Schatten norm

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. 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. Let H 1 {displaystyle H_{1}} , H 2 {displaystyle H_{2}} be separable Hilbert spaces, and T {displaystyle T} a (linear) bounded operator from H 1 {displaystyle H_{1}} to H 2 {displaystyle H_{2}} . For p ∈ [ 1 , ∞ ) {displaystyle pin [1,infty )} , define the Schatten p-norm of T {displaystyle T} as for s 1 ( T ) ≥ s 2 ( T ) ≥ ⋯ s n ( T ) ≥ ⋯ ≥ 0 {displaystyle s_{1}(T)geq s_{2}(T)geq cdots s_{n}(T)geq cdots geq 0} the singular values of T {displaystyle T} , i.e. the eigenvalues of the Hermitian operator | T | := ( T ∗ T ) {displaystyle |T|:={sqrt {(T^{*}T)}}} .From functional calculus on the positive operator T ∗ T {displaystyle T^{*}T} it follows that In the following we formally extend the range of p {displaystyle p} to [ 1 , ∞ ] {displaystyle } . The dual index to p = ∞ {displaystyle p=infty } is then q = 1 {displaystyle q=1} . (For matrices this can be generalized to ‖ S T ‖ r ≤ ‖ S ‖ p ‖ T ‖ q {displaystyle |ST|_{r}leq |S|_{p}|T|_{q}} for 1 r = 1 p + 1 q {displaystyle { frac {1}{r}}={ frac {1}{p}}+{ frac {1}{q}}} .) where ⟨ S , T ⟩ = t r ( S ∗ T ) {displaystyle langle S,T angle =mathrm {tr} (S^{*}T)} denotes the Hilbert–Schmidt inner product. Notice that ‖ ⋅ ‖ 2 {displaystyle |cdot |_{2}} is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), ‖ ⋅ ‖ 1 {displaystyle |cdot |_{1}} is the trace class norm (see trace class), and ‖ ⋅ ‖ ∞ {displaystyle |cdot |_{infty }} is the operator norm (see operator norm). For p ∈ ( 0 , 1 ) {displaystyle pin (0,1)} the function ‖ ⋅ ‖ p {displaystyle |cdot |_{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 ) {displaystyle S_{p}(H_{1},H_{2})} . With this norm, S p ( H 1 , H 2 ) {displaystyle S_{p}(H_{1},H_{2})} is a Banach space, and a Hilbert space for p = 2.