Inventiones mathematicae

, Volume 146, Issue 2, pp 329–363

Continuous family of invariant subspaces for R–diagonal operators

  • Piotr Śniady
  • Roland Speicher

DOI: 10.1007/s002220100166

Cite this article as:
Śniady, P. & Speicher, R. Invent. math. (2001) 146: 329. doi:10.1007/s002220100166


We show that every R–diagonal operator x has a continuous family of invariant subspaces relative to the von Neumann algebra generated by x. This allows us to find the Brown measure of x and to find a new conceptual proof that Voiculescu’s S–transform is multiplicative. Our considerations base on a new concept of R–diagonality with amalgamation, for which we give several equivalent characterizations.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Piotr Śniady
    • 1
  • Roland Speicher
    • 2
  1. 1.Institute of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland (e-mail:
  2. 2.Department of Mathematics and Statistics, Queens University, Kingston Ontario K7L 3N6, Canada (e-mail: