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
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.

Special cases

  • The Schatten 1-norm is the nuclear norm.
  • The Schatten 2-norm is the Frobenius norm.
  • The Schatten ∞-norm is the spectral norm.

    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.
  • The Schatten norms are unitarily invariant: for unitary operators and and,
  • They satisfy Hölder's inequality: for all and such that, and operators defined between Hilbert spaces and respectively,
If satisfy, then we have
The latter version of Hölder's inequality is proven in higher generality in.
  • Sub-multiplicativity: For all and operators defined between Hilbert spaces and respectively,
  • Monotonicity: For,
  • Duality: Let be finite-dimensional Hilbert spaces, and such that, then
  • Let be two orthonormal basis of the Hilbert spaces, then for

    Remarks

Notice that is the Hilbert–Schmidt norm, is the trace class norm, and is the 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.