What question did this study set out to answer?

To rigorously characterize continuous inner products in the Hilbert space S2 of Hilbert–Schmidt operators.

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March 15, 2026Open Access

Inner Products on the Hilbert Space S₂ of Hilbert–Schmidt Operators

Key Points

To rigorously characterize continuous inner products in the Hilbert space S2 of Hilbert–Schmidt operators.
Addressed continuous sesquilinear forms on a Hilbert space.
Identified necessary and sufficient conditions for positive operators in the context of S2.
Characterized all continuous inner products by means of positive operators in B(H).
Established criteria for characterizing inner products using positive operators.
Described continuous inner products on S2 rigorously.

Abstract

Abstract This work presents a rigorous characterization of continuous inner products on the Hilbert space S₂ S 2 of Hilbert–Schmidt operators. We begin by addressing the general framework of continuous sesquilinear forms on a Hilbert space H H and provide a characterization of all continuous inner products, by means of positive operators in B (H) B (H). Next, we establish necessary and sufficient conditions for an operator in B (S₂) B (S 2) to be positive. Identifying a continuous inner product with a positive operator enables us to rigorously describe inner products on S₂ S 2.

Inner Products on the Hilbert Space S₂ of Hilbert–Schmidt Operators

Key Points

Abstract

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