Generalized permutation matrix

In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.

Introduction and definition

Given two normed vector spaces V and W (over the same base field, either the real numbers R or the complex numbers C), a linear map A : V → W is continuous if and only if there exists a real number c such that

$\|Av\|\leq c\|v\|\quad {\mbox{ for all }}v\in V$

(the norm on the left is the one in W, the norm on the right is the one in V). Intuitively, the continuous operator A never "lengthens" any vector more than by a factor of c. Thus the 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 operators. In order to "measure the size" of A, it then seems natural to take the smallest number c such that the above inequality holds for all v in V. In other words, we measure the "size" of A by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of A as

$\|A\|_{op}=\min\{c\geq 0:\|Av\|\leq c\|v\|{\mbox{ for all }}v\in V\}$

(the minimum exists as the set of all such c is closed, nonempty, and bounded from below).

Examples

Every real m-by-n matrix yields a linear map from Rⁿ to R^m. One can put several different norms on these spaces, as explained in the article on norms. Each such choice of norms gives rise to an operator norm and therefore yields a norm on the space of all m-by-n matrices. Examples can be found in the article on matrix norms.

If we specifically choose the Euclidean norm on both Rⁿ and R^m, then we obtain the matrix norm which to a given matrix A assigns the square root of the largest eigenvalue of the matrix A^*A (where A^* denotes the conjugate transpose of A). This is equivalent to assigning the largest singular value of A.

Passing to a typical infinite dimensional example, consider the sequence space $l^{2}$ defined by

$l^{2}=\{(a_{n})_{n\geq 1}:\;a_{n}\in \mathbb {C} ,\;\sum _{n}|a_{n}|^{2}<\infty \}.$

This can be viewed as an infinite dimensional analogue of the Euclidean space Cⁿ. Now take a bounded sequence s = (s_n ). The sequence s is an element of the space l ^∞, with a norm given by

$\|s\|_{\infty }=\sup _{n}|s_{n}|.$

Define an operator T_s by simply multiplication:

$(a_{n}){\stackrel {T_{s}}{\longrightarrow }}(s_{n}\cdot a_{n}).$

The operator T _s is bounded with operator norm

$\|T_{s}\|_{op}=\|s\|_{\infty }.$

One can extend this discussion directly to the case where l ² is replaced by a general L^p space with p > 1 and l^∞ replaced by L^∞.

Equivalent definitions

One can show that the following definitions are all equivalent:

${\begin{aligned}\|A\|_{op}&=\inf\{c\geq 0:\|Av\|\leq c\|v\|{\mbox{ for all }}v\in V\}\\&=\sup\{\|Av\|:v\in V{\mbox{ with }}\|v\|\leq 1\}\\&=\sup\{\|Av\|:v\in V{\mbox{ with }}\|v\|<1\}\\&=\sup\{\|Av\|:v\in V{\mbox{ with }}\|v\|=1\}\\&=\sup \left\{{\frac {\|Av\|}{\|v\|}}:v\in V{\mbox{ with }}v\neq 0\right\}.\end{aligned}}$

Properties

The operator norm is indeed a norm on the space of all bounded operators between V and W. This means

$\|A\|_{op}\geq 0{\mbox{ and }}\|A\|_{op}=0{\mbox{ if and only if }}A=0,$

$\|aA\|_{op}=|a|\|A\|_{op}\quad {\mbox{ for every scalar }}a,$

$\|A+B\|_{op}\leq \|A\|_{op}+\|B\|_{op}.$

The following inequality is an immediate consequence of the definition:

$\|Av\|\leq \|A\|_{op}\|v\|\quad {\mbox{ for every }}v\in V.$

The operator norm is also compatible with the composition, or multiplication, of operators: if V, W and X are three normed spaces over the same base field, and A : V → W and B: W → X are two bounded operators, then

$\|BA\|_{op}\leq \|B\|_{op}\|A\|_{op}.$

For bounded operators on V, this implies that operator multiplication is jointly continuous.

It follows from the definition that a sequence of operators converge in operator norm means they converge uniformly on bounded sets.

Operators on a Hilbert space

Suppose H is a real or complex Hilbert space. If A : H → H is a bounded linear operator, then we have

$\|A\|_{op}=\|A^{*}\|_{op}$

and

$\|A^{*}A\|_{op}=\|A\|_{op}^{2}$

where A^* denotes the adjoint operator of A (which in Euclidean Hilbert spaces with the standard inner product corresponds to the conjugate transpose of the matrix A).

In general, the spectral radius of A is bounded above by the operator norm of A:

$\rho (A)\leq \|A\|_{op}.$

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 operators is one class of such examples. A nonzero quasinilpotent operator A has spectrum {0}. So ρ(A) = 0 while ||A||_op > 0.

However, when a matrix N is 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

$\rho (N)=\|N\|_{op}.$

The spectral theorem can be extended to normal operators in general. Therefore the above equality holds for any bounded normal operator N. This formula can sometimes be used to compute the operator norm of a given bounded operator A: define the Hermitian operator B = A^*A, determine its spectral radius, and take the square root to obtain the operator norm of A.

The space of bounded operators on H, with the topology induced by operator norm, is not separable. For example, consider the Hilbert space L²[0,1]. For 0 < t ≤ 1, let Ω_t be the characteristic function of [0,t], and P_t be the multiplication operator given by Ω_t, i.e.

$P_{t}(f)=f\cdot \Omega _{t}.$

Then each P_t is a bounded operator with operator norm 1 and

$\|P_{t}-P_{s}\|_{op}=1,\quad {\mbox{for all}}\quad t\neq s.$

But {P_t} is an uncountable set. This implies the space of bounded operators on L²[0,1] is not separable, in operator norm. One can compare this with the fact that the sequence space l ^∞ is not separable.

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

See also

• operator algebra
• topologies on the set of operators on a Hilbert space
• matrix norm

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010