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 Hilbert spaces, and T a (linear) bounded operator from H_1 to H_2. For, define the Schatten p-norm of T as where, using the operator square root. If T is compact and H_1,,H_2 are separable, then for the singular values of T, i.e. the eigenvalues of the Hermitian operator.

Properties

In the following we formally extend the range of p to [1,\infty] with the convention that is the operator norm. The dual index to p=\infty is then q=1. If satisfy, then we have The latter version of Hölder's inequality is proven in higher generality (for noncommutative L^p spaces instead of Schatten-p classes) in. (For matrices the latter result is found in .)

Remarks

Notice that |\cdot|_2 is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), |\cdot|_1 is the trace class norm (see trace class), and is the operator norm (see operator norm). For p\in(0,1) the function |\cdot|_p is an example of a quasinorm. An operator which has a finite Schatten norm is called a Schatten class operator and the space of such operators is denoted by. With this norm, is a Banach space, and a Hilbert space for p = 2. Observe that, 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). The case p = 1 is often referred to as the nuclear norm (also known as the trace norm, or the Ky Fan n-norm )