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Invariant Subspaces for Operators in a General II_1-factor

Uffe Haagerup·Hanne Schultz
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Abstract

It is shown that to every operator T in a general von Neumann factor M of type II_1 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\to\infty}[(T^n)* T^n]^{1/2n} exists in the strong operator topology and K_T(\bar{B(0,r)})=1_{[0,r]}(A), r>0.

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Uffe Haagerup, Hanne Schultz. 2006-11-09. Invariant Subspaces for Operators in a General II_1-factor. https://arxiv.org/abs/math/0611256

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