Operator norm

An operator norm is an object from the mathematical branch of functional analysis . The operator norm generalizes the idea of ​​assigning a length to an object to the set of linear operators . If the operators to be considered are continuous , then the operator norm is a real norm , otherwise the operator norm can take on the value infinite. The operator norm of a linear mapping between finite-dimensional vector spaces is a natural matrix norm after choosing a basis .

definition

Let and be normalized vector spaces and be a linear operator . Then the operator norm is ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle f\colon V\rightarrow W}$

${\displaystyle \|{\cdot }\|\;\colon \;\{f\colon V\to W\mid f\ {\text{linear}}\}\to \mathbb {R} _{0}^{+}\cup \{\infty \}}$

regarding the vector norms and by ${\displaystyle \|\cdot \|_{V}}$ ${\displaystyle \|\cdot \|_{W}}$

${\displaystyle \|f\|:=\inf \left\{c\geq 0\mid \forall x\in V\colon \|f(x)\|_{W}\leq c\,\|x\|_{V}\right\}}$

Are defined. This is equivalent to

${\displaystyle \|f\|=\sup _{x\in V\setminus \{0\}}{\frac {\|f(x)\|_{W}}{\|x\|_{V}}}=\sup _{\|x\|_{V}=1}\|f(x)\|_{W}=\sup _{\|x\|_{V}\leq 1}\|f(x)\|_{W}.}$

properties

In addition to the three properties that are characteristic of norms, definiteness , absolute homogeneity and triangular inequality , the operator norm has other properties . These are not least:

Validity of the fundamental inequality

If is a linear operator, then holds for always ${\displaystyle f\colon V\to W}$ ${\displaystyle x\in V}$

${\displaystyle \|f(x)\|_{W}\leq \|f\|\cdot {\|x\|_{V}}.}$

Sub-multiplicativity

If and are linear operators, then the respective operator norms are sub-multiplicative in addition to the usual norm properties . That is, it applies ${\displaystyle f\colon V\to W}$ ${\displaystyle g\colon X\rightarrow V}$

${\displaystyle \|f\circ g\|\leq \|f\|\cdot \|g\|.}$

Narrow-mindedness

The operator norm of linear mappings between finite-dimensional vector spaces is always finite, since the unit sphere is a compact set . Thus, in the finite-dimensional case, the operator norm is always a real norm. This does not always apply to infinite-dimensional vector spaces. Operators whose norm takes infinite as their value are called unbounded . Strictly speaking, the operator norm is not a real norm on spaces with such unrestricted operators. One can show that a linear operator between normalized spaces has a finite operator norm if and only if it is bounded and thus continuous . In particular, this turns the space of continuous linear operators into a normalized vector space .

completeness

If is complete , the operator space is complete. The room doesn't need to be complete. ${\displaystyle W}$ ${\displaystyle L(V,W)}$ ${\displaystyle V}$

Examples

Natural matrix norms

→ Main article : Natural matrix norm

Since every linear operator between finite-dimensional vector spaces can be represented as a matrix , special matrix norms , the natural or induced matrix norms , are obvious examples of operator norms . The most important of these natural matrix norms are the following three. ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$

• The column sum norm is the norm induced by the sum norm:

${\displaystyle \|A\|_{1}=\max _{\|x\|_{1}=1}\|Ax\|_{1}=\max _{j=1,\ldots ,n}\sum _{i=1}^{m}|a_{ij}|.}$

It corresponds to the maximum amount total of all columns in the matrix.

• The spectral norm is the norm induced by the Euclidean norm:

${\displaystyle \|A\|_{2}=\max _{\|x\|_{2}=1}\|Ax\|_{2}={\sqrt {\lambda _{\max }(A^{H}A)}}.}$

It corresponds to the square root of the absolute greatest eigenvalue of , whereby the adjoint matrix (in the real case transposed matrix ) is to. ${\displaystyle A^{H}A}$ ${\displaystyle A^{H}}$ ${\displaystyle A}$

• The row sum norm is the norm induced by the maximum norm:

${\displaystyle \|A\|_{\infty }=\max _{\|x\|_{\infty }=1}\|Ax\|_{\infty }=\max _{i=1,\ldots ,m}{\sum _{j=1}^{n}|a_{ij}|}.}$

It corresponds to the maximum amount total of all rows in the matrix.

However, not every matrix norm is an operator norm. For example, the overall norm and the Frobenius norm are not operator norms.

The sequence space l ²

Let be a bounded sequence and thus an element of the sequence space that is provided with the norm . Now define a multiplication operator by . Then applies to the corresponding operator norm ${\displaystyle s=(s_{i})_{i\in \mathbb {N} }}$ ${\displaystyle \ell ^{\infty }}$ ${\displaystyle \textstyle \|s\|_{\infty }=\sup _{n}|s_{n}|}$ ${\displaystyle T_{s}\colon \ell ^{2}\to \ell ^{2}}$ ${\displaystyle \textstyle a\mapsto (s_{i}\cdot a_{i})_{i\in \mathbb {N} }}$

${\displaystyle \|T_{s}\|=\sup _{\|a\|_{\ell ^{2}}\neq 0}{\frac {\|T_{s}a\|_{\ell ^{2}}}{\|a\|_{\ell ^{2}}}}=\sup _{\|a\|_{\ell ^{2}}=1}{\sqrt {\sum _{i=1}^{\infty }|s_{i}\cdot a_{i}|^{2}}}=\sup _{i}|s_{i}|=\|s\|_{\ell ^{\infty }}.}$

Norm of a (pseudo) differential operator

Let be and be a bounded linear operator between Sobolev spaces . Such operators can be represented as pseudo differential operators . Under certain circumstances, especially when the order of the Sobolev spaces is an integer, the pseudo differential operators are ( weak ) differential operators . The space of (pseudo) differential operators can be provided with an operator norm. Since the norm in Sobolev space is given by , the operator norm for the (pseudo) differential operators is by ${\displaystyle s,\alpha >0}$ ${\displaystyle P\colon H^{s}(\Omega )\to H^{s+\alpha }(\Omega )}$ ${\displaystyle \|f\|_{H^{s}}=\|(1+|\cdot |^{2})^{\frac {s}{2}}\cdot {\mathcal {F}}(f)\|_{L^{2}}}$

${\displaystyle \|P\|=\sup _{\|f\|_{H^{s}}\neq 0}{\frac {\|Pf\|_{H^{s+\alpha }}}{\|f\|_{H^{s}}}}=\sup _{\|f\|_{H^{s}}\neq 0}{\frac {\|(1+|\cdot |^{2})^{\frac {s+\alpha }{2}}\cdot {\mathcal {F}}(Pf)\|_{L^{2}}}{\|(1+|\cdot |^{2})^{\frac {s}{2}}\cdot {\mathcal {F}}(f)\|_{L^{2}}}}}$

given.

literature

• Dirk Werner : functional analysis , Springer-Verlag, Berlin, 2007, ISBN 978-3-540-72533-6