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On the Schur-Agler Norm

Michael Hartz, Yi Wang

Abstract

We establish a new description of the Schur-Agler norm of a holomorphic function on the polydisc as the solution of a convex optimization problem. Consequences of this description are explored both from a theoretical and from a practical point of view. Firstly, we give unified proofs of the known facts that the Schur-Agler norm can be tested with diagonalizable or nilpotent matrix tuples, as well as a new proof of the existence of Agler decompositions. Secondly, we describe the predual of the Schur-Agler space as a space of analytic functions on the polydisc. Thirdly, we give a unified treatment of existing counterexamples of von Neumann's inequality in our framework, and exhibit several methods for constructing counterexamples. On the practical side, we explain how the Schur-Agler norm of a homogeneous polynomial can be numerically approximated using semidefinite programming.

On the Schur-Agler Norm

Abstract

We establish a new description of the Schur-Agler norm of a holomorphic function on the polydisc as the solution of a convex optimization problem. Consequences of this description are explored both from a theoretical and from a practical point of view. Firstly, we give unified proofs of the known facts that the Schur-Agler norm can be tested with diagonalizable or nilpotent matrix tuples, as well as a new proof of the existence of Agler decompositions. Secondly, we describe the predual of the Schur-Agler space as a space of analytic functions on the polydisc. Thirdly, we give a unified treatment of existing counterexamples of von Neumann's inequality in our framework, and exhibit several methods for constructing counterexamples. On the practical side, we explain how the Schur-Agler norm of a homogeneous polynomial can be numerically approximated using semidefinite programming.
Paper Structure (14 sections, 25 theorems, 182 equations)

This paper contains 14 sections, 25 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardy Space $H^2(\mathbb{D}^d)$
  4. Weak products
  5. Schur--Agler space
  6. A Convex Description of the Schur--Agler Norm
  7. First Applications: New Proofs
  8. Testing the Schur--Agler Norm on Diagonal or Nilpotent Tuples
  9. The Agler Decomposition
  10. The Dual Schur--Agler Norm
  11. Methods of Constructing $L$
  12. The Weak Product Norm
  13. Semidefinite Programming
  14. Appendix: The classical counterexamples

Key Result

Lemma 1.1

For any $f\in H^2(\mathbb{D}^d)$, where both sides of the equality are allowed to be infinite.

Theorems & Definitions (63)

  • Lemma 1.1: Lemma \ref{['lem: SA norm with Lc']}
  • Proposition 1.2
  • Theorem 1.3: Theorem \ref{['thm:dual_norm_general']} (1)
  • Theorem 1.4: Theorem \ref{['thm: norm 1 and 2']}
  • Theorem 1.5: Theorem \ref{['thm:three_norm_two_norm']}
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 53 more