Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators

TL;DR

This paper rigorously characterizes all inner products on the Hilbert space of Hilbert--Schmidt operators using positive operators, providing necessary and sufficient conditions for positivity within this context.

Contribution

It offers a comprehensive characterization of inner products on $S_2$ via positive operators, extending the understanding of their structure in Hilbert spaces.

Findings

Characterization of all inner products on $S_2$ using positive operators

Necessary and sufficient conditions for positivity in $(S_2)$

Framework for describing inner products on Hilbert--Schmidt operators

Abstract

This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$.