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In [[mathematics]], particularly in [[functional analysis]], a  '''projection-valued measure (PVM)''' is a function defined on certain subsets of a fixed set and whose values are [[self-adjoint]] [[projection (mathematics)|projection]]s on a [[Hilbert space]]. Projection-valued measures are used to express results in [[spectral theory]], such as the spectral theorem for [[self-adjoint operator]]s. In [[quantum mechanics]], PVMs are the mathematical description of [[Quantum measurement|projective measurements]].  They are generalized by [[POVM|positive operator valued measures]] (POVMs) in the same sense that a [[mixed state (physics)|mixed state]] or [[density matrix]] generalizes the notion of a [[pure state]].
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== Formal definition ==
 
A projection-valued measure on a [[measurable space]]
(''X'', ''M''), where ''M'' is a [[σ-algebra]] of subsets of ''X'', is a [[function (mathematics)|mapping]] π from ''M'' to the set of [[self-adjoint]] [[projection operator|projections]] on a [[Hilbert space]] ''H'' such that
 
: <math>\pi(X) = \operatorname{id}_H \quad </math>
 
and for every ξ, η ∈ ''H'', the set-function
 
:<math>A \mapsto \langle \pi(A)\xi \mid \eta \rangle </math>
 
is a [[complex measure]] on ''M'' (that is, a complex-valued [[sigma additivity|countably additive]] function). We denote this measure by <math>\operatorname{S}_\pi(\xi, \eta)</math>.
 
If π is a projection-valued measure and
 
: <math>A \cap B = \emptyset,</math>
 
then π(''A''), π(''B'') are [[orthogonal projection]]s.  From this follows that in general,
 
: <math> \pi(A) \pi(B) = \pi(A \cap B). </math>
 
'''Example'''.  Suppose (''X'', ''M'', μ) is a measure space. Let π(''A'') be the operator of multiplication by the [[indicator function]] 1<sub>''A''</sub> on [[Lp space|''L''<sup>2</sup>(''X'')]]. Then π is a projection-valued measure.
 
== Extensions of projection-valued measures==
 
If π is an additive projection-valued measure on (''X'', ''M''), then the map
 
: <math> \mathbf{1}_A \mapsto \pi(A)</math>
 
extends to a linear map on the vector space of [[step function]]s on ''X''.  In fact, it is easy to check that this map is a [[ring homomorphism]].  In fact this map extends in a canonical way to all bounded complex-valued Borel functions on ''X''.
 
'''Theorem'''. For any bounded ''M''-measurable function ''f'' on ''X'', there is a unique bounded linear operator T<sub>π</sub>(''f'') such that
 
: <math> \langle \operatorname{T}_\pi(f) \xi \mid  \eta \rangle = \int_X f(x) d \operatorname{S}_\pi (\xi,\eta)(x) </math>
 
for all ξ, η ∈ ''H''.  The map
 
: <math>f \mapsto \operatorname{T}_\pi(f)</math>
 
is a homomorphism of rings.
 
== Structure of projection-valued measures==
 
First we provide a general example of projection-valued measure based on [[direct integral]]s.  Suppose (''X'', ''M'', μ) is a measure space and let {''H''<sub>''x''</sub>}<sub>''x'' ∈ ''X'' </sub> be a μ-measurable family of separable Hilbert spaces. For every  ''A'' ∈ ''M'', let π(''A'') be the operator of multiplication by 1<sub>''A''</sub> on the Hilbert space
 
:<math> \int_X^\oplus H_x \ d \mu(x). </math>
 
Then π is a projection-valued measure on (''X'', ''M'').
 
Suppose π, ρ  are projection-valued measures on  (''X'', ''M'') with values in the projections of ''H'', ''K''.   π,  ρ  are '''unitarily equivalent''' [[if and only if]] there is a unitary operator ''U'':''H'' → ''K'' such that
 
:<math> \pi(A) = U^* \rho(A) U \quad </math>
 
for every ''A'' ∈ ''M''.
 
'''Theorem'''. If (''X'', ''M'') is a [[Borel algebra#Standard Borel spaces and Kuratowski theorems|standard Borel space]], then for every projection-valued measure π on (''X'', ''M'') taking values in the projections of a ''separable'' Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces {''H''<sub>''x''</sub>}<sub>''x'' ∈ ''X'' </sub>, such that π is unitarily equivalent to multiplication by 1<sub>''A''</sub> on the Hilbert space
 
:<math> \int_X^\oplus H_x \ d \mu(x). </math>
 
The measure class of μ and the measure equivalence class of the multiplicity function ''x'' → dim ''H''<sub>''x''</sub> completely characterize the projection-valued measure up to unitary equivalence.
 
A projection-valued measure  π is ''homogeneous of multiplicity'' ''n'' if and only if the multiplicity function has constant value ''n''.  Clearly,
 
'''Theorem'''.  Any projection-valued measure π taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:
 
:<math> \pi = \bigoplus_{1 \leq n \leq \omega} (\pi | H_n) </math>
 
where
 
:<math> H_n = \int_{X_n}^\oplus H_x \ d (\mu | X_n) (x) </math>
 
and
 
:<math> X_n = \{x \in X: \operatorname{dim} H_x = n\}. </math>
 
==Generalizations==
The idea of a projection-valued measure is generalized by the [[positive operator-valued measure]] (POVM), where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal [[partition of unity]]. This generalization is motivated by applications to [[quantum information theory]].
 
== References ==
 
* G. W. Mackey, ''The Theory of Unitary Group Representations'', The University of Chicago Press, 1976
* [[Michael_C._Reed|M. Reed]] and [[Barry Simon|B. Simon]], ''Methods of Mathematical Physics'', vols I–IV, Academic Press 1972.
* [[Gerald Teschl|G. Teschl]], ''Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators'', http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
* V. S. Varadarajan, ''Geometry of Quantum Theory'' V2, Springer Verlag, 1970.
 
[[Category:Linear algebra]]
[[Category:Spectral theory]]
[[Category:Measures (measure theory)]]

Latest revision as of 10:22, 24 July 2014

No one is aware of the Singapore real estate trade better than My Next Leap. My Subsequent Leap also gives serviced places of work for lease in leading areas around the nation, providing you with the perfect location and infrastructure that meet what you are promoting needs.

2.1. If you have a $10,000 rental price range, no babies and your partner works in Woodlands, will you need to reside close to to the American Membership or the American School? In Singapore, most expats can afford one car as automobile possession is prohibitive. Much is determined by the partner who's caring for the children or the working spouse who must commute >half-hour by subway to downtown Singapore's office. 2.4. Teenaged kids studying at the American School. Will the chief govt change the minds of the spouse and daughter whose hearts are on the waterfront penthouse? LTV limits for 3rd houses onwards had been cut from 60% to forty%; or 20% if the mortgage tenure is more than 30 years or extends past age sixty five. Jun 10 在菲律宾投资物业 pic real property - by dealer

Most contracts for the sale and purchase of property in Singapore will incorporatestandard phrases referred to as the Singapore Law Society's Situations of Sale 1999. If theterms of the sale and purchase agreement or of the option are silent on certainpoints, then the provisions of these situations will apply. These conditions mayalso be modified by categorical terms within the contract - for example, it is not uncommon forthe sale to be subject to the property being unaffected by highway widening schemes. Purchase from an Unlicensed Housing Developer

NeWest @ Hong Leong Garden is exclusively 999 years leasehold Mix Development consists of 12 storey residential and commercial shops within landed residential enclave in the West Coast neighborhood brought to you by developer Oxley Holdings. MIDTOWN, a blended development consisting of each residential and business units, is excellently positioned inside minutes stroll to Hougang MRT and Hougang Central. Midtown consists of 160 residential models and 107 commercial models jointly developed by Oxley Holdings and Lian Beng. Positioned on the junction of Tampines Ave 1 and 10 with shut proximity to Tampines Regional Centre and future Tampines West MRT station, Qbay comes with full condominium services and has good connectivity to all main expressways.

Submit and listing your house, apartment, home for Rental or Sale in our on-line property classifieds. Usually, the value of a property is pricey in Singapore hence comes the excessive worth of properties for lease Properties for rent are anicart.com overall extra expensive and the chance of landlords having a change of heart and deciding to demand emptiness with out proper discover are a number of the distinct causes that end result to people preferring to personal their own residence. The rental prices for private condos can vary significantly. A condo in prime areas fetch highest rental prices and these rental market prices usually rely on its standard of luxurious, facilities supplied and the accessibility of the condominium. Excessive-end condominium near Orchard Street Condominium For Sale – Paterson Residence (D09)

3Rm HDB For Rent Close to Tampines MRT Blk 267 Tampines St 21  2+1 HDB unit, Fully furnished, Aircon 3Rm HDB For Lease Close to Bedok Reservoir Blk 526 Bedok North St 3  2+1 HDB unit, Absolutely furnished, Obtainable Immed $2.1k Nego With my vast experience in sales and leases of residential property, you will be assured that your curiosity is well taken care of. View my complete profile Joseph, thanks in your guidences and sufferers. Thank god on your advise else I'd have bought the wrong property with out realizing what difficulties it could give me." Jerry, your sufferers and non-pushy style is why we chose you over others. Well Achieved and Thank You for recommending such a stunning house to us" Pointers for Property Auction Sale (Sellers)

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