On the closure of absolutely norm attaining operators
On the closure of absolutely norm attaining operators
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DOI：
10.1080/03081087.2022.2126426
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发表时间：
2022
影响因子：
1.1
通讯作者：
Shanola S. Sequeira
中科院分区：
文献类型：
--
作者：
G. Ramesh;Shanola S. Sequeira
ABSTRACT Let $ H_1 $ H1 and $ H_2 $ H2 be complex Hilbert spaces and $ T:H_1\rightarrow H_2 $ T:H1→H2 be a bounded linear operator. We say $ T $ T is norm attaining if there exists $ x\in H_1 $ x∈H1 with $ \|x\|=1 $ ‖x‖=1 such that $ \|Tx\|=\|T\| $ ‖Tx‖=‖T‖. If for every non-zero closed subspace $ M $ M of $ H_1 $ H1, the restriction $ T|_{M}:M\rightarrow H_2 $ T|M:M→H2 is norm attaining, then $ T $ T is called an absolutely norm attaining operator or $ \mathcal {AN} $ AN-operator. If we replace the norm of the operator by the minimum modulus $ m(T)=\inf {\{\|Tx\|:x\in H_1,\; \|x\|=1}\} $ m(T)=inf{‖Tx‖:x∈H1,‖x‖=1} in the above definitions, then $ T $ T is called a minimum attaining and an absolutely minimum attaining operator or $ \mathcal {AM} $ AM-operator, respectively. In this article, we discuss the operator norm closure of $ \mathcal {AN} $ AN-operators. We completely characterize operators in this closure and study several important properties. We mainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for $ \mathcal {AM} $ AM-operators and prove that the closure of $ \mathcal {AM} $ AM-operators is the same as the closure of $ \mathcal {AN} $ AN-operators. Consequently, we prove similar results for operators in the norm closure of $ \mathcal {AM} $ AM-operators.
ABSTRACT Let $ H_1 $ H1 and $ H_2 $ H2 be complex Hilbert spaces and $ T:H_1\rightarrow H_2 $ T:H1→H2 be a bounded linear operator. We say $ T $ T is norm attaining if there exists $ x\in H_1 $ x∈H1 with $ \|x\|=1 $ ‖x‖=1 such that $ \|Tx\|=\|T\| $ ‖Tx‖=‖T‖. If for every non-zero closed subspace $ M $ M of $ H_1 $ H1, the restriction $ T|_{M}:M\rightarrow H_2 $ T|M:M→H2 is norm attaining, then $ T $ T is called an absolutely norm attaining operator or $ \mathcal {AN} $ AN-operator. If we replace the norm of the operator by the minimum modulus $ m(T)=\inf {\{\|Tx\|:x\in H_1,\; \|x\|=1}\} $ m(T)=inf{‖Tx‖:x∈H1,‖x‖=1} in the above definitions, then $ T $ T is called a minimum attaining and an absolutely minimum attaining operator or $ \mathcal {AM} $ AM-operator, respectively. In this article, we discuss the operator norm closure of $ \mathcal {AN} $ AN-operators. We completely characterize operators in this closure and study several important properties. We mainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for $ \mathcal {AM} $ AM-operators and prove that the closure of $ \mathcal {AM} $ AM-operators is the same as the closure of $ \mathcal {AN} $ AN-operators. Consequently, we prove similar results for operators in the norm closure of $ \mathcal {AM} $ AM-operators.