Fine topology (potential theory): Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Ultra snozbarg
 
No edit summary
 
Line 1: Line 1:
Greetings. The author's name is Phebe and she feels comfortable when people use the complete name. One of over the counter std test ([http://www.buzzbit.net/blog/333162 Recommended Resource site]) things she loves most is to study comics and she'll be starting something else alongside with it. North Dakota is our birth place. For many years he's been operating as a receptionist.
In [[operator theory]], '''[[Mark Naimark|Naimark]]'s dilation theorem''' is a result that characterizes [[POVM|positive operator valued measures]]. It can be viewed as a consequence of [[Stinespring factorization theorem|Stinespring's dilation theorem]].
 
== Note ==
 
 
In the mathematical literature, one may also find other results that bear Naimark's name.
 
== Some preliminary notions ==
 
Let ''X'' be a [[Compact space|compact]] [[Hausdorff space]], ''H'' be a [[Hilbert space]], and ''L(H)'' the [[Banach space]] of [[bounded operator]]s on ''H''. A mapping ''E'' from the [[Borel σ-algebra]] on ''X'' to <math>L(H)</math> is called a '''operator-valued measure''' if it is weakly countably additive, that is, for any disjoint sequence of Borel sets <math>\{ B_i \}</math>, we have
 
:<math>
\langle E (\cup _i B_i) x, y \rangle = \sum_i \langle E (B_i) x, y \rangle
</math>
 
for all ''x'' and ''y''. Some terminology for describing such measures are:
 
* ''E'' is called ''regular'' if the scalar valued measure
 
:<math>
B \rightarrow \langle E (B) x, y \rangle
</math>
 
is a regular Borel measure, meaning all compact sets have finite total variation and the measure of a set can be approximated by those of open sets.
 
* ''E'' is called ''bounded'' if <math>|E| = \sup_B \|E(B) \| < \infty</math>.
 
* ''E'' is called ''positive'' if ''E(B)'' is a positive operator for all ''B''. 
 
* ''E'' is called ''self-adjoint '' if ''E(B)'' is self-adjoint for all ''B''.
 
* ''E'' is called ''spectral'' if <math>E (B_1 \cap B_2) = E(B_1) E(B_2)</math>.
 
We will assume throughout that ''E'' is regular.
 
Let ''C(X)'' denote the abelian C*-algebra of continuous functions on ''X''. If ''E'' is regular and bounded, it induces a map <math>\Phi _E : C(X) \rightarrow L(H)</math> in the obvious way:
 
:<math>\langle \Phi _E (f) h_1 , h_2 \rangle = \int _X f d \langle E(B) h_1, h_2 \rangle</math>
 
The boundedness of ''E'' implies, for all ''h'' of unit norm
 
:<math>
\langle \Phi _E (f) h , h \rangle = \int _X f d \langle E(B) h, h \rangle \leq \| f \| \cdot |E| .
</math>
 
This shows <math>\; \Phi _E (f)</math> is a bounded operator for all ''f'', and <math>\Phi _E</math> itself is a bounded linear map as well.
 
The properties of <math>\Phi_E</math> are directly related to those of ''E'':
 
* If ''E'' is positive, then <math>\Phi_E</math>, viewed as a map between C*-algebras, is also positive.
 
* <math>\Phi_E</math> is a homomorphism if, by definition, for all continuous ''f'' on ''X'' and <math>h_1, h_2 \in H</math>,
 
:<math>
\langle \Phi_E (fg) h_1, h_2 \rangle = \int _X f \cdot g \; d \langle E(B) h_1, h_2 \rangle
= \langle \Phi_E (f) \Phi_E (g) h_1 , h_2 \rangle.
</math>
 
Take ''f'' and ''g'' to be indicator functions of Borel sets and we see that <math>\Phi _E</math> is a homomorphism if and only if ''E'' is spectral.
 
* Similarly, to say <math>\Phi_E</math> respects the * operation means
 
:<math>
\langle \Phi_E ( {\bar f} ) h_1, h_2 \rangle = \langle \Phi_E (f) ^* h_1 , h_2 \rangle.
</math>
 
The LHS is
 
:<math>
\int _X {\bar f} \; d \langle E(B) h_1, h_2 \rangle,
</math>
 
and the RHS is
 
:<math>
\langle h_1, \Phi_E (f) h_2 \rangle = \int _X {\bar f} \; d \langle E(B) h_2, h_1 \rangle
</math>
 
So, for all ''B'', <math>\langle E(B) h_1, h_2 \rangle = \langle E(B) h_2, h_1 \rangle</math>, i.e. ''E(B)'' is self adjoint.
 
* Combining the previous two facts gives the conclusion that <math>\Phi _E</math> is a *-homomorphism if and only if ''E'' is spectral and self adjoint. (When ''E'' is spectral and self adjoint, ''E'' is said to be a [[projection-valued measure]] or PVM.)
 
== Naimark's theorem ==
 
The theorem reads as follows: Let ''E'' be a positive ''L(H)''-valued measure on ''X''. There exists a Hilbert space ''K'', a bounded operator <math>V: K \rightarrow H</math>, and a self-adjoint, spectral ''L(K)''-valued measure on ''X'', ''F'', such that
 
:<math>\; E(B) = V F(B) V^*.</math>
 
=== Proof ===
 
We now sketch the proof. The argument passes ''E'' to the induced map <math>\Phi_E</math> and uses [[Stinespring factorization theorem|Stinespring's dilation theorem]]. Since ''E'' is positive, so is <math>\Phi_E</math> as a map between C*-algebras, as explained above. Furthermore, because the domain of <math>\Phi _E</math>, ''C(X)'', is an abelian C*-algebra, we have that <math>\Phi_E</math> is [[Choi's theorem on completely positive maps|completely positive]]. By Stinespring's result, there exists a Hilbert space ''K'', a *-homomorphism <math>\pi : C(X) \rightarrow L(K)</math>, and operator <math>V: K \rightarrow H</math> such that
 
:<math>\; \Phi_E(f) = V \pi (f) V^*.</math>
 
Since π is a *-homomorphism, its corresponding operator-valued measure ''F'' is spectral and self adjoint. It is easily seen that ''F'' has the desired properties.
 
== Finite dimensional case ==
 
In the finite dimensional case, there is a somewhat more explicit formulation.
 
Suppose now <math>X = \{1, \cdots, n \}</math>, therefore ''C(X)'' is the finite dimensional algebra <math>\mathbb{C}^n</math>, and ''H'' has finite dimension ''m''. A positive operator-valued measure ''E'' then assigns each ''i'' a positive semidefinite ''m X m'' matrix <math>E_i</math>. Naimark's theorem now says there
is a projection valued measure on ''X'' whose restriction is ''E''.
 
Of particular interest is the special case when <math>\; \sum _i E_i = I</math> where ''I'' is the identity operator. (See the article on [[POVM]] for relevant applications.) This would mean the induced map <math>\Phi _E</math> is unital. It can be assumed with no loss of generality that each <math>E_i</math> is a rank-one projection onto some <math>x_i \in \mathbb{C}^m</math>. Under such assumptions, the case <math>n < m</math> is excluded and we must have either:
 
1) <math>n = m</math> and ''E'' is already a projection valued measure. (Because <math>\sum _{i=1}^n x_i x_i^* = I </math> if and only if <math>\{ x_i\}</math> is an orthonormal basis.)
,or
 
2) <math>n > m</math> and <math>\{ E_i \}</math> does not consist of mutually orthogonal projections.
 
For the second possibility, the problem of finding a suitable PVM now becomes the following: By assumption, the non-square matrix
 
:<math> M = \begin{bmatrix} x_1 & \cdots x_n \end{bmatrix}</math>
 
is an isometry, i.e. <math>M M^* = I</math>. If we can find a <math>(n-m) \times n</math> matrix ''N'' where
 
:<math>U = \begin{bmatrix} M \\ N \end{bmatrix} </math>
 
is a ''n X n'' unitary matrix, the PVM whose elements are projections onto the column vectors of ''U'' will then have the desired properties. In principle, such a ''N'' can always be found.
 
==References==
 
*V. Paulsen, ''Completely Bounded Maps and Operator Algebras'', Cambridge University Press, 2003.
 
[[Category:Operator theory]]
[[Category:Measure theory]]
[[Category:Theorems in functional analysis]]

Latest revision as of 21:49, 5 January 2014

In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring's dilation theorem.

Note

In the mathematical literature, one may also find other results that bear Naimark's name.

Some preliminary notions

Let X be a compact Hausdorff space, H be a Hilbert space, and L(H) the Banach space of bounded operators on H. A mapping E from the Borel σ-algebra on X to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(H)} is called a operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ B_i \}} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle E (\cup _i B_i) x, y \rangle = \sum_i \langle E (B_i) x, y \rangle }

for all x and y. Some terminology for describing such measures are:

  • E is called regular if the scalar valued measure
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \rightarrow \langle E (B) x, y \rangle }

is a regular Borel measure, meaning all compact sets have finite total variation and the measure of a set can be approximated by those of open sets.

  • E is called bounded if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |E| = \sup_B \|E(B) \| < \infty} .
  • E is called positive if E(B) is a positive operator for all B.
  • E is called self-adjoint if E(B) is self-adjoint for all B.
  • E is called spectral if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E (B_1 \cap B_2) = E(B_1) E(B_2)} .

We will assume throughout that E is regular.

Let C(X) denote the abelian C*-algebra of continuous functions on X. If E is regular and bounded, it induces a map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi _E : C(X) \rightarrow L(H)} in the obvious way:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Phi _E (f) h_1 , h_2 \rangle = \int _X f d \langle E(B) h_1, h_2 \rangle}

The boundedness of E implies, for all h of unit norm

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Phi _E (f) h , h \rangle = \int _X f d \langle E(B) h, h \rangle \leq \| f \| \cdot |E| . }

This shows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \; \Phi _E (f)} is a bounded operator for all f, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi _E} itself is a bounded linear map as well.

The properties of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_E} are directly related to those of E:

  • If E is positive, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_E} , viewed as a map between C*-algebras, is also positive.
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_E} is a homomorphism if, by definition, for all continuous f on X and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_1, h_2 \in H} ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Phi_E (fg) h_1, h_2 \rangle = \int _X f \cdot g \; d \langle E(B) h_1, h_2 \rangle = \langle \Phi_E (f) \Phi_E (g) h_1 , h_2 \rangle. }

Take f and g to be indicator functions of Borel sets and we see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi _E} is a homomorphism if and only if E is spectral.

  • Similarly, to say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_E} respects the * operation means
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Phi_E ( {\bar f} ) h_1, h_2 \rangle = \langle \Phi_E (f) ^* h_1 , h_2 \rangle. }

The LHS is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int _X {\bar f} \; d \langle E(B) h_1, h_2 \rangle, }

and the RHS is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle h_1, \Phi_E (f) h_2 \rangle = \int _X {\bar f} \; d \langle E(B) h_2, h_1 \rangle }

So, for all B, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle E(B) h_1, h_2 \rangle = \langle E(B) h_2, h_1 \rangle} , i.e. E(B) is self adjoint.

  • Combining the previous two facts gives the conclusion that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi _E} is a *-homomorphism if and only if E is spectral and self adjoint. (When E is spectral and self adjoint, E is said to be a projection-valued measure or PVM.)

Naimark's theorem

The theorem reads as follows: Let E be a positive L(H)-valued measure on X. There exists a Hilbert space K, a bounded operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V: K \rightarrow H} , and a self-adjoint, spectral L(K)-valued measure on X, F, such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \; E(B) = V F(B) V^*.}

Proof

We now sketch the proof. The argument passes E to the induced map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_E} and uses Stinespring's dilation theorem. Since E is positive, so is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_E} as a map between C*-algebras, as explained above. Furthermore, because the domain of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi _E} , C(X), is an abelian C*-algebra, we have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_E} is completely positive. By Stinespring's result, there exists a Hilbert space K, a *-homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi : C(X) \rightarrow L(K)} , and operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V: K \rightarrow H} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \; \Phi_E(f) = V \pi (f) V^*.}

Since π is a *-homomorphism, its corresponding operator-valued measure F is spectral and self adjoint. It is easily seen that F has the desired properties.

Finite dimensional case

In the finite dimensional case, there is a somewhat more explicit formulation.

Suppose now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \{1, \cdots, n \}} , therefore C(X) is the finite dimensional algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}^n} , and H has finite dimension m. A positive operator-valued measure E then assigns each i a positive semidefinite m X m matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_i} . Naimark's theorem now says there is a projection valued measure on X whose restriction is E.

Of particular interest is the special case when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \; \sum _i E_i = I} where I is the identity operator. (See the article on POVM for relevant applications.) This would mean the induced map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi _E} is unital. It can be assumed with no loss of generality that each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_i} is a rank-one projection onto some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i \in \mathbb{C}^m} . Under such assumptions, the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n < m} is excluded and we must have either:

1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = m} and E is already a projection valued measure. (Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum _{i=1}^n x_i x_i^* = I } if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ x_i\}} is an orthonormal basis.) ,or

2) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ E_i \}} does not consist of mutually orthogonal projections.

For the second possibility, the problem of finding a suitable PVM now becomes the following: By assumption, the non-square matrix

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = \begin{bmatrix} x_1 & \cdots x_n \end{bmatrix}}

is an isometry, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M M^* = I} . If we can find a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n-m) \times n} matrix N where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \begin{bmatrix} M \\ N \end{bmatrix} }

is a n X n unitary matrix, the PVM whose elements are projections onto the column vectors of U will then have the desired properties. In principle, such a N can always be found.

References

  • V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Fine_topology_(potential_theory)&oldid=13870"