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
Let
,
be Hilbert spaces, and
a (linear) bounded operator from
to
. For
, define the Schatten p-norm of
as
![{\displaystyle \|T\|_{p}=[\operatorname {Tr} (|T|^{p})]^{1/p},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cda85cbc741f7fdf34db91aa5f7c23f080cd097)
where
, using the operator square root.
If
is compact and
are separable, then
![{\displaystyle \|T\|_{p}:={\bigg (}\sum _{n\geq 1}s_{n}^{p}(T){\bigg )}^{1/p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/831c8693457e0951141d6e8415a4dda8608c440f)
for
the singular values of
, i.e. the eigenvalues of the Hermitian operator
.
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
,
![{\displaystyle \|UTV\|_{p}=\|T\|_{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de0571e6985748981280fff7b386e0240821fd1f)
- They satisfy Hölder's inequality: for all
and
such that
, and operators
defined between Hilbert spaces
and
respectively,
![{\displaystyle \|ST\|_{1}\leq \|S\|_{p}\|T\|_{q}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026a5e0d639bea3e27e4622385786fa903390a60)
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,
![{\displaystyle \|ST\|_{p}\leq \|S\|_{p}\|T\|_{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f99da41167c3afd8e8013a2ed267324f94bf8cc5)
- Monotonicity: For
,
![{\displaystyle \|T\|_{1}\geq \|T\|_{p}\geq \|T\|_{p'}\geq \|T\|_{\infty }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed83af77c7f4db2e747ebc20097d25bea0d833e)
- Duality: Let
be finite-dimensional Hilbert spaces,
and
such that
, then
![{\displaystyle \|S\|_{p}=\sup \lbrace |\langle S,T\rangle |\mid \|T\|_{q}=1\rbrace ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5c783ebb2df4d090e30e21c9c2b63d6634c78d)
- where
denotes the Hilbert–Schmidt inner product.
- Let
be two orthonormal basis of the Hilbert spaces
, then for ![{\displaystyle p=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924)
![{\displaystyle \|T\|_{1}\leq \sum _{k,k'}\left|T_{k,k'}\right|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f97c226653b0602d90fbad8f1bdfccb4675754)
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
Matrix norms
References
- ^ Fack, Thierry; Kosaki, Hideki (1986). "Generalized
-numbers of
-measurable operators" (PDF). Pacific Journal of Mathematics. 123 (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.
- ^ 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.