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Gateaux derivative of matrix norms in operator spaces and operator systems

Sushil Singla

TL;DR

This paper determines the Gateaux derivative of matrix norms $\|\cdot\|_n$ arising from operator spaces and operator systems by introducing a support-mapping framework and the sets $M_0(\phi_n(v))$. The main results express the derivative as a maximum over admissible support mappings and unit vectors, linking first-order norm behavior to matrix-range data and completely positive maps. The approach blends convex analysis, duality, and dilation techniques to obtain explicit derivative formulas, and extends to operator systems, yielding Birkhoff-James-type orthogonality results and matrix-state characterizations. Applications include special cases for operator norms, orthogonality criteria, quantum-measure formulations via POVMs, and extensions to commutative and real operator space settings, highlighting the impact on the geometry of operator spaces and their quantum-analytic interpretations.

Abstract

We find expressions for the Gateaux derivative of the matrix norms in operator spaces, and operator systems. Some applications of the results to quantum probability measures, states on C$^*$-algebras, and Birkhoff-James orthogonality are also presented.

Gateaux derivative of matrix norms in operator spaces and operator systems

TL;DR

This paper determines the Gateaux derivative of matrix norms arising from operator spaces and operator systems by introducing a support-mapping framework and the sets . The main results express the derivative as a maximum over admissible support mappings and unit vectors, linking first-order norm behavior to matrix-range data and completely positive maps. The approach blends convex analysis, duality, and dilation techniques to obtain explicit derivative formulas, and extends to operator systems, yielding Birkhoff-James-type orthogonality results and matrix-state characterizations. Applications include special cases for operator norms, orthogonality criteria, quantum-measure formulations via POVMs, and extensions to commutative and real operator space settings, highlighting the impact on the geometry of operator spaces and their quantum-analytic interpretations.

Abstract

We find expressions for the Gateaux derivative of the matrix norms in operator spaces, and operator systems. Some applications of the results to quantum probability measures, states on C-algebras, and Birkhoff-James orthogonality are also presented.

Paper Structure

This paper contains 5 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Proofs of the Main Results
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['cor3']}
  5. Applications of the Main Results

Key Result

Theorem 1.1

If $(V, \|\cdot\|_n)$ is an operator space, and $v,w\in M_n(V)$. Then there exists a support mapping for $v$ such that $M_0(\phi_n(v))\neq \emptyset$ and

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 7 more