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On Orthogonal Projections of Symplectic balls

Nuno Costa Dias, Maurice A. de Gosson, Joao Nuno Prata

Abstract

We study the orthogonal projections of symplectic balls in $\mathbb{R}^{2n}$ on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov's non-squeezing theorem. We use a conceptually simpler approach where the Schur complement of a matrix plays a central role.

On Orthogonal Projections of Symplectic balls

Abstract

We study the orthogonal projections of symplectic balls in on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov's non-squeezing theorem. We use a conceptually simpler approach where the Schur complement of a matrix plays a central role.

Paper Structure

This paper contains 5 sections, 1 theorem, 7 equations.

Table of Contents

  1. Introduction
  2. What is known
  3. What we will do
  4. Preliminaries
  5. Williamson's symplectic diagonalization

Key Result

Lemma \oldthetheorem

The Hermitian matrix $M+iJ$ is positive semi-definite: $M+iJ\geq0$ if and only if $\lambda_{j}^{\sigma}(M)\geq1$ for $1\leq j\leq n$.

Theorems & Definitions (1)

  • Lemma \oldthetheorem