# Journal of Operator Theory

Volume 75, Issue 1, Winter 2016 pp. 139149.

Three observations regarding Schatten $p$ classes

**Authors**:
Gideon Schechtman

**Author institution:** Department of Mathematics, Weizmann Institute of Science,
Rehovot 76100, Israel

**Summary: **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 elements 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.

**DOI: **http://dx.doi.org/10.7900/jot.2014dec07.2048

**Keywords: **Schatten classes, complemented subspaces, tight embedding, paving

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