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In mathematics, in the field of functional analysis, the Cotlar–Stein almost orthogonality lemma is named after mathematicians Mischa Cotlar and Elias Stein. It may be used to obtain information on the operator norm on an operator, acting from one Hilbert space into another when the operator can be decomposed into almost orthogonal pieces. The original version of this lemma (for self-adjoint and mutually commuting operators) was proved by Mischa Cotlar in 1955 and allowed him to conclude that the Hilbert transform is a continuous linear operator in $L^2$ without using the Fourier transform. A more general version was proved by Elias Stein.

Cotlar–Stein almost orthogonality lemma

Let $E,F$ be two Hilbert spaces. Consider a family of operators $T_j$, $j\ge 1$, with each $T_j$ a bounded linear operator from $E$ to $F$.

Denote

$a_{jk}=\Vert T_j{T}_k^{\ast }\Vert ,b_{jk}=\Vert T_j^{\ast }{T}_k\Vert .$

The family of operators $T_j:E\to F$, $j\ge 1,$ is almost orthogonal if

$A={\sup }_j\sum _k\sqrt{a_{jk}}<\mathrm{\infty },B={\sup }_j\sum _k\sqrt{b_{jk}}<\mathrm{\infty }.$

The Cotlar–Stein lemma states that if $T_j$ are almost orthogonal, then the series $\sum _j{T}_j$ converges in the strong operator topology, and that

$\Vert \sum _j{T}_j\Vert \le \sqrt{AB}.$

Proof

If R₁, ..., R_n is a finite collection of bounded operators, then

$\sum _{i,j}|(R_{i}v,R_{j}v)|\le ({\max }_i\sum _j\Vert R_i^{*}{R}_j{\Vert }^{\frac{1}{2}})({\max }_i\sum _j\Vert R_i{R}_j^{*}{\Vert }^{\frac{1}{2}})\Vert v{\Vert }^2.$

So under the hypotheses of the lemma,

$\sum _{i,j}|(T_{i}v,T_{j}v)|\le AB\Vert v{\Vert }^2.$

It follows that

$\Vert \sum _{i=1}^n{T}_{i}v{\Vert }^2\le AB\Vert v{\Vert }^2,$

and that

$\Vert \sum _{i=m}^n{T}_{i}v{\Vert }^2\le \sum _{i,j\ge m}|(T_{i}v,T_{j}v)|.$

Hence the partial sums

$s_n=\sum _{i=1}^n{T}_{i}v$

form a Cauchy sequence.

The sum is therefore absolutely convergent with limit satisfying the stated inequality.

To prove the inequality above set

$R=\sum a_{ij}{R}_i^{*}{R}_j$

with |a_ij| ≤ 1 chosen so that

$(Rv,v)=|(Rv,v)|=\sum |(R_{i}v,R_{j}v)|.$

Then

$\Vert R{\Vert }^{2m}=\Vert (R^{*}R{)}^m\Vert \le \sum \Vert R_{i_1}^{*}{R}_{i_2}{R}_{i_3}^{*}{R}_{i_4}\cdots R_{i_{2m}}\Vert \le \sum {(\Vert R_{i_1}^{*}\Vert \Vert R_{i_1}^{*}{R}_{i_2}\Vert \Vert R_{i_2}{R}_{i_3}^{*}\Vert \cdots \Vert R_{i_{2m-1}}^{*}{R}_{i_{2m}}\Vert \Vert R_{i_{2m}}\Vert )}^{\frac{1}{2}}.$

Hence

$\Vert R{\Vert }^{2m}\le n\cdot \max \Vert R_i\Vert {({\max }_i\sum _j\Vert R_i^{*}{R}_j{\Vert }^{\frac{1}{2}})}^{2m}{({\max }_i\sum _j\Vert R_i{R}_j^{*}{\Vert }^{\frac{1}{2}})}^{2m}.$

Taking 2mth roots and letting m tend to ≈,

$\Vert R\Vert \le ({\max }_i\sum _j\Vert R_i^{*}{R}_j{\Vert }^{\frac{1}{2}})({\max }_i\sum _j\Vert R_i{R}_j^{*}{\Vert }^{\frac{1}{2}}),$

which immediately implies the inequality.

Generalization

There is a generalization of the Cotlar–Stein lemma with sums replaced by integrals. Let X be a locally compact space and μ a Borel measure on X. Let T(x) be a map from X into bounded operators from E to F which is uniformly bounded and continuous in the strong operator topology. If

$A={\sup }_x\underset{X}{\int }\Vert T(x{)}^{*}T(y){\Vert }^{\frac{1}{2}}d\mu (y),B={\sup }_x\underset{X}{\int }\Vert T(y)T(x{)}^{*}{\Vert }^{\frac{1}{2}}d\mu (y),$

are finite, then the function T(x)v is integrable for each v in E with

$\Vert \underset{X}{\int }T(x)vd\mu (x)\Vert \le \sqrt{AB}\cdot \Vert v\Vert .$

The result can be proved by replacing sums by integrals in the previous proof or by using Riemann sums to approximate the integrals.

Example

Here is an example of an orthogonal family of operators. Consider the inifite-dimensional matrices

$T=\left[\begin{array}{cccc}1& 0& 0& ⋮\\ 0& 1& 0& ⋮\\ 0& 0& 1& ⋮\\ \cdots & \cdots & \cdots & \ddots \end{array}\right]$

and also

$T_1=\left[\begin{array}{cccc}1& 0& 0& ⋮\\ 0& 0& 0& ⋮\\ 0& 0& 0& ⋮\\ \cdots & \cdots & \cdots & \ddots \end{array}\right],T_2=\left[\begin{array}{cccc}0& 0& 0& ⋮\\ 0& 1& 0& ⋮\\ 0& 0& 0& ⋮\\ \cdots & \cdots & \cdots & \ddots \end{array}\right],T_3=\left[\begin{array}{cccc}0& 0& 0& ⋮\\ 0& 0& 0& ⋮\\ 0& 0& 1& ⋮\\ \cdots & \cdots & \cdots & \ddots \end{array}\right],\dots .$

Then $\Vert T_j\Vert =1$ for each $j$, hence the series $\sum _{j\in \mathbb{\mathbb{N}}}{T}_j$ does not converge in the uniform operator topology.

Yet, since $\Vert T_j{T}_k^{\ast }\Vert =0$ and $\Vert T_j^{\ast }{T}_k\Vert =0$ for $j\ne k$, the Cotlar–Stein almost orthogonality lemma tells us that

$T=\sum _{j\in \mathbb{\mathbb{N}}}{T}_j$

converges in the strong operator topology and is bounded by 1.

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.
  
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  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
• 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