Absolutely norm attaining Toeplitz and absolutely minimum attaining Hankel operators

Ramesh, G. and Sequeira, Shanola S. (2022) Absolutely norm attaining Toeplitz and absolutely minimum attaining Hankel operators. Journal of Mathematical Analysis and Applications, 516 (1). ISSN 0022-247X

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Abstract

Let H1 and H2 be complex Hilbert spaces. A bounded linear operator T:H1→H2 is called norm attaining if ‖Tx‖=‖T‖ for some unit vector x∈H1. If for every closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator (or AN-operator). In the above definitions, if we replace the norm of the operator by the minimum modulus m(T)=inf⁡{‖Tx‖:x∈H1,‖x‖=1}, then T is called a minimum attaining and an absolutely minimum attaining operator (or AM-operator), respectively. In this article, we characterize Toeplitz AN-operators and discuss a few results on the minimum modulus of Toeplitz operator Tφ, φ∈L∞(T). We further characterize the minimum attaining Hankel operators and deduce that the only Hankel AM-operators are finite rank operators. While proving our results, we also obtained the following result; If φ∈L∞(T), then m(Lφ)=ess inf|φ| and there exists ψ∈L∞(T) such that γ(Lψ)>ess inf|ψ|, which improves a result from [15]. © 2022 Elsevier Inc.

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IITH Creators:

IITH Creators	ORCiD
Ramesh, G.	UNSPECIFIED

Item Type:

Article

Additional Information:

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

Uncontrolled Keywords:

Absolutely minimum attaining operator; Absolutely norm attaining operator; Hankel operator; Inner function; Invariant subspace; Toeplitz operator

Subjects:

Mathematics

Divisions:

Department of Mathematics

Depositing User:

. LibTrainee 2021

Date Deposited:

01 Aug 2022 10:23

Last Modified:

01 Aug 2022 10:23