Inner products on the Hilbert space S₂ of Hilbert--Schmidt operators

2025/06/05 by Josué I. Rios-Cangas, Rios-Cangas, Josué I.
Mathematics · #47A65 #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Primary 46C50 #Secondary 46C05 #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2506.04541

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This work presents a rigorous characterization of inner products on the Hilbert space S₂ 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(S₂) to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on S₂.

Citations

• Characterisation of equivalent norms on a linear space using exponential vector space
• On Hilbert-Schmidt operator formulation of noncommutative quantum mechanics

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