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.
be Hilbert spaces, and
a (linear) bounded operator from
, define the Schatten p-norm of
, using the operator square root.
the singular values of
, i.e. the eigenvalues of the Hermitian operator
In the following we formally extend the range of
with the convention that
is the operator norm.
The dual index to
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.
[2] (For matrices the latter result is found in[3].)
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).
Note that the matrix p-norm is often also written as
, but it is not the same as Schatten norm.
matrix p-norm
Schatten
{\displaystyle \|A\|_{{\text{matrix p-norm}},2}=\|A\|_{{\text{Schatten}},\infty }}
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
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).
Matrix norms