Schatten norm

From Wikipedia, the free encyclopedia

In the theory of operators the Schatten norm arises as a generalization of p-integrability similar to the trace class norm and the Hilbert-Schmidt norm. The norm is defined as

\|T\| _{S_p}^p := \sum _{x\in \sigma (T^*T)} x^{p/2}

for p\in [1,\infty[ and an operator T on the Hilbert space X. Here σ(T * T) denotes the spectrum of the positive operator T*T. This should be interpreted as a multiset. An operator which has a finite Schatten norm is called a Schatten class operator and the space of such operators is denoted by Sp(X). With this norm, Sp(X) is a Banach space, and a Hilbert space for p=2.

Observe that S_p(X) \subseteq \mathcal{K} (X), 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). From functional calculus on the positive operator T*T it follows that

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

When p=2 this is the Hilbert-Schmidt norm (see Hilbert-Schmidt operator) and when p = 1 it is the trace class norm (see trace class).