Schatten norm

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In mathematics, specifically functional analysis, the Schatten norm (or Schatten- Von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm.

Definition

Let $H_{1}$ , $H_{2}$ be separable Hilbert spaces, and $T$ a (linear) bounded operator from $H_{1}$ to $H_{2}$ . For $p\in [1,\infty )$ , define the Schatten p-norm of $T$ as

$\|T\|_{{p}}:={\bigg (}\sum _{{n\geq 1}}s_{n}^{p}(T){\bigg )}^{{1/p}}$

for $s_{1}(T)\geq s_{2}(T)\geq \cdots s_{n}(T)\geq \cdots \geq 0$ the singular values of $T$ , i.e. the eigenvalues of the Hermitian matrix $|T|:={\sqrt {(T^{*}T)}}$ . From functional calculus on the positive operator T*T it follows that

$\|T\|_{{p}}^{p}={\mathrm {tr}}(|T|^{p})$

Remarks

The Schatten norm is unitarily invariant: for $U$ and $V$ unitary operators,

$\|UTV\|_{{p}}=\|T\|_{{p}}$

Notice that $\|\ \|_{{2}}$ is the Hilbert-Schmidt norm (see Hilbert-Schmidt operator) and $\|\ \|_{{1}}$ is the trace class norm (see trace class).

An operator which has a finite Schatten norm is called a Schatten class operator and the space of such operators is denoted by $S_{p}(H_{1},H_{2})$ . With this norm, $S_{p}(H_{1},H_{2})$ is a Banach space, and a Hilbert space for p=2.

Observe that $S_{p}(H_{1},H_{2})\subseteq {\mathcal {K}}(H_{1},H_{2})$ , the algebra of compact operators. This follows from the fact that if the sum is finite the spectrum will be finite or countable with the origin as limit point, and hence a compact operator (see compact operator on Hilbert space).

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