Schatten class

Remarks

Example: on $\ell^2$, the diagonal operator $T e_n = e_n / n$ has $s_n(T) = 1/n$, so $T \in S_p$ for every $p > 1$. Non-example: the same operator is not in $S_1$, because $\sum 1/n$ diverges. $S_1$ is the trace class, and $S_2$ is the class of Hilbert–Schmidt operators, with $\norm{T}_2^2 = \sum_{i} \norm{T e_i}^2$ for any orthonormal basis $(e_i)$ of $H$. For $p \le q$, $S_p \subseteq S_q$ and $\norm{T}_q \le \norm{T}_p$. Each $S_p(H) = S_p(H, H)$ is a two-sided ideal of the algebra of bounded operators on $H$, and a Banach space under $\norm{\cdot}_p$. The finite-rank operators are dense in each $S_p$, and the operator norm satisfies $\norm{T} \le \norm{T}_p$. Some authors also allow $0 < p < 1$, where $\norm{\cdot}_p$ is only a quasi-norm.