A NONCOMMUTATIVE BEURLING THEOREM WITH RESPECT TO UNITARILY INVARIANT NORMS

作者：Chen, Yanni^*; Hadwin, Don; Shen, Junhao

来源：Journal of Operator Theory, 2016, 75(2): 497-523.

DOI：10.7900/jot.2015jul13.2080

• Normalized
• unitarily invariant
• parallel to . parallel to(1)-dominating
• continuous norm
• maximal subdiagonal algebra
• dual space
• Beurling theorem
• noncommutative Hardy space

In 1967, Arveson invented a noncommutative generalization of classical H-infinity, known as finite maximal subdiagonal subalgebras, for a finite von Neumann algebra M with a faithful normal tracial state tau. In 2008, Blecher and Labuschagne proved a version of Beurling theorem on H-infinity-right invariant subspaces in a noncommutative L-p (M, tau) space for 1 <= p <= infinity. In the present paper, we define and study a class of norms N-c(M, tau) on M, called normalized, unitarily invariant, parallel to . parallel to (1)-dominating, continuous norms, which properly contains the class {parallel to . parallel to(p) : 1 <= p < infinity} and the class of rearrangement invariant quasi Banach function norms studied by Bekjan. For alpha is an element of N-c( M, tau), we define a noncommutative L-alpha( M, tau) space and a noncommutative H a space. Then we obtain a version of the Blecher-Labuschagne-Beurling invariant subspace theorem on H-infinity-right invariant subspaces in L-alpha(M, tau) spaces and H a spaces. Key ingredients in the proof of our main result include a characterization theorem of H a and a density theorem for L-alpha(M, tau).

• 出版日期2016
• 单位陕西师范大学