Schatten norm

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.


Let , be separable Hilbert spaces, and a (linear) bounded operator from to . For , define the Schatten p-norm of as

for the singular values of , i.e. the eigenvalues of the Hermitian operator . From functional calculus on the positive operator it follows that


In the following we formally extend the range of to . The dual index to is then .

where denotes the Hilbert-Schmidt inner product.


Notice that is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), is the trace class norm (see trace class), and is the operator norm (see 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 (see compact operator on Hilbert space).


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