(z) := {j(g^{-1})(z)}^{v/p}f(g^{-1})(z)) g∈G, z∈D, where p is the genus of D and v is in the Wallach set D. We identify the minimal and the maximal U_{v} (G)-invariant Banach spaces of holomorphic functions on D in a very explicit way: The minimal space m_{v} is a Besov-1 space, and the maximal space M_{v} is a weighted H^{∞}-space. Moreover, with respect to the pairing under the (unique) U_{v}(G)- invariant inner product we have M_{v}* =M_{v}. In the second part of the paper we consider invariant Banach spaces of vector-valued holomorphic functions and obtain analogous descriptions of the unique maximal and minimal space, in particular for the important special case of "constant" partitions which arises naturally in connection with nontube type domains.

Original language | English |
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Title of host publication | A Panorama of Modern Operator Theory and Related Topics |

Subtitle of host publication | The Israel Gohberg Memorial Volume |

Publisher | Springer Basel |

Pages | 19-49 |

Number of pages | 31 |

ISBN (Electronic) | 9783034802215 |

ISBN (Print) | 9783034802208 |

DOIs | |

State | Published - 1 Jan 2012 |

### Bibliographical note

Publisher Copyright:© 2012 Springer Basel AG. All rights reserved.

## Keywords

- Banach spaces
- Holomorphic functions
- Symmetric domains

## ASJC Scopus subject areas

- General Mathematics