Core (group)

In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space ${\mathcal {H}}$ and one-parameter families

$(U_{t})_{t\in \mathbb {R} }$

of unitary operators that are strongly continuous, i.e.,

$\forall t_{0}\in \mathbb {R} ,~\xi \in {\mathcal {H}}:\quad \lim _{t\rightarrow t_{0}}U_{t}\xi =U_{t_{0}}\xi$

and are homomorphisms, i.e.,

$\forall s,t\in \mathbb {R} :\quad U_{t+s}=U_{t}U_{s}.$

Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

The theorem was proved by Template:Harvs, and Template:Harvtxt showed that the requirement that $(U_{t})_{t\in \mathbb {R} }$ be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.

This is a very stunning theorem, as it allows to define the derivative of the mapping $t\mapsto U_{t}$, which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.

Formal statement

Let $(U_{t})_{t\in \mathbb {R} }$ be a strongly continuous one-parameter unitary group. Then there exists a unique (not necessarily bounded) self-adjoint operator $A$ such that

$\forall t\in \mathbb {R} :\quad U_{t}=e^{itA}.$

Conversely, let $A$ be a (not necessarily bounded) self-adjoint operator on a Hilbert space ${\mathcal {H}}$. Then the one-parameter family $(U_{t})_{t\in \mathbb {R} }$ of unitary operators defined by (using the Spectral Theorem for Self-Adjoint Operators)

$\forall t\in \mathbb {R} :\quad U_{t}:=e^{itA}$

is a strongly continuous one-parameter group.

The infinitesimal generator of $(U_{t})_{t\in \mathbb {R} }$ is defined to be the operator $iA$. This mapping is a bijective correspondence. Furthermore, $A$ will be a bounded operator if and only if the operator-valued mapping $t\mapsto U_{t}$ is norm-continuous.

Stone's Theorem can be recast using the language of the Fourier transform. The real line $\mathbb {R}$ is a locally compact abelian group. Non-degenerate $*$-representations of the group C*-algebra ${C^{*}}(\mathbb {R} )$ are in one-to-one correspondence with strongly continuous unitary representations of $\mathbb {R}$, i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a $*$-isomorphism from ${C^{*}}(\mathbb {R} )$ to ${C_{0}}(\mathbb {R} )$, the C*-algebra of complex-valued continuous functions on the real line that vanish at infinity. Hence, there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and $*$-representations of ${C_{0}}(\mathbb {R} )$. As every $*$-representation of ${C_{0}}(\mathbb {R} )$ corresponds uniquely to a self-adjoint operator, Stone's Theorem holds.

Therefore, the procedure for obtaining the infinitesimal generator of a strongly continuous one-parameter unitary group is as follows.

• Let $(U_{t})_{t\in \mathbb {R} }$ be a strongly continuous unitary representation of $\mathbb {R}$ on a Hilbert space ${\mathcal {H}}$.
• Integrate this unitary representation to yield a non-degenerate $*$-representation $\rho$ of ${C^{*}}(\mathbb {R} )$ on ${\mathcal {H}}$ by first defining $\displaystyle \rho (f):=\int _{\mathbb {R} }f(t)U_{t}\,\mathrm {d} {t}$ for all $f\in {C_{c}}(\mathbb {R} )$ and then extending $\rho$ to all of ${C^{*}}(\mathbb {R} )$ by continuity.
• Use the Fourier transform to obtain a non-degenerate $*$-representation $\tau$ of ${C_{0}}(\mathbb {R} )$ on ${\mathcal {H}}$.
• By the Riesz-Markov Theorem, $\tau$ gives rise to a projection-valued measure on $\mathbb {R}$ that is the resolution of the identity of a unique self-adjoint operator $A$, which may be unbounded.
• Then $iA$ is the infinitesimal generator of $(U_{t})_{t\in \mathbb {R} }$.

The precise definition of ${C^{*}}(\mathbb {R} )$ is as follows. Consider the $*$-algebra ${C_{c}}(\mathbb {R} )$, the complex-valued continuous functions on $\mathbb {R}$ with compact support, where the multiplication is given by convolution. The completion of this $*$ algebra with respect to the $L^{1}$-norm is a Banach $*$-algebra, denoted by $({L^{1}}(\mathbb {R} ),\star )$. Then ${C^{*}}(\mathbb {R} )$ is defined to be the enveloping C*-algebra of $({L^{1}}(\mathbb {R} ),\star )$, i.e., its completion with respect to the largest possible C*-norm. It is a non-trivial fact that, via the Fourier transform, ${C^{*}}(\mathbb {R} )$ is isomorphic to ${C_{0}}(\mathbb {R} )$. A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps ${L^{1}}(\mathbb {R} )$ to ${C_{0}}(\mathbb {R} )$.

Example

The family of translation operators

$[T_{t}\psi ](x)=\psi (x+t)\quad$

is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator

${\frac {d}{dx}}=i{\frac {1}{i}}{\frac {d}{dx}}$

defined on the space of complex-valued continuously differentiable functions of compact support on R. Thus

$T_{t}=e^{t\,{d}/{dx}}.\quad$

In other words, motion on the line is generated by the momentum operator.

Applications

Stone's theorem has numerous applications in quantum mechanics. For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group on H. The infinitesimal generator of this group is the system Hamiltonian.

Generalizations

The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self-adjoint operators, Q, P satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on L²(R).

The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.

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
• 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

• K. Yosida, Functional Analysis, Springer-Verlag, (1968)