Direct sums of irreducible operators

Jun Shen Fang ¹   ; Chun-Lan Jiang ²   ; Pei Yuan Wu ³

¹ Department of Mathematics Hebei University of Technology Tianjin 300130, China
² Department of Mathematics Hebei Nomal University Shijiazhuang 050016, China
³ Department of Applied Mathematics National Chiao Tung University Hsinchu 300, Taiwan

Studia Mathematica, Tome 155 (2003) no. 1, pp. 37-49

Résumé

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 $T$.

Détail

DOI : 10.4064/sm155-1-3

Keywords: known every operator separable hilbert space direct integral irreducible operators every direct sum irreducible operator have either finitely uncountably many reducing subspaces former holds only operator direct sum finitely many irreducible operators which unitarily equivalent characterize operators which direct sums irreducible operators terms c* structure commutant von neumann algebra generated