On the denseness of minimum attaining operator-valued functions

Abstract

Let $\mathcal{U}$ be a bounded open subset of $\mathbb{C}$ and $\mathcal{H}$ be a complex separable Hilbert space. We define the following classes of functions on $\overline{\mathcal{U}}.$ $\begin{array}{rl}& \mathcal{C}(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))=\{f:\overline{\mathcal{U}}\to \mathcal{B}(H):f\text{ is continuous}\}\\ & {\mathcal{C}}_m(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))=\{f\in \mathcal{C}(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H})):f(z)\text{ is minimum}\\ & \text{attaining for every }z\in \mathcal{U}\}\\ & {\mathcal{C}}_{rm}(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))=\{f\in \mathcal{C}(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H})):f(z)\text{ is reduced minimum}\\ & \text{attaining for every }z\in \mathcal{U}\}\\ & {\mathcal{C}}_h(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))=\{f\in \mathcal{C}(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H})):|f|\text{ is harmonic}\}\end{array}$Along with a few finer results on denseness of minimum attaining operators, this article primarily deals with denseness of (i) ${\mathcal{C}}_m(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))$ in $\mathcal{C}(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))$ and (ii) ${\mathcal{C}}_{rm}(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))$ in ${\mathcal{C}}_h(\overline{\mathcal{U}},\mathcal{B}(\mathcal{H}))$ with respect to the supremum norm.

Keywords:

• Minimum attaining operators
• reduced minimum attaining operators
• operator-valued functions

2020 Mathematics Subject Classifications:

• Primary: 47A55
• 47A56
• 47B02
• Secondary:47A05
• 47A10
• 47A56
• 47A58

Disclosure statement

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