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Isometric Embeddability of Schatten Classes Revisited

Arup Chattopadhyay, Chandan Pradhan, Anna Skripka

Abstract

In this note, we summarize known results and open questions on the existence of isometric embeddings between different Schatten classes as well as obtain a new non-embeddability result using a novel method. We also provide a brief overview of the relevant methods.

Isometric Embeddability of Schatten Classes Revisited

Abstract

In this note, we summarize known results and open questions on the existence of isometric embeddings between different Schatten classes as well as obtain a new non-embeddability result using a novel method. We also provide a brief overview of the relevant methods.
Paper Structure (6 sections, 14 theorems, 28 equations)

This paper contains 6 sections, 14 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Results on embeddability.
  3. Embeddability of sequence spaces.
  4. Embeddability of function spaces.
  5. Embeddability of Schatten classes.
  6. Open problems

Key Result

Theorem 2.1

( Ba32) Let $U$ be a linear onto isometry on $\ell_p(\mathbb{C})$, where $1\le p\neq 2$. Then there exist a function $\phi:\mathbb{N}\to\mathbb{N}$ and a sequence $\{\epsilon_n\}_{n\in\mathbb{N}}$ such that Conversely, for any $\phi$ and $\{\epsilon_n\}$ satisfying (a) and (b), the operator $U$ defined by (c) is a linear isometry on $\ell_p$.

Theorems & Definitions (20)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • proof
  • Theorem 2.8
  • ...and 10 more