Schatten norm

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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.

Definition[edit]

Let , be Hilbert spaces, and a (linear) 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[edit]

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,

(For matrices this can be generalized to for .[1])

  • 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

where denotes the Hilbert–Schmidt inner product.

Remarks[edit]

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).

The case p = 1 is often referred to as the nuclear norm (also known as the trace norm, or the Ky Fan 'n'-norm[2])

See also[edit]

Matrix Norms

References[edit]

  1. ^ Ball, Keith; Carlen, Eric A.; Lieb, Elliott H. (1994). "Sharp uniform convexity and smoothness inequalities for trace norms". Inventiones Mathematicae. 115: 463–482. doi:10.1007/BF01231769.
  2. ^ Fan, Ky. (1951). "Maximum properties and inequalities for the eigenvalues of completely continuous operators". Proceedings of the National Academy of Sciences of the United States of America. 37 (11): 760–766. Bibcode:1951PNAS...37..760F. doi:10.1073/pnas.37.11.760. PMC 1063464. PMID 16578416.
  • Rajendra Bhatia, Matrix analysis, Vol. 169. Springer Science & Business Media, 1997.
  • John Watrous, Theory of Quantum Information, 2.3 Norms of operators, lecture notes, University of Waterloo, 2011.
  • Joachim Weidmann, Linear operators in Hilbert spaces, Vol. 20. Springer, New York, 1980.