Direct sums of irreducible operators

Volume 155 / 2003

Jun Shen Fang, Chun-Lan Jiang, Pei Yuan Wu Studia Mathematica 155 (2003), 37-49 MSC: 47A15, 47C15. DOI: 10.4064/sm155-1-3


It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators $T$ which are direct sums of irreducible operators in terms of the $C$*-structure of the commutant of the von Neumann algebra generated by $T$.


  • Jun Shen FangDepartment of Mathematics
    Hebei University of Technology
    Tianjin 300130, China
  • Chun-Lan JiangDepartment of Mathematics
    Hebei Nomal University
    Shijiazhuang 050016, China
  • Pei Yuan WuDepartment of Applied Mathematics
    National Chiao Tung University
    Hsinchu 300, Taiwan

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