Three observations regarding Schatten \(p\) classes (Q2835229)

Recall that, for \(1\leq p <\infty\), \(C_p\) denotes the Banach space of all compact operators \(A\) on \(\ell_2\) for which the norm \(| A|_p = (\operatorname{trace}(A^*A)^{p/2})^{1/p}\) is finite. One of the main result of this paper is to introduce a new complemented subspace of \(C_p\) (\(p\not=2\)), answering a question of \textit{J. Arazy} and \textit{J. Lindenstrauss} [Compos. Math. 30, 81--111 (1975; Zbl 0302.47034)]. The new complemented subspace is defined as follows. Let \(A\) be or a finite or infinite matrix \(A\) and let \(A(k, l)\) denote its \(k,l\) entry. Let \(Z_p\) denote the Banach space of matrices with the norm NEWLINE\[CARRIAGE_RETURNNEWLINE | A|_{Z_p}=\left(\sum_{k=1}^\infty\left(\sum_{l=1}^\infty | A(k.l) |^2\right)^{p/2}\right)^{1/p}<\infty. CARRIAGE_RETURNNEWLINE\]NEWLINE For \(p>2\), let \(\tilde{Z}_p\) denote the Banach space of matrices with the norm NEWLINE\[CARRIAGE_RETURNNEWLINE | A|_{\tilde{Z}_p}=(| A|_{Z_p}^p+| A^*|_{\tilde{Z}_p}^p)^ {1/p}<\infty. CARRIAGE_RETURNNEWLINE\]NEWLINE For \(1\leq q<2\), let \(\tilde{Z}_q\) denote the Banach space of matrices with the norm NEWLINE\[CARRIAGE_RETURNNEWLINE | A|_{\tilde{Z}_q}=\inf\{(| B|_{Z_q}^q+| C^*|_ {\tilde{Z}_q}^q)^{1/q};\;A=B+C\}<\infty. CARRIAGE_RETURNNEWLINE\]NEWLINE Finally, let \(C^n_p\), \(Z_p^n\) and \(\tilde{Z}_p^n\) denote the spaces of \(n\times n\) matrices with the norms inherited from \(C_p\), \(Z_p\) and \(\tilde{Z}_p\), respectively. The author shows that the space NEWLINE\[CARRIAGE_RETURNNEWLINE D_p=(\oplus_{n=1}^\infty \tilde{Z}_p^n)_p CARRIAGE_RETURNNEWLINE\]NEWLINE is complemented in \(C_p\), \(1<p \leq \infty\), and is not isomorphic to any of the previously known complemented subspaces of \(C_p\). The author shows two other results about the space \(C_p\). Precisely, the second result relates to tight embeddings of finite dimensional subspaces of \(C_p\) in \(C^n_p\) with small \(n\) and shows that \(\ell_p^k\) nicely embeds into \(C^n_p\) only if \(n\) is at least proportional to \(k\). The third result concerns single elements of \(C^n_p\) 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.