In mathematics, specifically functional analysis, the Schatten norm (or SchattenâÂÂvon-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the HilbertâÂÂSchmidt norm.
Let , be Hilbert spaces, and a (linear) bounded operator from to . For , define the Schatten p-norm of as
where , using the operator square root.
If is compact and are separable, then
for the singular values of , i.e. the eigenvalues of the Hermitian operator .
In the following we formally extend the range of to with the convention that is the operator norm. The dual index to is then .
If satisfy , then we have
The latter version of Hölder's inequality is proven in higher generality (for noncommutative spaces instead of Schatten-p classes) in. (For matrices the latter result is found in.)
Notice that is the HilbertâÂÂSchmidt norm (see HilbertâÂÂSchmidt operator), is the trace class norm (see trace class), and is the operator norm (see operator norm).
Note that the matrix p-norm is often also written as , but it is not the same as Schatten norm. In fact, we have .
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 (see compact operator on Hilbert space).