Three observations regarding Schatten p classes

TL;DR

This paper presents three new results on Schatten p classes, including a complemented subspace construction, embedding dimension bounds, and paving properties for matrices, advancing understanding of their structure and embeddings.

Contribution

It introduces a new complemented subspace of C_p, establishes dimension bounds for embeddings of ℓ_p^k into C_p^n, and shows paving properties for matrices in C_p for p>2.

Findings

Constructed a new complemented subspace of C_p for p≠2.

Proved lower bounds on n for embedding ℓ_p^k into C_p^n.

Demonstrated paving properties for matrices in C_p when p>2.

Abstract

The paper contains three results, the common feature of which is that they deal with the Schatten $p$ class. The first is a presentation of a new complemented subspace of $C_p$ in the reflexive range (and $p\not= 2$). This construction answers a question of Arazy and Lindestrauss from 1975. The second result relates to tight embeddings of finite dimensional subspaces of $C_p$ in $C_p^n$ with small $n$ and shows that $\ell_p^k$ nicely embeds into $C_p^n$ only if $n$ is at least proportional to $k$ (and then of course the dimension of $C_p^n$ is at least of order $k^2$). The third result concerns single element of $C_p^n$ and shows that for $p>2$ any $n\times n$ matrix of $C_p$ norm one and zero diagonal admits, for every $\varepsilon>0$, a $k$-paving of $C_p$ norm at most $\varepsilon$ with $k$ depending on $\varepsilon$ and $p$ only.