Invariant subspaces for operators in a general II1-factor

Haagerup, Uffe; Schultz, Hanne

Publications Mathématiques de l'IHÉS, Tome 109 (2009), pp. 19-111.

Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦={𝒦}_{\mathrm{T}}\left(B\right)$ affiliated with ℳ, such that the Brown measure of ${\mathrm{T}|}_{𝒦}$ is concentrated on B and the Brown measure of ${\mathrm{P}}_{{𝒦}^{\perp }}{\mathrm{T}|}_{{𝒦}^{\perp }}$ is concentrated on ℂ∖B. Moreover, $𝒦$ is T-hyperinvariant and the trace of ${\mathrm{P}}_{𝒦}$ is equal to μ T(B). In particular, if T∈ℳ 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∈ℳ the limit $A:={lim}_{n\to \infty }{\left[{\left({\mathrm{T}}^n\right)}^{*}{\mathrm{T}}^n\right]}^{\frac{1}{2n}}$ exists in the strong operator topology, and the projection onto ${𝒦}_{\mathrm{T}}\left(\overline{B(0,r)}\right)$ is equal to 1[0,r](A), for every r>0.

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