Schatten-class operator - Imported Knowlpedia Content

Definition

Let $H$ and $K$ be Hilbert spaces and let $0<p<\infty$ . A Schatten-class operator of order $p$ is a compact operator $T:H\to K$ whose singular values satisfy

\sum_{n\geq 1}s_n(T)^p<\infty.

The space of all such operators is denoted $\mathcal S^p(H,K)$ , or $\mathcal L^p(H,K)$ , and carries the norm or quasinorm

\lVert T\rVert_p=\left(\sum_{n\geq1}s_n(T)^p\right)^{1/p}.

It is a norm when $p\geq1$ and a quasinorm when $0<p<1$ . When $H=K$ , $\mathcal S^p(H)$ is a two-sided ideal in the bounded operators on $H$ . Thus Schatten membership is a quantitative compactness condition.

Ideal and inclusion properties

If $A$ and $B$ are bounded operators of compatible sizes and $T\in\mathcal S^p$ , then

\lVert ATB\rVert_p\leq\lVert A\rVert\,\lVert T\rVert_p\,\lVert B\rVert.

For $0<p<q<\infty$ , one has $\mathcal S^p\subseteq\mathcal S^q\subseteq\mathcal K$ . Hölder's inequality also has an operator-ideal form: if $1/r=1/p+1/q$ , then products of $\mathcal S^p$ - and $\mathcal S^q$ -operators lie in $\mathcal S^r$ , with the usual qualification $r\geq1$ when a Banach norm is desired Simon, Chapters 1–2.

Distinguished cases

The class $\mathcal S^1$ is the trace class, and $\mathcal S^2$ is the Hilbert--Schmidt class. For a diagonal compact operator on $\ell^2$ with diagonal $(\lambda_n)$ , membership in $\mathcal S^p$ is exactly the condition $(\lambda_n)\in\ell^p$ . Finite-rank operators belong to every $\mathcal S^p$ and are dense there in the $p$ -norm.

Conventions and scope

For $0<p<1$ , “Schatten class” is sometimes replaced by “quasi-Schatten ideal” to emphasize the lack of a norm. Some authors extend the notation by setting $\mathcal S^\infty(H)=\mathcal K(H)$ with the operator norm. Others reserve “Schatten class” for finite $p$ , as in the core definition. These conventions should be stated when endpoints or subunit exponents matter.

References

Barry Simon, Trace Ideals and Their Applications, 2nd ed., American Mathematical Society, 2005. DOI record. Relevant: Chapters 1--2 on singular values, Schatten ideals, and ideal inequalities.