Invariant Subspaces for Operators in a General II₁-factor

2006/11/09 by Uffe Haagerup, Haagerup, Uffe, Hanne Schultz +1 · 2 citations
Mathematics · #46L54 #47A15 #47C15 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.math/0611256

openalex publication_date 2006/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that to every operator T in a general von Neumann factor M of type II₁ and to every Borel set B in the complex plane, one can associate a largest, closed, T-invariant subspace, K = K_T(B), affiliated with M, such that the Brown measure of T|_K is concentrated on B. Moreover, K is T-hyperinvariant, and the Brown measure of (1-P_K)T|_(1-P_K)(H) is concentrated on C\B. In particular, if T has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T in M, the limit A=lim_n→∞[(Tⁿ)* Tⁿ]^1/2n exists in the strong operator topology and K_T(B(0,r))=1_[0,r](A), r>0.

Cited by

• Von Neumann algebraic H^p theory
• The single ring theorem

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