Semi-inner product and angles in Schatten ideals

TL;DR

The paper investigates the geometry of Schatten $p$-class ideals $\,\mathcal{B}_p(\mathcal{H})$ ($1<p<\infty$) through the lens of semi-inner products, defining a unique SIP $[\cdot,\cdot]_p$ and establishing a CS-type inequality that underpins operator orthogonality and parallelism. It then develops a rich angle theory by introducing several angle notions, notably a $p$-angle based on $|\alpha_{Y,X}|$ and $|\alpha_{X,Y}|$, along with gg- and Heinz-mean based angles, and analyzes their relationships to $p$-orthogonality and $p$-parallelism. The work provides a unified framework to study operator geometry in Schatten classes, including conditions for equivalence and symmetry of semi-inner products, and it proposes generalized mean-based angles to capture broader geometric behavior. Overall, these results deepen the understanding of the geometric structure and operator behavior in semi-inner product spaces associated with Schatten ideals, with potential implications for spectral theory and non-Hilbertian operator analysis.

Abstract

In this paper, we investigate the Schatten $p$-class ideals for $p >1$ as semi-inner product spaces in the sense of Giles and Lumer. Within this framework, we explore several geometric and analytic notions such as Birkhoff-James orthogonality, $p$-parallelism, and related properties that naturally arise when these structures are interpreted through the lens of the associated semi-inner product. Furthermore, we introduce a novel notion of angle adapted to this context, which generalizes and unifies existing angle definitions in normed spaces. Our results contribute to a deeper understanding of the geometry of the $p$-Schatten class and offer new perspectives on operator behavior in semi-inner product spaces.

Theorems & Definitions (38)

• Lemma 2.1
• Remark 2.1
• Definition 2.1
• Example 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof