# Invariant subspaces for operators in a general II1-factor

Publications Mathématiques de l'IHÉS (2009)

- Volume: 109, page 19-111
- ISSN: 0073-8301

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topHaagerup, Uffe, and Schultz, Hanne. "Invariant subspaces for operators in a general II1-factor." Publications Mathématiques de l'IHÉS 109 (2009): 19-111. <http://eudml.org/doc/274348>.

@article{Haagerup2009,

abstract = {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 $\{\mathcal \{K\}\}=\{\mathcal \{K\}\}_\{\mathrm \{T\}\}(B)$ 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 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 $A:=\lim _\{n\rightarrow \infty \}[(\mathrm \{T\}^\{n\})^\{*\}\mathrm \{T\}^\{n\}]^\{\frac\{1\}\{2n\}\}$ 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 rgt;0.},

author = {Haagerup, Uffe, Schultz, Hanne},

journal = {Publications Mathématiques de l'IHÉS},

keywords = {von Neumann factors; Brown measures; invariant subspace; T-hyperinvariant subspace},

language = {eng},

pages = {19-111},

publisher = {Springer-Verlag},

title = {Invariant subspaces for operators in a general II1-factor},

url = {http://eudml.org/doc/274348},

volume = {109},

year = {2009},

}

TY - JOUR

AU - Haagerup, Uffe

AU - Schultz, Hanne

TI - Invariant subspaces for operators in a general II1-factor

JO - Publications Mathématiques de l'IHÉS

PY - 2009

PB - Springer-Verlag

VL - 109

SP - 19

EP - 111

AB - 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 ${\mathcal {K}}={\mathcal {K}}_{\mathrm {T}}(B)$ 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 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 $A:=\lim _{n\rightarrow \infty }[(\mathrm {T}^{n})^{*}\mathrm {T}^{n}]^{\frac{1}{2n}}$ 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 rgt;0.

LA - eng

KW - von Neumann factors; Brown measures; invariant subspace; T-hyperinvariant subspace

UR - http://eudml.org/doc/274348

ER -

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