Operators That Attain their Minima

TL;DR

This paper investigates operators on complex Hilbert spaces that reach their minimum in the unit sphere, providing characterizations of specific classes like N* and AN* operators, emphasizing the role of injectivity.

Contribution

It introduces new characterizations of N* and AN* operators on Hilbert spaces, highlighting the significance of injectivity in their properties.

Findings

Characterization of N* operators

Characterization of AN* operators

Injectivity's role in operator properties

Abstract

In this paper we study the theory of operators on complex Hilbert spaces, which attain their minimum in the unit sphere. We prove some important results concerning the characterization of the N*, and also AN* operators, see respectively Definition 1.1 and Definition 1.3. The injective property plays an important role in these operators, and shall be established by these classes.