Schatten norm


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

Definition

Let, be Hilbert spaces, and a bounded operator from
to. For, define the Schatten p-norm of as
If is compact and are separable, then
for
the singular values of, i.e. the eigenvalues of the Hermitian operator.

Properties

In the following we formally extend the range of to with the convention that is the operator norm. The dual index to is then.
where denotes the Hilbert–Schmidt inner product.

Remarks

Notice that is the Hilbert–Schmidt norm, is the trace class norm, and is the operator norm.
For the function 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.
The case p = 1 is often referred to as the nuclear norm

See Also