Schatten norm

Summarize Timeline Fact Check

Let ${\displaystyle H_{1}}$, ${\displaystyle H_{2}}$ be Hilbert spaces, and ${\displaystyle T}$ a (linear) bounded operator from ${\displaystyle H_{1}}$ to ${\displaystyle H_{2}}$. For ${\displaystyle p\in [1,\infty )}$, define the Schatten p-norm of ${\displaystyle T}$ as

${\displaystyle \|T\|_{p}=[\operatorname {Tr} (|T|^{p})]^{1/p},}$

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

If ${\displaystyle T}$ is compact and ${\displaystyle H_{1},\,H_{2}}$ are separable, then

${\displaystyle \|T\|_{p}:={\bigg (}\sum _{n\geq 1}s_{n}^{p}(T){\bigg )}^{1/p}}$

for ${\displaystyle s_{1}(T)\geq s_{2}(T)\geq \cdots \geq s_{n}(T)\geq \cdots \geq 0}$ the singular values of ${\displaystyle T}$, i.e. the eigenvalues of the Hermitian operator ${\displaystyle |T|:={\sqrt {(T^{*}T)}}}$.

• The Schatten 1-norm is the nuclear norm (also known as the trace norm, or the Ky Fan n-norm^[1]).
• 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).

Summarize Timeline Top Qs Fact Check

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

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

${\displaystyle \|UTV\|_{p}=\|T\|_{p}.}$

• They satisfy Hölder's inequality: for all ${\displaystyle p\in [1,\infty ]}$ and ${\displaystyle q}$ such that ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}$, and operators ${\displaystyle S\in {\mathcal {L}}(H_{2},H_{3}),T\in {\mathcal {L}}(H_{1},H_{2})}$ defined between Hilbert spaces ${\displaystyle H_{1},H_{2},}$ and ${\displaystyle H_{3}}$ respectively,

${\displaystyle \|ST\|_{1}\leq \|S\|_{p}\|T\|_{q}.}$

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

${\displaystyle \|ST\|_{r}\leq \|S\|_{p}\|T\|_{q}}$.

The latter version of Hölder's inequality is proven in higher generality (for noncommutative ${\displaystyle L^{p}}$ spaces instead of Schatten-p classes) in.^[2] (For matrices the latter result is found in^[3].)

• Sub-multiplicativity: For all ${\displaystyle p\in [1,\infty ]}$ and operators ${\displaystyle S\in {\mathcal {L}}(H_{2},H_{3}),T\in {\mathcal {L}}(H_{1},H_{2})}$ defined between Hilbert spaces ${\displaystyle H_{1},H_{2},}$ and ${\displaystyle H_{3}}$ respectively,

${\displaystyle \|ST\|_{p}\leq \|S\|_{p}\|T\|_{p}.}$

• Monotonicity: For ${\displaystyle 1\leq p\leq p'\leq \infty }$,

${\displaystyle \|T\|_{1}\geq \|T\|_{p}\geq \|T\|_{p'}\geq \|T\|_{\infty }.}$

• Duality: Let ${\displaystyle H_{1},H_{2}}$ be finite-dimensional Hilbert spaces, ${\displaystyle p\in [1,\infty ]}$ and ${\displaystyle q}$ such that ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}$, then

${\displaystyle \|S\|_{p}=\sup \lbrace |\langle S,T\rangle |\mid \|T\|_{q}=1\rbrace ,}$

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

• Let ${\displaystyle (e_{k})_{k},(f_{k'})_{k'}}$ be two orthonormal basis of the Hilbert spaces ${\displaystyle H_{1},H_{2}}$, then for ${\displaystyle p=1}$

${\displaystyle \|T\|_{1}\leq \sum _{k,k'}\left|T_{k,k'}\right|.}$