Schatten norm

Let $H\_1$, $H\_2$ be Hilbert spaces, and $T$ a (linear) bounded operator from

$H\_1$ to $H\_2$. For $p\backslash in\; [1,\backslash infty)$, define the Schatten p-norm of $T$ as

: $\backslash |T\backslash |\; \_p\; =\; [\backslash operatorname\{Tr\}\; (|T|^p)]^\{1/p\},$

where $|T|:=\backslash sqrt\{(T^*T)\}$, using the operator square root.

If $T$ is compact and $H\_1,\backslash ,H\_2$ are separable, then

: $\backslash |T\backslash |\; \_p\; :=\; \backslash bigg(\; \backslash sum\_\{n\backslash ge\; 1\}\; s^p\_n(T)\backslash bigg)^\{1/p\}$

for $s\_1(T)\; \backslash ge\; s\_2(T)\; \backslash ge\; \backslash cdots\; \backslash ge\; s\_n(T)\; \backslash ge\; \backslash cdots\; \backslash ge\; 0$ the singular values of $T$, i.e. the eigenvalues of the Hermitian operator $|T|:=\backslash sqrt\{(T^*T)\}$.

• The Schatten 1-norm is the nuclear norm (also known as the trace norm, or the Ky Fan n-norm{{Cite journal |last=Fan |first=Ky. |date=1951 |title=Maximum properties and inequalities for the eigenvalues of completely continuous operators |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=37 |issue=11 |pages=760–766 |bibcode=1951PNAS...37..760F |doi=10.1073/pnas.37.11.760 |pmc=1063464 |pmid=16578416 |doi-access=free}}).
• The Schatten 2-norm is the Frobenius norm.
• The Schatten ∞-norm is the spectral norm (also known as the operator norm, or the largest singular value).

In the following we formally extend the range of $p$ to $[1,\backslash infty]$ with the convention that $\backslash |\backslash cdot\backslash |\_\{\backslash infty\}$ is the operator norm. The dual index to $p=\backslash infty$ is then $q=1$.

• The Schatten norms are unitarily invariant: for unitary operators $U$ and $V$ and $p\backslash in\; [1,\backslash infty]$,

:: $\backslash |U\; T\; V\backslash |\_p\; =\; \backslash |T\backslash |\_p.$

• They satisfy Hölder's inequality: for all $p\backslash in\; [1,\backslash infty]$ and $q$ such that $\backslash frac\{1\}\{p\}\; +\; \backslash frac\{1\}\{q\}\; =\; 1$, and operators $S\backslash in\backslash mathcal\{L\}(H\_2,H\_3),\; T\backslash in\backslash mathcal\{L\}(H\_1,H\_2)$ defined between Hilbert spaces $H\_1,\; H\_2,$ and $H\_3$ respectively,

:: $\backslash |ST\backslash |\_1\; \backslash leq\; \backslash |S\backslash |\_p\; \backslash |T\backslash |\_q.$

If $p,q,r\backslash in\; [1,\backslash infty]$ satisfy $\backslash tfrac\{1\}\{p\}\; +\; \backslash tfrac\{1\}\{q\}\; =\; \backslash tfrac\{1\}\{r\}$, then we have

:: $\backslash |ST\backslash |\_r\; \backslash leq\; \backslash |S\backslash |\_p\; \backslash |T\backslash |\_q$.

The latter version of Hölder's inequality is proven in higher generality (for noncommutative $L^p$ spaces instead of Schatten-p classes) in.{{cite journal |last1=Fack |first1=Thierry |last2=Kosaki |first2=Hideki |title=Generalized $s$-numbers of $\backslash tau$-measurable operators. |journal=Pacific Journal of Mathematics |date=1986 |volume=123 |issue=2 |url=https://msp.org/pjm/1986/123-2/pjm-v123-n2-p03-s.pdf}}

(For matrices the latter result is found in{{Cite journal |doi = 10.1007/BF01231769|title = Sharp uniform convexity and smoothness inequalities for trace norms|journal = Inventiones Mathematicae|volume = 115|pages = 463–482|year = 1994|last1 = Ball|first1 = Keith|last2 = Carlen|first2 = Eric A.|last3 = Lieb|first3 = Elliott H.| bibcode=1994InMat.115..463B |s2cid = 189831705}}.)

• Sub-multiplicativity: For all $p\backslash in\; [1,\backslash infty]$ and operators $S\backslash in\backslash mathcal\{L\}(H\_2,H\_3),\; T\backslash in\backslash mathcal\{L\}(H\_1,H\_2)$ defined between Hilbert spaces $H\_1,\; H\_2,$ and $H\_3$ respectively,

:: $\backslash |ST\backslash |\_p\; \backslash leq\; \backslash |S\backslash |\_p\; \backslash |T\backslash |\_p\; .$

• Monotonicity: For $1\backslash leq\; p\backslash leq\; p\text{'}\backslash leq\backslash infty$,

:: $\backslash |T\backslash |\_1\; \backslash geq\; \backslash |T\backslash |\_p\; \backslash geq\; \backslash |T\backslash |\_\{p\text{'}\}\; \backslash geq\; \backslash |T\backslash |\_\backslash infty.$

• Duality: Let $H\_1,\; H\_2$ be finite-dimensional Hilbert spaces, $p\backslash in\; [1,\backslash infty]$ and $q$ such that $\backslash frac\{1\}\{p\}\; +\; \backslash frac\{1\}\{q\}\; =\; 1$, then

:: $\backslash |S\backslash |\_p\; =\; \backslash sup\backslash lbrace\; |\backslash langle\; S,T\backslash rangle\; |\; \backslash mid\; \backslash |T\backslash |\_q\; =\; 1\backslash rbrace,$

: where $\backslash langle\; S,T\backslash rangle\; =\; \backslash operatorname\{tr\}(S^*T)$ denotes the Hilbert–Schmidt inner product.

• Let $(e\_k)\_k,(f\_\{k\text{'}\})\_\{k\text{'}\}$ be two orthonormal basis of the Hilbert spaces $H\_1,\; H\_2$, then for $p=1$

:: $\backslash |T\backslash |\_1\; \backslash leq\; \backslash sum\_\{k,k\text{'}\}\backslash left|T\_\{k,k\text{'}\}\backslash right|.$

Notice that $\backslash |\backslash cdot\backslash |\_2$ is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), $\backslash |\backslash cdot\backslash |\_1$ is the trace class norm (see trace class), and $\backslash |\backslash cdot\backslash |\_\backslash infty$ is the operator norm (see operator norm).

Note that the matrix p-norm is often also written as $\backslash |\backslash cdot\backslash |\_p$, but it is not the same as Schatten norm. In fact, we have $\backslash |A\backslash |\_\{\backslash text\{matrix\; p-norm\},\; 2\}\; =\; \backslash |A\backslash |\_\{\backslash text\{Schatten\},\; \backslash infty\}$.

For $p\backslash in(0,1)$ the function $\backslash |\backslash cdot\backslash |\_p$ 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 $S\_p(H\_1,H\_2)$. With this norm, $S\_p(H\_1,H\_2)$ is a Banach space, and a Hilbert space for p = 2.

Observe that $S\_p(H\_1,H\_2)\; \backslash subseteq\; \backslash mathcal\{K\}\; (H\_1,H\_2)$, 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

{{reflist}}

• Rajendra Bhatia, Matrix analysis, Vol. 169. Springer Science & Business Media, 1997.
• John Watrous, Theory of Quantum Information, [https://web.archive.org/web/20160304053759/https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/02.pdf 2.3 Norms of operators], lecture notes, University of Waterloo, 2011.
• Joachim Weidmann, Linear operators in Hilbert spaces, Vol. 20. Springer, New York, 1980.

Category:Operator theory