On the closure of absolutely norm attaining operators

G, Ramesh and Sequeira, Shanola S. (2022) On the closure of absolutely norm attaining operators. Linear and Multilinear Algebra. pp. 1-21. ISSN 0308-1087

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Abstract

Let H-1 and H-2 be complex Hilbert spaces and T : H-1 -> H-2 be a bounded linear operator. We say T is norm attaining if there exists x is an element of H-1 with parallel to x parallel to = 1 such that parallel to Tx parallel to = parallel to T parallel to. If for every non-zero closed subspaceMof H-1, the restriction T|(M) : M -> H-2 is norm attaining, then T is called an absolutely norm attaining operator or ANoperator. If we replace the norm of the operator by the minimum modulus m(T) = inf {parallel to Tx parallel to : x is an element of H-1, parallel to x parallel to = 1} in the above definitions, then T is called a minimum attaining and an absolutely minimumattaining operator orAM-operator, respectively. In this article, we discuss the operator norm closure of AN-operators. We completely characterize operators in this closure and study several important properties. Wemainly 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 AMoperators and prove that the closure ofAM-operators is the same as the closure ofAN-operators. Consequently, we prove similar results for operators in the norm closure ofAM-operators.

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IITH Creators:

IITH Creators	ORCiD
G, Ramesh	https://orcid.org/0000-0002-8005-3262

Item Type:

Article

Additional Information:

The first author is supported by SERB Grant No. MTR/2019/001307, Govt. of India. The second author is supported by the Department of Science and Technology-INSPIRE Fellowship (Grant No. DST/INSPIRE FELLOWSHIP/2018/IF180107).

Uncontrolled Keywords:

Absolutely norm attaining operatorabsolutely minimum attaining operatoressential spectrum compact operatorpartial isometry

Subjects:

Mathematics

Divisions:

Department of Mathematics

Depositing User:

. LibTrainee 2021

Date Deposited:

12 Oct 2022 07:39

Last Modified:

12 Oct 2022 07:39