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

 

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  .

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,
 

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.[1] (For matrices the latter result is found in [2].)

  • 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.
  • Let   be two orthonormal basis of the Hilbert spaces  , then for  
 

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[3])

See also edit

Matrix norms

References edit

  1. ^ Fack, Thierry; Kosaki, Hideki (1986). "Generalized  -numbers of  -measurable operators" (PDF). Pacific Journal of Mathematics. 123 (2).
  2. ^ 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. S2CID 189831705.
  3. ^ 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.