Journal of Operator Theory

Volume 75, Issue 1, Winter 2016 pp. 139-149.

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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