论文标题
$ \ MATHCAL {an} $ - 运算符的不正常运算符的表示和正态性 - 运算符
Representation and normality of Hyponormal operators in the closure of $\mathcal{AN}$-operators
论文作者
论文摘要
令$ h_1 $，$ h_2 $为复杂的希尔伯特空间。有限的线性运算符$ t：h_1 \ to h_2 $，如果存在单位向量$ x \ in H_1 $中的单位向量$ x \，则可以实现$ \ | tx \ | = \ | t \ | $。如果$ t | _ {m}：m \ to h_2 $是每一个封闭的子空间$ m $ $ h_1 $的范围实现，那么我们说$ t $是绝对的标准实现（$ \ nathcal {an} $ - 运算符）。如果运算符的标准被最小模量$ m（t）= \ inf \ {\ | tx \ | | ：x \ in H_1，\ | x \ | = 1 \} $，然后$ t $分别是最低限度的，并且分别是最低限度的操作员（$ \ nathcal {am} $ - 运营商）。 在本文中，我们给出了准$ \ Mathcal {an} $，$ \ Mathcal {am} $ - 运算符和操作员的代表，以关闭这两个类。稍后，我们将这些结果扩展到了$ \ Mathcal {an} $ - 运算符的关闭，并进一步查看这些操作员正常的一些足够条件。
Let $H_1$, $H_2$ be complex Hilbert spaces. A bounded linear operator $T : H_1 \to H_2$ is said to be norm attaining if there exists a unit vector $x \in H_1$ such that $\|Tx\| = \|T\|$. If $T|_{M} : M \to H_2$ is norm attaining for every closed subspace $M$ of $H_1$, then we say that $T$ is an absolutely norm attaining ($\mathcal{AN}$-operator). If the norm of the operator is replaced by the minimum modulus $m(T) = \inf\{\|Tx\| : x \in H_1, \|x\| =1\}$, then $T$ is said to be a minimum attaining and an absolutely minimum attaining operator ($\mathcal{AM}$-operator), respectively. In this article, we give representations of quasinormal $\mathcal{AN}$, $\mathcal{AM}$-operators and the operators in the closure of these two classes. Later we extend these results to the class of hyponormal operators in the closure of $\mathcal{AN}$-operators and a further look at some sufficient conditions under which these operators become normal.
