Invariant subspaces for operators in a general II1-factor

By Uffe Haagerup, Hanne Schultz. 2006

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AbstractLet 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 μT (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 ${\mathcal{K}}={\mathcal{K}}_{\mathrm{T}}(B)$ affiliated with ℳ, such that the Brown measure of $\mathrm {T}|_{{\mathcal{K}}}$ is concentrated on B and the Brown measure of $\mathrm{P}_{{\mathcal{K}}^{\bot}}\mathrm{T}|_{{\mathcal{K}}^{\bot}}$ is concentrated on ℂ∖B. Moreover, ${\mathcal{K}}$ is T-hyperinvariant and the trace of $\mathrm{P}_{\mathcal{K}}$ is equal to μT(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 $A:=\lim _{n\rightarrow\infty}[(\mathrm{T}^{n})^{*}\mathrm{T}^{n}]^{\frac{1}{2n}}$ exists in the strong operator topology, and the projection onto ${\mathcal{K}}_{\mathrm{T}}(\overline{B(0,r)})$ is equal to 1[0,r](A), for every r>0.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a4766f8f267e0f3e865dfbff763c387ea4fbf09a

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Citations in the corpus

Brown measures of unbounded operators affiliated with a finite von Neumann algebra (2006, background): “Since the results of the remaining sections of this paper and of our forthcoming paper [10] rely heavily on these statements, we have decided to include complete proofs.”
Free Probability and Random Matrices (2014, background): “This was strengthened by Haagerup and Schultz [87] in the following way: If Minv denotes the invertible elements in M, then we actually have for all a ∈M and every p > 0 that ∫”
The single ring theorem (2009, methodology): “) R-diagonal operators were introduced by Nica and Speicher [21] in the context of free probability; they represent the weak*-limit (or more precisely, the limit in ∗-moments) of operators of the form UT with U unitary with size going to infinity and T diagonal, and were intensively studied in the l”