On the closure of absolutely norm attaining operators

ABSTRACT

Let $H_1$ and $H_2$ be complex Hilbert spaces and $T:H_1\to H_2$ be a bounded linear operator. We say $T$  is norm attaining if there exists $x\in H_1$ with $\Vert x\Vert =1$ such that $\Vert Tx\Vert =\Vert T\Vert$. If for every non-zero closed subspace $M$ of $H_1$, the restriction $T{|}_M:M\to H_2$ is norm attaining, then $T$ is called an absolutely norm attaining operator or $\mathcal{A}\mathcal{N}$-operator. If we replace the norm of the operator by the minimum modulus $m\left(T\right)=inf\{\Vert Tx\Vert :x\in H_1,\Vert x\Vert =1\}$ in the above definitions, then $T$ is called a minimum attaining and an absolutely minimum attaining operator or $\mathcal{A}\mathcal{M}$-operator, respectively. In this article, we discuss the operator norm closure of $\mathcal{A}\mathcal{N}$-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 $\mathcal{A}\mathcal{M}$-operators and prove that the closure of $\mathcal{A}\mathcal{M}$-operators is the same as the closure of $\mathcal{A}\mathcal{N}$-operators. Consequently, we prove similar results for operators in the norm closure of $\mathcal{A}\mathcal{M}$-operators.

KEYWORDS:

• Absolutely norm attaining operator
• absolutely minimum attaining operator
• essential spectrum
• compact operator
• partial isometry

Acknowledgments

We thank all the referees for their valuable suggestions which improved the clarity of the paper and the handling editor for the help during the editorial process.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

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).