论文标题
关闭绝对的规范运营商
On the closure of Absolutely Norm attaining Operators
论文作者
论文摘要
令$ h_1 $和$ h_2 $为复杂的希尔伯特空间，$ t：h_1 \ rightarrow h_2 $是有限的线性运算符。我们说，如果存在$ x \ in h_1 $，则$ x \ | x \ | = 1 $，以便$ x \ |如果对于每个闭合的子空间$ m $ $ h_1 $，则限制$ t | _ {m}：m \ rightArrow h_2 $ n n n norm inst n n n n n n n n n n and the，$ t $被称为绝对norm norm达到操作员或$ \ Mathcal {an} $ - 运营商。如果我们将操作员的标准替换为最小模量$ m（t）= \ inf {\ {\ | tx \ |：x \ in H_1，\; \ | x \ | = 1} \} $，然后$ t $分别称为最低达到最低限度，并且分别获得最小达到运算符（或$ \ Mathcal {am} $ - 运算符）。 在本文中，我们讨论了$ \ Mathcal {an} $ - 运算符的运营商规范闭合。我们在此关闭中完全表征了操作员，并研究了几种重要特性。我们主要给出该类别中正运算符的光谱表征，并在运算符正常时给出表示。稍后，我们还研究了$ \ Mathcal {am} $ - 运算符的类似属性，并证明了$ \ Mathcal {am} $ - 运算符的关闭与$ \ Mathcal {an} $ - 运营商的关闭相同。结果，我们在$ \ Mathcal {am} $ - 运算符的规范关闭中证明了运营商的相似结果。
Let $H_1$ and $H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a bounded linear operator. We say $T$ to be norm attaining, if there exists $x\in H_1$ with $\|x\|=1$ such that $\|Tx\|=\|T\|$. If for every closed subspace $M$ of $H_1$, the restriction $T|_{M}:M\rightarrow H_2$ is norm attaining then, $T$ is called absolutely norm attaining operator or $\mathcal{AN}$-operator. If we replace the norm of the operator by the minimum modulus $m(T)=\inf{\{\|Tx\|:x\in H_1,\; \|x\|=1}\}$, then $T$ is called the minimum attaining and the absolutely minimum attaining operator (or $\mathcal{AM}$-operator) respectively. In this article, we discuss about the operator norm closure of the $\mathcal{AN}$-operators. We completely characterize operators in this closure and study several important properties. We mainly give the spectral characterization of the positive operators in this class and give the representation when the operator is normal. Later we also study the analogous properties for $\mathcal{AM}$-operators and prove that the closure of $\mathcal{AM}$-operators is same as that of the closure of $\mathcal{AN}$-operators. As a consequence, we prove similar results for operators in the norm closure of $\mathcal{AM}$-operators.
