Completely Reducible Operator Algebras and Spectral Synthesis
Completely Reducible Operator Algebras and Spectral Synthesis
复制标题
DOI：
10.4153/cjm-1982-074-9
复制
发表时间：
1982-10
期刊：
影响因子：
--
通讯作者：
Shlomo Rosenoer
中科院分区：
文献类型：
--
作者：
Shlomo Rosenoer
希尔伯特空间H上的有界算子代数称为约化的，如果它是酉的，弱闭的，并且具有如下性质：如果M H对中的每个算子都是（闭）子空间不变量，那么M也是。Loginov和Šul'man [6]及Rosenthal [9]证明了：如果是一个交换约化代数，它与一个具有稠密值域的紧算子K交换，则是一个vonNeumann代数。注意，在这种情况下，的每个不变子空间都被一维不变子空间所张成。事实上，运营商KK * 通勤。因此，它的特征空间是不变的，所以H是有限维不变子空间的正交和。
An algebra of bounded operators on a Hilbert space H is said to be reductive if it is unital, weakly closed and has the property that if M ⊂ H is a (closed) subspace invariant for every operator in , then so is M ⊥. Loginov and Šul'man [6] and Rosenthal [9] proved that if is an abelian reductive algebra which commutes with a compact operator K having a dense range, then is a von Neumann algebra. Note that in this case every invariant subspace of is spanned by one-dimensional invariant subspaces. Indeed, the operator KK * commutes with . Hence its eigenspaces are invariant for , so that H is an orthogonal sum of the finite-dimensional invariant subspaces of From this our claim easily follows.