论文标题
高音希尔伯特空间操作员的能力
Powers of posinormal Hilbert-space operators
论文作者
论文摘要
如果存在正面操作员$ p $，则有限的线性运算符$ a $ $ a $是正常的，因此是正常的。我们表明，如果$ a $是正态的，则封闭范围，则$ a^n $是正常的，并且所有整数$ n \ ge 1 $均具有闭合范围。由于正态操作员的收集包括所有不正常的操作员，因此我们获得的推论是，闭合范围的不正常操作员的能力继续具有封闭范围。我们还提供了一个简单的示例$ t：\ Mathcal {h} \ to \ Mathcal {h} $，这样$ t^2 $没有封闭范围。
A bounded linear operator $A$ on a Hilbert space $\mathcal{H}$ is posinormal if there exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. We show that if $A$ is posinormal with closed range, then $A^n$ is posinormal and has closed range for all integers $n\ge 1$. Because the collection of posinormal operators includes all hyponormal operators, we obtain as a corollary that powers of closed-range hyponormal operators continue to have closed range. We also present a simple example of a closed-range operator $T: \mathcal{H}\to \mathcal{H}$ such that $T^2$ does not have closed range.
