Minimal and maximal invariant spaces of holomorphic functions on bounded symmetric domains

Jonathan Arazy, Harald Upmeier

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


Let D be a Cartan domain in ℂd 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 (Uv(g)f)(z) := {j(g-1)(z)}v/pf(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 Uv (G)-invariant Banach spaces of holomorphic functions on D in a very explicit way: The minimal space mv is a Besov-1 space, and the maximal space Mv is a weighted H-space. Moreover, with respect to the pairing under the (unique) Uv(G)- invariant inner product we have Mv* =Mv. 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 languageEnglish
Title of host publicationA Panorama of Modern Operator Theory and Related Topics
Subtitle of host publicationThe Israel Gohberg Memorial Volume
PublisherSpringer Basel
Number of pages31
ISBN (Electronic)9783034802215
ISBN (Print)9783034802208
StatePublished - 1 Jan 2012

Bibliographical note

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


  • Banach spaces
  • Holomorphic functions
  • Symmetric domains

ASJC Scopus subject areas

  • Mathematics (all)


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