View source for Operator norm

You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 44.200.0.0/16 has been blocked by ‪Blablubbs‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider or colocation provider. To prevent abuse, web hosts and colocation providers may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a colocation provider's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: {{unblock|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~}}. You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 16:45, 10 February 2025. Your current IP address is 44.200.52.12.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
This IP address range has been globally blocked.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

This block affects editing on all Wikimedia wikis.
The IP address or range 44.200.0.0/16 has been globally blocked by ‪AntiCompositeNumber‬ for the following reason(s):

Open proxy/Webhost: Visit the FAQ if you are affected

This block will expire on 13:55, 18 March 2025. Your current IP address is 44.200.52.12.

Even while globally blocked, you will usually still be able to edit pages on Meta-Wiki.

If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

Other useful links: Global blocks · Help:I have been blocked

You can view and copy the source of this page:

{{Short description|Measure of the "size" of linear operators}}
In [[mathematics]], the '''operator norm''' measures the "size" of certain [[linear operator]]s by assigning each a [[real number]] called its {{em|operator norm}}. Formally, it is a [[Norm (mathematics)|norm]] defined on the space of [[bounded linear operator]]s between two given [[normed vector space]]s. Informally, the operator norm <math>\|T\|</math> of a linear map <math>T : X \to Y</math> is the maximum factor by which it "lengthens" vectors.

== Introduction and definition ==

Given two normed vector spaces <math>V</math> and <math>W</math> (over the same base [[Field (mathematics)|field]], either the [[real number]]s <math>\R</math> or the [[complex number]]s <math>\Complex</math>), a [[linear map]] <math>A : V \to W</math> is continuous [[if and only if]] there exists a real number <math>c</math> such that<ref>{{Citation|last1=Kreyszig|first1=Erwin|title=Introductory functional analysis with applications|publisher=John Wiley & Sons|year=1978|isbn=9971-51-381-1|page=97}}</ref>
<math display="block">\|Av\| \leq c \|v\| \quad \text{ for all } v\in V.</math>

The norm on the left is the one in <math>W</math> and the norm on the right is the one in <math>V</math>. 
Intuitively, the continuous operator <math>A</math> never increases the length of any vector by more than a factor of <math>c.</math> Thus the [[Image (mathematics)|image]] of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as [[bounded operator]]s. 
In order to "measure the size" of <math>A,</math> one can take the [[infimum]] of the numbers <math>c</math> such that the above inequality holds for all <math>v \in V.</math> 
This number represents the maximum scalar factor by which <math>A</math> "lengthens" vectors. 
In other words, the "size" of <math>A</math> is measured by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of <math>A</math> as
<math display="block">\|A\|_\text{op} = \inf\{ c \geq 0 : \|Av\| \leq c \|v\| \text{ for all } v \in V \}.</math>

The infimum is attained as the set of all such <math>c</math> is [[Closed set|closed]], [[Empty set|nonempty]], and [[Bounded set|bounded]] from below.<ref>See e.g. Lemma 6.2 of {{harvtxt|Aliprantis|Border|2007}}.</ref>

It is important to bear in mind that this operator norm depends on the choice of norms for the normed vector spaces <math>V</math> and <math>W</math>.

== Examples ==

Every real <math>m</math>-by-<math>n</math> [[matrix (mathematics)|matrix]] corresponds to a linear map from <math>\R^n</math> to <math>\R^m.</math> Each pair of the plethora of (vector) [[norm (mathematics)|norms]] applicable to real vector spaces induces an operator norm for all <math>m</math>-by-<math>n</math> matrices of real numbers; these induced norms form a subset of [[matrix norm]]s.

If we specifically choose the [[Euclidean norm]] on both <math>\R^n</math> and <math>\R^m,</math> then the matrix norm given to a matrix <math>A</math> is the [[square root]] of the largest [[eigenvalue]] of the matrix <math>A^{*} A</math> (where <math>A^{*}</math> denotes the [[conjugate transpose]] of <math>A</math>).<ref>{{Cite web|url=https://mathworld.wolfram.com/OperatorNorm.html|title=Operator Norm|last=Weisstein|first=Eric W.|authorlink = Eric W. Weisstein|website=mathworld.wolfram.com|language=en|access-date=2020-03-14}}</ref> 
This is equivalent to assigning the largest [[singular value]] of <math>A.</math>

Passing to a typical infinite-dimensional example, consider the [[sequence space]] <math>\ell^2,</math> which is an [[Lp space|L<sup>''p''</sup> space]], defined by
<math display="block">\ell^2 = \left\{ (a_n)_{n \geq 1} : \; a_n \in \Complex, \; \sum_n |a_n|^2 < \infty \right\}.</math>

This can be viewed as an infinite-dimensional analogue of the [[Euclidean space]] <math>\Complex^n.</math> 
Now consider a bounded sequence <math>s_{\bull} = \left(s_n\right)_{n=1}^\infty.</math> The sequence <math>s_{\bull}</math> is an element of the space <math>\ell^\infty,</math> with a norm given by
<math display="block">\left\|s_{\bull}\right\|_\infty = \sup _n \left|s_n\right|.</math>

Define an operator <math>T_s</math> by pointwise multiplication:
<math display="block">\left(a_n\right)_{n=1}^{\infty} \;\stackrel{T_s}{\mapsto}\;\ \left(s_n \cdot a_n\right)_{n=1}^{\infty}.</math>

The operator <math>T_s</math> is bounded with operator norm
<math display="block">\left\|T_s\right\|_\text{op} = \left\|s_{\bull}\right\|_\infty.</math>

This discussion extends directly to the case where <math>\ell^2</math> is replaced by a general <math>L^p</math> space with <math>p > 1</math> and <math>\ell^\infty</math> replaced by <math>L^\infty.</math>

==Equivalent definitions==

Let <math>A : V \to W</math> be a linear operator between normed spaces. The first four definitions are always equivalent, and if in addition <math>V \neq \{0\}</math> then they are all equivalent:
:<math>
\begin{alignat}{4}
\|A\|_\text{op} &= \inf &&\{ c \geq 0 ~&&:~ \| A v \| \leq c \| v \| ~&&~ \text{ for all } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| \leq 1 ~&&~\mbox{ and } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| < 1 ~&&~\mbox{ and } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| \in \{0,1\}  ~&&~\mbox{ and } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| = 1 ~&&~\mbox{ and } ~&&v \in V \}  \;\;\;\text{ this equality holds if and only if } V \neq \{ 0 \} \\
&= \sup &&\bigg\{ \frac{\| Av \|}{\| v \|} ~&&:~ v \ne 0 ~&&~\mbox{ and } ~&&v \in V \bigg\}  \;\;\;\text{ this equality holds if and only if } V \neq \{ 0 \}. \\
\end{alignat}
</math>
If <math>V = \{0\}</math> then the sets in the last two rows will be empty, and consequently their [[supremum]]s over the set <math>[-\infty, \infty]</math> will equal <math>-\infty</math> instead of the correct value of <math>0.</math> If the supremum is taken over the set <math>[0, \infty]</math> instead, then the supremum of the empty set is <math>0</math> and the formulas hold for any <math>V.</math> 

Importantly, a linear operator <math>A : V \to W</math> is not, in general, guaranteed to achieve its norm <math>\|A\|_\text{op} = \sup \{\|A v\| : \|v\| \leq 1, v \in V\}</math> on the closed unit ball <math>\{v \in V : \|v\| \leq 1\},</math> meaning that there might not exist any vector <math>u \in V</math> of norm <math>\|u\| \leq 1</math> such that <math>\|A\|_\text{op} = \|A u\|</math> (if such a vector does exist and if <math>A \neq 0,</math> then <math>u</math> would necessarily have unit norm <math>\|u\| = 1</math>). R.C. James proved [[James's theorem]] in 1964, which states that a [[Banach space]] <math>V</math> is [[reflexive space|reflexive]] if and only if every [[bounded linear functional]] <math>f \in V^*</math> achieves its [[Dual norm|norm]] on the closed unit ball.{{sfn|Diestel|1984|p=6}} 
It follows, in particular, that every non-reflexive Banach space has some bounded linear functional (a type of bounded linear operator) that does not achieve its norm on the closed unit ball.

If <math>A : V \to W</math> is bounded then{{sfn|Rudin|1991|pp=92-115}} 
<math display="block">\|A\|_\text{op} = \sup \left\{\left|w^*(A v)\right| : \|v\| \leq 1, \left\|w^*\right\| \leq 1 \text{ where } v \in V, w^* \in W^*\right\}</math>
and{{sfn|Rudin|1991|pp=92-115}}  
<math display="block">\|A\|_\text{op} = \left\|{}^tA\right\|_\text{op}</math>
where <math>{}^t A : W^* \to V^*</math> is the [[Transpose of a linear map|transpose]] of <math>A : V \to W,</math> which is the linear operator defined by <math>w^* \,\mapsto\, w^* \circ A.</math>

== Properties ==

The operator norm is indeed a norm on the space of all [[bounded operator]]s between <math>V</math> and <math>W</math>. This means
<math display="block">\|A\|_\text{op} \geq 0 \mbox{ and } \|A\|_\text{op} = 0 \mbox{ if and only if } A = 0,</math>
<math display="block">\|aA\|_\text{op} = |a| \|A\|_\text{op} \mbox{ for every scalar } a ,</math>
<math display="block">\|A + B\|_\text{op} \leq \|A\|_\text{op} + \|B\|_\text{op}.</math>

The following inequality is an immediate consequence of the definition:
<math display="block">\|Av\| \leq \|A\|_\text{op} \|v\| \ \mbox{ for every }\ v \in V.</math>

The operator norm is also compatible with the composition, or multiplication, of operators: if <math>V</math>, <math>W</math> and <math>X</math> are three normed spaces over the same base field, and <math>A : V \to W</math> and <math>B : W \to X</math> are two bounded operators, then it is a [[sub-multiplicative norm]], that is: 
<math display="block">\|BA\|_\text{op} \leq \|B\|_\text{op} \|A\|_\text{op}.</math>

For bounded operators on <math>V</math>, this implies that operator multiplication is jointly continuous.

It follows from the definition that if a sequence of operators converges in operator norm, it [[converges uniformly]] on bounded sets.

== Table of common operator norms ==

By choosing different norms for the codomain, used in computing <math>\|Av\|</math>, and the domain, used in computing <math>\|v\|</math>, we obtain different values for the operator norm. Some common operator norms are easy to calculate, and others are [[NP-hard]]. 
Except for the NP-hard norms, all these norms can be calculated in <math>N^2</math> operations (for an <math>N \times N</math> matrix), with the exception of the <math>\ell_2 - \ell_2</math> norm (which requires <math>N^3</math> operations for the exact answer, or fewer if you approximate it with the [[Power iteration|power method]] or [[Lanczos algorithm|Lanczos iterations]]).

{| class="wikitable" style="text-align: center; width: 500px; height: 200px;"
|+ Computability of Operator Norms<ref>section 4.3.1, [[Joel Tropp]]'s PhD thesis, [http://users.cms.caltech.edu/~jtropp/papers/Tro04-Topics-Sparse.pdf]</ref>
|-
! scope="col" colspan = "2" rowspan = "2" |
! scope="col" colspan = "3" | Co-domain
|-
! scope="col" | <math>\ell_1</math>
! scope="col" | <math>\ell_2</math>
! scope="col" | <math>\ell_\infty</math>
|-
! scope = "row" rowspan="3" | Domain
! scope="row" | <math>\ell_1</math>
| Maximum <math>\ell_1</math> norm of a column || Maximum <math>\ell_2</math> norm of a column || Maximum <math>\ell_{\infty}</math> norm of a column
|-
! scope="row" | <math>\ell_2</math>
| NP-hard || Maximum singular value || Maximum <math>\ell_2</math> norm of a row
|-
! scope="row" | <math>\ell_\infty</math>
| NP-hard || NP-hard || Maximum <math>\ell_1</math> norm of a row
|}

The norm of the [[Conjugate transpose|adjoint]] or transpose can be computed as follows. 
We have that for any <math>p, q,</math> then <math>\|A\|_{p\rightarrow q} = \|A^*\|_{q'\rightarrow p'}</math> where <math>p', q'</math> are [[Hölder's inequality|Hölder conjugate]] to <math>p, q,</math> that is, <math>1/p + 1/p' = 1</math> and <math>1/q + 1/q' = 1.</math>

== Operators on a Hilbert space ==

Suppose <math>H</math> is a real or complex [[Hilbert space]]. If <math>A : H \to H</math> is a bounded linear operator, then we have
<math display="block">\|A\|_\text{op} = \left\|A^*\right\|_\text{op}</math>
and
<math display="block">\left\|A^* A\right\|_\text{op} = \|A\|_\text{op}^2,</math>
where <math>A^{*}</math> denotes the [[adjoint operator]] of <math>A</math> (which in [[Euclidean space]]s with the standard [[inner product]] corresponds to the [[conjugate transpose]] of the matrix <math>A</math>).

In general, the [[spectral radius]] of <math>A</math> is bounded above by the operator norm of <math>A</math>:
<math display="block">\rho(A) \leq \|A\|_\text{op}.</math>

To see why equality may not always hold, consider the [[Jordan canonical form]] of a matrix in the finite-dimensional case. Because there are non-zero entries on the superdiagonal, equality may be violated. The [[quasinilpotent operator]]s is one class of such examples. A nonzero quasinilpotent operator <math>A</math> has spectrum <math>\{0\}.</math> So <math>\rho(A) = 0</math> while <math>\|A\|_\text{op} > 0.</math>

However, when a matrix <math>N</math> is [[Normal matrix|normal]], its [[Jordan canonical form]] is diagonal (up to unitary equivalence); this is the [[spectral theorem]]. In that case it is easy to see that
<math display="block">\rho(N) = \|N\|_\text{op}.</math>

This formula can sometimes be used to compute the operator norm of a given bounded operator <math>A</math>: define the [[Hermitian operator]] <math>B = A^{*} A,</math> determine its spectral radius, and take the [[square root of a matrix|square root]] to obtain the operator norm of <math>A.</math>

The space of bounded operators on <math>H,</math> with the [[Topological space|topology]] induced by operator norm, is not [[Separable space|separable]]. 
For example, consider the [[Lp space]] <math>L^2[0, 1],</math> which is a Hilbert space. 
For <math>0 < t \leq 1,</math> let <math>\Omega_t</math> be the [[Indicator function|characteristic function]] of <math>[0, t],</math> and <math>P_t</math> be the [[multiplication operator]] given by <math>\Omega_t,</math> that is,
<math display="block">P_t (f) = f \cdot \Omega_t.</math>

Then each <math>P_t</math> is a bounded operator with operator norm 1 and
<math display="block">\left\|P_t - P_s\right\|_\text{op} = 1 \quad \mbox{ for all } \quad t \neq s.</math>

But <math>\{P_t : 0 < t \leq 1\}</math> is an [[uncountable set]]. 
This implies the space of bounded operators on <math>L^2([0, 1])</math> is not separable, in operator norm. 
One can compare this with the fact that the sequence space <math>\ell^{\infty}</math> is not separable.

The [[associative algebra]] of all bounded operators on a Hilbert space, together with the operator norm and the adjoint operation, yields a [[C*-algebra]].

==See also==

* {{annotated link|Banach–Mazur compactum}}
* {{annotated link|Continuous linear operator}}
* {{annotated link|Contraction (operator theory)}}
* {{annotated link|Discontinuous linear map}}
* {{annotated link|Dual norm}}
* {{annotated link|Matrix norm}}
* {{annotated link|Norm (mathematics)}}
* {{annotated link|Normed space}}
* {{annotated link|Operator algebra}}
* {{annotated link|Operator theory}}
* {{annotated link|Topologies on the set of operators on a Hilbert space}}
* {{annotated link|Unbounded operator}}

==Notes==

{{reflist}}

==References==

* {{citation|last1=Aliprantis|first1=Charalambos D.|last2=Border|first2=Kim C.|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|year=2007|publisher=Springer|isbn=9783540326960|page=229|url=https://books.google.com/books?id=4hIq6ExH7NoC&pg=PA229}}.
* {{Citation|last=Conway|first=John B.|authorlink=John B. Conway|year=1990|contribution=III.2 Linear Operators on Normed Spaces|title=A Course in Functional Analysis|pages=67–69|isbn=0-387-97245-5|publisher=Springer-Verlag|location=New York|url=https://books.google.com/books?id=ix4P1e6AkeIC&pg=PA67}}
* {{cite book|last=Diestel|first=Joe|title=Sequences and series in Banach spaces|publisher=Springer-Verlag|publication-place=New York|date=1984|isbn=0-387-90859-5|oclc=9556781}} <!--{{sfn|Diestel|1984|p=}}-->
* {{Rudin Walter Functional Analysis|edition=2}} <!--{{sfn|Rudin|1991|p=}}-->

{{Banach spaces}}
{{Hilbert space}}
{{Functional analysis}}
{{Duality and spaces of linear maps}}

[[Category:Functional analysis]]
[[Category:Norms (mathematics)]]
[[Category:Operator theory]]

Pages transcluded onto the current version of this page (help):

Banach–Mazur compactum (edit)
Continuous linear operator (edit)
Contraction (operator theory) (edit)
Discontinuous linear map (edit)
Dual norm (edit)
Matrix norm (edit)
Norm (mathematics) (edit)
Normed space (edit)
Normed vector space (edit)
Operator algebra (edit)
Operator theory (edit)
Operator topologies (edit)
Topologies on the set of operators on a Hilbert space (edit)
Unbounded operator (edit)
Template:Annotated link (view source) (template editor protected)
Template:Banach spaces (edit)
Template:Category link (view source) (template editor protected)
Template:Category link/core (view source) (template editor protected)
Template:Citation (view source) (protected)
Template:Cite book (view source) (protected)
Template:Cite web (view source) (protected)
Template:Cl (view source) (template editor protected)
Template:Duality and spaces of linear maps (edit)
Template:Em (view source) (template editor protected)
Template:Functional analysis (view source) (semi-protected)
Template:Harvard citation text (view source) (template editor protected)
Template:Harvtxt (view source) (template editor protected)
Template:Hilbert space (edit)
Template:Hlist/styles.css (view source) (protected)
Template:Icon (view source) (template editor protected)
Template:Main other (view source) (protected)
Template:Navbox (view source) (template editor protected)
Template:Pagetype (view source) (protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Template:Rudin Walter Functional Analysis (edit)
Template:SDcat (view source) (protected)
Template:Sfn (view source) (template editor protected)
Template:Short description (view source) (protected)
Template:Short description/lowercasecheck (view source) (protected)
Template:Small (view source) (protected)
Template:Smaller (view source) (template editor protected)
Template:Yesno (view source) (protected)
Template:\ (view source) (template editor protected)
Module:Annotated link (view source) (template editor protected)
Module:Arguments (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:Disambiguation/templates (view source) (protected)
Module:Footnotes (view source) (template editor protected)
Module:Footnotes/anchor id list (view source) (template editor protected)
Module:Footnotes/anchor id list/data (view source) (template editor protected)
Module:Footnotes/whitelist (view source) (template editor protected)
Module:GetShortDescription (view source) (template editor protected)
Module:Icon (view source) (template editor protected)
Module:Icon/data (view source) (template editor protected)
Module:Lang (view source) (template editor protected)
Module:Lang/ISO 639 synonyms (view source) (template editor protected)
Module:Lang/data (view source) (template editor protected)
Module:Lang/data/iana languages (view source) (template editor protected)
Module:Lang/data/iana regions (view source) (template editor protected)
Module:Lang/data/iana scripts (view source) (template editor protected)
Module:Lang/data/iana suppressed scripts (view source) (template editor protected)
Module:Lang/data/iana variants (view source) (template editor protected)
Module:Navbar (view source) (protected)
Module:Navbar/configuration (view source) (protected)
Module:Navbar/styles.css (view source) (protected)
Module:Navbox (view source) (template editor protected)
Module:Navbox/configuration (view source) (template editor protected)
Module:Navbox/styles.css (view source) (template editor protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Pagetype/disambiguation (view source) (protected)
Module:Pagetype/rfd (view source) (protected)
Module:Pagetype/setindex (view source) (protected)
Module:Pagetype/softredirect (view source) (protected)
Module:SDcat (view source) (protected)
Module:String (view source) (protected)
Module:TableTools (view source) (protected)
Module:Unicode data (view source) (template editor protected)
Module:Wikitext Parsing (view source) (protected)
Module:Yesno (view source) (protected)

Return to Operator norm.

Retrieved from "https://en.wikipedia.org/wiki/Operator_norm"