for the singular values of , i.e. the eigenvalues of the Hermitian operator .
Propertiesedit
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
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 Fann-norm[3])
^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. PMC1063464. PMID 16578416.