Invariant subspaces for operators in a general II1-factor
Invariant subspaces for operators in a general II1-factor
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一般 II1 因子中算子的不变子空间
DOI：
10.1007/s10240-009-0018-7
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
Hanne Schultz
中科院分区：
文献类型：
--
作者：
U. Haagerup;Hanne Schultz
摘要：设z为具有归一化迹τ的II1型冯·诺伊曼因子。1983年，L. G. Brown证明了对于每一个算子T∈<e:1>，可以用一种自然的方式将一个谱分布测度μT（现称为T的Brown测度）与T的谱σ(T)上的支持联系起来。本文证明了对于每一个T∈z和每一个Borel集合B，存在一个唯一的闭T不变子空间
${\mathcal{K}}={\mathcal{K}}_{\mathrm{T}}(B)$
与之相关联的，使得布朗测量
$\mathrm {T}|_{{\mathcal{K}}}$
集中在B和布朗测量
$\mathrm{P}_{{\mathcal{K}}^{\bot}}\mathrm{T}|_{{\mathcal{K}}^{\bot}}$
都集中在了∈B上。而且，
${\mathcal{K}}$
t是超不变的
$\mathrm{P}_{\mathcal{K}}$
等于μT(B)。特别地，如果T∈<e:1>具有不集中于单态的Brown测度，则存在一个非平凡的、闭的、T超不变的子空间。进一步证明，对于每一个T∈z，都有极限
$A:=\lim _{n\rightarrow\infty}[(\mathrm{T}^{n})^{*}\mathrm{T}^{n}]^{\frac{1}{2n}}$
存在于强算子拓扑中，并且投影到
${\mathcal{K}}_{\mathrm{T}}(\overline{B(0,r)})$
等于1[0，r](A)对于每一个r>0。
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.