Diagonalization and representation results for nonpositive sesquilinear form measures
aa r X i v : . [ m a t h . F A ] J un DIAGONALIZATION AND REPRESENTATION RESULTS FORNONPOSITIVE SESQUILINEAR FORM MEASURES
Tuomas Hyt¨onenDepartment of Mathematics and StatisticsUniversity of HelsinkiGustaf H¨allstr¨omin katu 2bFI-00014 Helsinki, Finland [email protected]
Juha-Pekka Pellonp¨a¨a Department of PhysicsUniversity of TurkuFI-20014 Turku, Finland [email protected]
Kari YlinenDepartment of MathematicsUniversity of TurkuFI-20014 Turku, Finland [email protected]
Abstract.
We study decompositions of operator measures and more general sesquilinear formmeasures E into linear combinations of positive parts, and their diagonal vector expansions.The underlying philosophy is to represent E as a trace class valued measure of bounded vari-ation on a new Hilbert space related to E . The choice of the auxiliary Hilbert space fixes aunique decomposition with certain properties, but this choice itself is not canonical. We presentrelations to Naimark type dilations and direct integrals. Mathematics Subject Classification.
Date : November 19, 2018. Corresponding author. Telephone: +358-2-333 5737, telefax: +358-2-333 5070.
Keywords and phrases.
Sesquilinear form, operator measure, bounded variation, diagonal-ization, Naimark dilation, direct integral.1.
Introduction
The idea of rigged Hilbert spaces arises in attempts to develop mathematically rigorousinterpretations of the intuitively appealing Dirac formalism of Quantum Mechanics. Withthe help of generalized eigenvectors lying outside the Hilbertian state space, one is able towrite eigenvalue expansions, with formal similarity to the finite-dimensional case, even for self-adjoint operators with a continuous spectrum. By the Spectral Theorem, self-adjoint operatorsmay be identified with spectral measures on the real line, and they are the mathematicalrepresentatives of physical observables in the traditional von Neumann approach to QuantumMechanics. It is, however, well known that this point of view becomes too restrictive alreadywhen considering such basic physical examples as phase-like quantities (see e.g. [6]), but theycan still be incorporated into the mathematical formalism by allowing more general positiveoperator measures in place of spectral measures. Also for them, and for the yet larger class ofpositive sesquilinear form measures, generalized eigenvalue expansions have been obtained inthe literature. See e.g. [3] and the references therein.The present note is concerned with similar results for sesquilinear form measures without anypositivity conditions. Besides purely mathematical interest, motivation comes from importantphysical questions. Let us consider an example.
Example 1.1.
Let H be a complex Hilbert space with an orthonormal basis ( e n ) ∞ n =0 . Let z ∈ C \ { } and define a coherent state ψ z := e −| z | / ∞ X n =0 z n √ n ! e n . It describes quasimonochromatic laser light (in a single-mode quantum optical system), where | z | is the energy parameter and z/ | z | ∈ T is the phase parameter of the laser light (see e.g.[8]). The vector e n , the so-called number state or Fock state , describes an optical field whichcontains n photons of the same frequency.A measurement of the phase parameter can be described by using a phase shift covariantsemispectral measure [6, p. 23] E ( X ) := ∞ X m,n =0 c mn Z X w n − m dµ ( w ) | e m i h e n | ONPOSITIVE SESQUILINEAR FORM MEASURES 3 where X is a Borel set of T , µ is the normalized Haar measure of T and ( c mn ) is a positivesemidefinite complex matrix with the unit diagonal; the probability of getting a value w froma set X when the system is prepared in a state ψ z is thus h ψ z | E ( X ) ψ z i .In realistic physical situations we cannot produce arbitrarily high photon numbers, that is,we cannot prepare number states e n for an arbitrarily large n . In fact, as of 2004, a methoddescribed in [11] “still remains the only experiment in principle capable of providing an arbitraryFock state (at least up to n = 4) on demand”. But still we need the whole Hilbert space H to define coherent states. Hence, we can relax the definition of ( c mn ): we need only assumethat the first, say, 10 ×
10 block of ( c mn ) is positive semidefinite (so that we get probabilitydistributions also for superpositions of number states e n , n ≤ ψ z with sufficiently low energy | z | ≤ r ∈ R we can define a probabilitymeasure X e −| z | P ∞ m,n =0 c mn R X w n − m dµ ( w ) z n − m √ n ! m ! . Further restrictions can be imposed, if weassume that some superpositions of coherent states can be measured. If ( c mn ) is not assumed tobe positive semidefinite, then E ( X ) may be a nonpositive operator, or even a sesquilinear formon V = lin { e n | n = 0 , , . . . } , for some X . Then the mapping X E ( X ) can be understood asa (nonpositive) sesquilinear form measure . It can be shown that some (phase shift covariant)sesquilinear form measures give more accurate phase distributions in coherent states than any(covariant) positive semispectral measures [7].The sesquilinear form measures we study here generalize operator measures which have al-ready received a fair amount of attention in the mathematical literature. For example, wemay quote a well-known decomposition result from [5], pp. 104–105: A regular Borel operatormeasure on a compact Hausdorff space (with values in the space of bounded operators on aHilbert space) is, as a consequence of Wittstock’s decomposition theorem, completely boundedif, and only if, it can be expressed as a linear combination of positive operator valued measures.In this paper an analogous decomposition problem in the setting of sesquilinear form measuresis in a central role. We consider a σ -algebra Σ, a vector space V with a countable Hamelbasis, and measures E : Σ → S ( V ) where S ( V ) is the space of sesquilinear forms on V . Thisgeneralizes the more standard setting of operator measures in the context of a separable Hilbertspace, and it turns out that our more flexible framework yields new information even there:An operator measure may be decomposed into a linear combination of positive parts withoutthe condition of complete boundedness. Of course there is a price to pay: these positive partsare not necessarily operator valued but only sesquilinear form valued. While this on the one HYT ¨ONEN, PELLONP ¨A ¨A, AND YLINEN hand may be seen as a drawback, on the other hand it highlights the usefulness of generalsesquilinear form measures.The paper is organized as follows. After the setting is explained in Section 2, the next sectionestablishes a connection with operator measures taking their values in the trace class L ( H )of a separable Hilbert space. Since L ( H ) has the Radon-Nikod´ym property, a sesquilinearform measure can be expressed in terms of integrating an L ( H )-valued density function withrespect to a basic positive scalar measure. In Section 4 the desired decomposition is effectedby utilizing the operator density found in Section 3. While the basic idea is straightforwardenough, one must take care of rather delicate measurability issues. To this end, a classicalresult of Kuratowski and Ryll-Nardzewski on measurable selectors is used. The final Section 5deals with an analogue of the Naimark dilation theorem: The decomposition of a sesquilinearform measure into positive parts also yields a spectral dilation in a generalized sense involving aunitary operator W on the dilation space where the spectral measure acts. The characateristicfeature of W is that W = I . The paper concludes with a remark on formulating the dilationresult in terms of a direct integral representation.2. Basics
We write Z + := { , , , . . . } , N := { } ∪ Z + and Z − := Z \ N . For p > I an index set, ℓ p ( I ) is the space of the complex families c = ( c n ) n ∈I such that P n ∈I | c n | p < ∞ .Let V be a vector space. The scalar field is always C . A mapping Φ : V × V → C is calleda sesquilinear form (SF), if it is antilinear (i.e., conjugate linear) in the first and linear in thesecond variable. It is symmetric if Φ( φ, ψ ) = Φ( ψ, φ ) =: Φ ∗ ( φ, ψ ) and positive if Φ( φ, φ ) ≥ φ, ψ ∈ V . Any positive SF is symmetric, and any SF Φ is a linear combination of twosymmetric SFs:(2.1) Φ = 12 (Φ + Φ ∗ ) + i i Φ ∗ − i Φ) . We let S ( V ) (resp. P S ( V )) denote the set of sesquilinear forms (resp. positive sesquilinearforms) on V × V .Our basic reference on measure and (vector) integration is [2]. Measurability means µ -measurability where µ is a fixed positive measure. Let (Ω , Σ) be a measurable space, i.e., Σ isa σ -algebra of subsets of Ω. Definition 2.2.
Let E : Σ → S ( V ) be a mapping and denote E ( X ) = E X for X ∈ Σ. Wecall E a sesquilinear form measure (SFM) if the mapping X E X ( φ, ψ ) is σ -additive, i.e. a ONPOSITIVE SESQUILINEAR FORM MEASURES 5 complex measure, for all φ, ψ ∈ V . If in addition E ( X ) is symmetric (resp. positive) for all X ∈ Σ, E is called a symmetric (resp. positive) sesquilinear form measure .The inner product of any Hilbert space H is linear in the second variable and denoted by h · | · i . We let L ( H ) stand for the bounded linear operators on H , L s ( H ) ⊂ L ( H ) for theself-adjoint operators, and L + ( H ) ⊂ L s ( H ) for the positive ones. The trace class is denoted by L ( H ), and L s ( H ) := L ( H ) ∩ L s ( H ), L ( H ) := L ( H ) ∩ L + ( H ). Definition 2.3.
Let H be a Hilbert space and E : Σ → L ( H ) a mapping. We call E an operator measure (OM) if it is weakly σ -additive, i.e. the mapping X
7→ h φ | E ( X ) ψ i is σ -additive for all φ, ψ ∈ H . If in addition E (Σ) ⊂ L s ( H ) (resp. E (Σ) ⊂ L + ( H )) we say that E is a self-adjoint (resp. positive) operator measure , and if E ( X ) = E ( X ) = E ( X ) ∗ forall X ∈ Σ, E is called a projection measure . An OM E : Σ → L ( H ) is called normalized if E (Ω) = I , the identity operator on H . A normalized positive OM is also called a semispectralmeasure and a normalized projection measure a spectral measure .Every (self-adjoint or positive) OM E can be identified with a (symmetric or positive) SFM E by setting E X ( φ, ψ ) := h φ | E ( X ) ψ i .3. Reduction to trace-class operator measures
For the rest of the note, we assume that V has a countably infinite Hamel basis ( e n ) ∞ n =0 , and H is the Hilbert space completion of V such that ( e n ) ∞ n =0 is an orthonormal basis of H . Forany SF Φ on V we write (formally) Φ = ∞ X m,n =0 Φ mn | e m i h e n | where Φ mn := Φ( e m , e n ). If Φ is bounded with respect to the norm of H , it determines a uniquebounded linear operator ˜Φ ∈ L ( H ) satisfying h φ | ˜Φ ψ i = Φ( φ, ψ ). Then the series above is notjust formal; when | e m i h e n | denotes as usual the rank one operator φ
7→ h e n | φ i e m , the seriesconverges with respect to the weak operator topology to ˜Φ. We may identify Φ and ˜Φ, andthen Φ mn = h e m | Φ e n i . Lemma 3.1.
Let Φ ∈ S ( V ) be represented by an infinite matrix (Φ mn ) ∞ m,n =0 ∈ ℓ ( N × N ) . Then Φ has a unique extension Φ ∈ L ( H ) and k Φ k L ( H ) ≤ P ∞ m,n =0 | Φ mn | . HYT ¨ONEN, PELLONP ¨A ¨A, AND YLINEN
Proof.
Since L ( H ) is the dual of the space of finite rank operators on H , the first claim isequivalent to requiring that sup | tr(ΦΛ) | < ∞ where Φ is interpreted as a matrix (Φ mn ) and Λranges over the matrices (Λ mn ) of finite rank operators of norm ≤
1. But | tr(ΦΛ) | = (cid:12)(cid:12)(cid:12) ∞ X m,n =0 Φ mn Λ nm (cid:12)(cid:12)(cid:12) ≤ ∞ X m,n =0 | Φ mn | · | Λ nm | ≤ ∞ X m,n =0 | Φ mn | , since | Λ nm | ≤ k Λ k L ( H ) ≤ (cid:3) Theorem 3.2.
For any SFM E : Σ → S ( V ) there exist an L ( H ) -valued measure F of boundedvariation, and an injective operator D ∈ L + ( H ) such that DV = V and E X ( Dφ, Dψ ) = h φ | F ( X ) ψ i , φ, ψ ∈ V. There further exist a finite positive measure µ : Σ → [0 , ∞ ) and a function T ∈ L (Ω , Σ , µ ; L ( H )) such that E X ( Dφ, Dψ ) = Z X h φ | T ( ω ) ψ i dµ ( ω ) , φ, ψ ∈ V. Defining C ω ( φ, ψ ) := h D − φ | T ( ω ) D − ψ i , we also obtain the integral representation E X ( φ, ψ ) = Z X C ω ( φ, ψ ) dµ ( ω ) , φ, ψ ∈ V. Proof.
We denote E mn ( X ) := E X ( e m , e n ), and write | E mn | ( X ) for its total variation on X .Choose any bounded positive sequence ( d m ) ∞ m =0 such that δ := ∞ X m,n =0 d m d n | E mn | (Ω) < ∞ . For example, we may take d m = α m / max { , p | E kl | (Ω) | ≤ k, l ≤ m } where ( α m ) ∞ m =0 is anysummable positive sequence.Let D be the diagonal operator Dφ := ∞ X n =0 d n | e n i h e n | φ i . Then for φ, ψ ∈ V ,(3.3) E X ( Dφ, Dψ ) = ∞ X m,n =0 h φ | e m i d m d n E mn ( X ) h e n | ψ i =: h φ | F ( X ) ψ i , ONPOSITIVE SESQUILINEAR FORM MEASURES 7 and we have F ( X ) ∈ L ( H ) with k F ( X ) k L ( H ) ≤ δ by Lemma 3.1. If ( X k ) ∞ k =0 is any countablepartition of X ⊂ Ω, then(3.4) ∞ X k =0 k F ( X k ) k L ( H ) ≤ ∞ X k =0 ∞ X m,n =0 d m d n | E mn ( X k ) | ≤ ∞ X m,n =0 d m d n | E mn | ( X ) ≤ δ. This justifies the computation ∞ X k =0 F ( X k ) = ∞ X k =0 ∞ X m,n =0 d m d n E mn ( X k ) | e m i h e n | = ∞ X m,n =0 d m d n E mn ( X ) | e m i h e n | = F ( X ) , which shows that F is σ -additive, and (3.4) with X = Ω also shows that F is of boundedvariation.For the measure µ one can take any finite positive measure with respect to which the vectormeasure F , or equivalently E , is absolutely continuous (i.e., whenever µ ( X ) = 0, we have also F ( X ) = 0, or equivalently E X = 0 as a sesquilinear form). To be specific, we take µ to be thetotal variation of F , | F | ( X ) := sup N X k =1 k F ( X k ) k L ( H ) where the supremum is over all finite Σ-partitions of X . As in the proof of Proposition 7.1of [3], the existence of T then follows from the vector-valued Radon–Nikod´ym theorem, since L ( H ) (as a separable dual space) has the Radon–Nikod´ym property. (cid:3) Remark 3.5.
The above theorem shows that a sesquilinear form measure on V can always beviewed as an operator measure on a new Hilbert space. In fact, let us denote by H D the rangeof D ∈ L ( H ) equipped with the inner product h η | θ i D := h D − η | D − θ i and the induced norm.Then D : H → H D is an isometric Hilbert space isomorphism. Observe that in (3.3) the seriesin the middle is absolutely convergent, and the right-hand side makes sense, for all φ, ψ ∈ H .Thus E X extends continuously to a sesquilinear form on H D , and for η = Dφ, θ = Dψ ∈ H D we have E X ( η, θ ) = h D − η | F ( X ) D − θ i = h D − η | D − DF ( X ) D − θ i = h η | DF ( X ) D − θ i D =: h η | ˜ E ( X ) θ i D . Due to the operator-ideal property of the trace class, we find that X ∈ Σ ˜ E ( X ) = DF ( X ) D − is an L ( H D )-valued measure of bounded variation. By the Radon–Nikod´ym the-orem, it can be written as˜ E ( X ) = Z X S ( ω ) dµ ( ω ) , S ∈ L (Ω , Σ , µ ; L ( H D )) . HYT ¨ONEN, PELLONP ¨A ¨A, AND YLINEN
Remark 3.6.
In the rest of the paper we take µ to be the measure constructed in the aboveproof. Assume now that { ω } ∈ Σ for all ω ∈ Ω. If we let µ = µ + µ be the decompositionof µ as the sum of a discrete measure µ and a continuous measure µ , the integral formulain the above theorem may be used to decompose E as E = E + E where E is a discreteSFM, i.e. vanishes outside a countable set, and the SFM E is continuous, i.e., vanishes at everysingleton. Clearly such a decomposition is unique.4. Diagonalization; positive and negative parts
By formula (2.1) we may decompose the measures E and F as well as the operator density T into linear combinations of two symmetric parts, and by linearity the representation formulaeof Theorem 3.2 remain true for these parts. In this section we obtain a further decompositionof these symmetric parts. We will need the following classical result on measurable selectors from [4]; it is also stated in [1], Lemma 1.9: Lemma 4.1.
Let E be a compact metric space and let ψ : E × Ω → R be a mapping such that ψ ( x, · ) is measurable for arbitrary x ∈ E and ψ ( · , ω ) is continuous for arbitrary ω ∈ Ω . Thenthere exists a measurable ξ : Ω → E such that ψ ( ξ ( ω ) , ω ) = max x ∈ E ψ ( x, ω ) , ω ∈ Ω . Corollary 4.2.
Let T : Ω → L ( H ) be a measurable function. Then there exists a measurable Φ : Ω → ¯ B H , the closed unit ball of H , such that |h Φ( ω ) | T ( ω )Φ( ω ) i| = max φ ∈ ¯ B H |h φ | T ( ω ) φ i| , ω ∈ Ω . Proof.
It is well known that the unit ball ¯ B H of a separable Hilbert space, when equipped withthe weak topology, is a compact metrizable space. We consider the mapping ψ : ¯ B H × Ω → R , ( φ, ω )
7→ h φ | T ( ω ) φ i , and it suffices to check the conditions of Lemma 4.1.That ψ ( φ, · ) is measurable is clear from the assumptions. To see that ψ ( · , ω ) is continuous,denote Λ := T ( ω ) ∈ L ( H ). Assume first that Λ = | ψ i h ψ | has rank 1. The mappings φ
7→ h ψ i | φ i are obviously continuous in the topology in question, and so is their product. Ingeneral, we have Λ = ∞ X k =1 | ψ k i h ρ k | , ∞ X k =1 k ψ k k · k ρ k k < ∞ . ONPOSITIVE SESQUILINEAR FORM MEASURES 9
Since uniformly convergent series of continuous functions are continuous, we have reached theconclusion. (cid:3)
We can now prove a measurable diagonalization of an L s ( H )-valued function. The prooffollows closely the same pattern as the special case for L ( H )-valued functions given in [1],Proposition 1.8, but we include the details for the reader’s convenience. Theorem 4.3.
Given a measurable function T : Ω → L s ( H ) , there exist measurable functions φ k : Ω → H and λ k : Ω → R , k ∈ Z + , such that for any fixed ω ∈ Ω there holds h φ k ( ω ) | φ ℓ ( ω ) i = δ kℓ , | λ k ( ω ) | ≥ | λ ℓ ( ω ) | if k ≤ ℓ,T ( ω ) = ∞ X k =1 λ k ( ω ) | φ k ( ω ) i h φ k ( ω ) | , k T ( ω ) k L ( H ) = ∞ X k =1 λ k ( ω ) . Proof.
This representation of T ( ω ) for each fixed ω ∈ Ω is just the usual spectral representation,but the point is to obtain this with a measurable dependence on ω . To see this, we recall analgorithm for computing the spectral representation. An eigenvalue λ of Λ ∈ L s ( H ) of largestmodulus satisfies | λ | = max φ ∈ ¯ B H |h φ | Λ φ i| , and any φ ∈ ¯ B H , which gives the maximum, is an eigenvector related to ± λ . By Corollary 4.2,there is a measurable function φ : Ω → ¯ B H such that λ ( ω ) := h φ ( ω ) | T ( ω ) φ ( ω ) i , which is also a measurable function of ω by the above formula, is an eigenvalue of T ( ω ) ofmaximal modulus, with the eigenvector φ ( ω ).We then repeat the same procedure with T ( ω ) := T ( ω ) − λ ( ω ) | φ ( ω ) i h φ ( ω ) | in placeof T ( ω ), obtaining new measurable functions λ ( ω ) and φ ( ω ). Proceeding inductively, weobtain sequences of measureable functions ( λ k ( ω )) ∞ k =1 and ( φ k ( ω )) ∞ k =1 . At each fixed ω ∈ Ω,these give the spectral decomposition of T ( ω ) by standard results about compact selfadjointoperators. (cid:3) It is now also easy to separate the positive and negative parts of the operator density in ameasurable way:
Corollary 4.4.
Given a measurable function T : Ω → L s ( H ) , there exist measurable functions g k : Ω → H , k ∈ Z \ { } , such that for any fixed ω ∈ Ω there holds h g k ( ω ) | g ℓ ( ω ) i = δ kℓ k g k ( ω ) k , k g k ( ω ) k ≥ k g ℓ ( ω ) k if < k < ℓ or > k > ℓ,T ( ω ) = X k ∈ Z \{ } sgn( k ) | g k ( ω ) i h g k ( ω ) | , k T ( ω ) k L ( H ) = X k ∈ Z \{ } k g k ( ω ) k . Proof.
With the notation of Theorem 4.3, we define the measurable functions n ( ω ) := 0 , g ( ω ) := 0 ,n ± k ( ω ) := inf { n ∈ Z + | n > n ± ( k − ( ω ) , ± λ n ( ω ) > } , k ∈ Z + g ± k ( ω ) := | λ n ± k ( ω ) ( ω ) | / φ n ± k ( ω ) ( ω ) , k ∈ Z + , where it is understood that inf ∅ := ∞ and λ ∞ ( ω ) := 0 =: φ ∞ ( ω ). (cid:3) Corollary 4.5.
Given a measurable function T : Ω → L s ( H ) , there exists a pair of measurablefunctions T ± : Ω → L ( H ) , such that for any fixed ω ∈ Ω we have (i) T ( ω ) = T + ( ω ) − T − ( ω ) , (ii) T + ( ω ) T − ( ω ) = 0 , and (iii) k T ( ω ) k L = k T + ( ω ) k L + k T − ( ω ) k L . Moreover, if (i) and (ii), or alternatively (i) and (iii), hold for all ω ∈ Ω , the functions T + and T − are uniquely determined.Proof. For existence, it suffices to set T ± ( ω ) := X k ∈ Z ± | g k ( ω ) i h g k ( ω ) | . The uniqueness statement assuming (i) and (ii) follows e.g. from Corollary 2.10 in [9]. Assuming(i) and (iii), the uniqueness claim is a consequence of Theorem 4.2 in [10], since L ( H ) with itsnorm and order may be identified with the predual of L ( H ). (cid:3) ONPOSITIVE SESQUILINEAR FORM MEASURES 11
In the case of a symmetric
SFM E , its trace-class density T is self-adjoint operator valuedand, using the above corollaries, we get E X ( φ, ψ ) = Z X h D − φ | [ T + ( ω ) − T − ( ω )] D − ψ i dµ ( ω )= Z X X k ∈ Z \{ } sgn( k ) h D − φ | g k ( ω ) ih g k ( ω ) | D − ψ i dµ ( ω )= Z X X k ∈ Z \{ } sgn( k ) h φ | d k ( ω ) ih d k ( ω ) | ψ i dµ ( ω ) , (4.6)where we have defined d k ( ω ) := D − g k ( ω ) ∈ H D − , and H D − is the Hilbert space consisting of all the formal sums P ∞ n =0 c n e n such that P ∞ n =0 d n | c n | < ∞ . Note that we have a Hilbert space triplet H D ⊂ H ⊂ H D − , where H D − isthe topological antidual of H D . Note that the conclusion of (4.6) could also have been reachedby applying Corollary 4.4 to (the symmetric parts of) the function S : Ω → L (Ω , Σ , µ ; L ( H D ))from Remark 3.5.Denoting E ± X ( φ, ψ ) := Z X X k ∈ Z ± h φ | d k ( ω ) ih d k ( ω ) | ψ i dµ ( ω )we obtain a splitting(4.7) E X = E + X − E − X of an arbitrary symmetric sesquilinear form measure into a difference of two positive sesquilinearform measures. Despite the above notation, this splitting is not canonical, and a different choiceof the operator D typically yields a different decomposition. (The choice of µ is less important:it only affects the normalization of the vectors d k ( ω ).) However, by Corollary 4.5, given thechoice of D , there is a unique splitting with the stated properties. In particular, the L ( H D )-valued extension ˜ E (cf. Remark 3.5) has a canonical splitting into L ( H D )-valued operatormeasures. Also, if E is already positive in the beginning, then the process used in the proof ofthe decomposition only gives T + = T and E + = E .Let then E : Σ → S ( V ) be an arbitrary SFM. Definition 4.8.
The family ( E ( k ) ) k =0 of positive SFMs E ( k ) : Σ → P S ( V ) is a decompositionof E (into positive parts) if E = X k =0 i k E ( k ) . From eqs. (2.1) and (4.7) one sees easily that for any SFM E there exists a decompositionof E into positive parts. 5. Dilations
Definition 5.1.
Let K be a Hilbert space, F : Σ → L ( K ) a spectral measure, and W ∈ L ( K )a unitary operator whose spectrum σ ( W ) is contained in { , − , i, − i } . Let J : V → K bea linear map. We say that the quadruple ( K, F, W, J ) is a (spectral W -)dilation of a SFM E : Σ → S ( V ) if the following conditions hold:(1) h J φ | F ( X ) W J ψ i = E X ( φ, ψ ) for all X ∈ Σ and φ, ψ ∈ V ,(2) W F ( X ) = F ( X ) W for all X ∈ Σ.(3) the linear span of the set { W k F ( X ) J φ | k = 0 , , , , X ∈ Σ , φ ∈ V } is dense in K .For k ∈ { , , , } , let K k be the eigenspace of W corresponding to i k (define K k = { } if i k / ∈ σ ( W )), I k the identity of K k , P k the projection of K onto K k , J k := P k ◦ J , F k : Σ → L ( K k )the restriction F k ( X ) := F ( X ) | K k , and E ( k ) : Σ → S ( V ) the positive SFM defined by(5.2) E ( k ) X ( φ, ψ ) := h J k φ | F k ( X ) J k ψ i . Theorem 5.3.
Let E be a SFM. Any dilation ( K, F, W, J ) of E defines by (5.2) a decomposition ( E ( k ) ) k =0 of E into positive parts. Conversely, for any decomposition ( E ( k ) ) k =0 , there exists adilation ( K, F, W, J ) such that (5.2) holds. In particular, any SFM has a spectral W -dilation. In the situation of Theorem 5.3, we saythat ( E ( k ) ) k =0 is the decomposition of E associated to the dilation ( K, F, W, J ). Proof.
Given a dilation (
K, F, W, J ), it follows from 5.1(2) that each K k is invariant under F ( X ), and ( K k , F k , I k , J k ) is a spectral dilation of E ( k ) . Then 5.1(1) implies that ( E ( k ) ) k =0 is adecomposition of E .Conversely, let ( E ( k ) ) k =0 be a decomposition of E . Then each E ( k ) is a positive SFM, for whichthere exists a spectral dilation of the form ( K k , F k , I k , J k ) by Theorem 3.6 of [3]. Define K := K ⊕ K ⊕ K ⊕ K , F ( X ) := F ( X ) ⊕ F ( X ) ⊕ F ( X ) ⊕ F ( X ), W := I ⊕ ( iI ) ⊕ ( − I ) ⊕ ( − iI ),and J := J ⊕ J ⊕ J ⊕ J . To check that ( K, F, W, J ) is a dilation of E , conditions 5.1(1)and 5.1(2) are clear and 5.1(3) follows from lin { W l | l = 0 , . . . , } = lin { P k | k = 0 , . . . , , } . Itis also clear that (5.2) holds. (cid:3) Let M = ( K, F, W, J ) and M ′ = ( K ′ , F ′ , W ′ , J ′ ) be two dilations of E . The quantities K ′ k , F ′ k , P ′ k , I ′ k and J ′ k related to M ′ are defined as before in the obvious way. ONPOSITIVE SESQUILINEAR FORM MEASURES 13
Definition 5.4.
The dilations M and M ′ of E are unitarily equivalent if there exists a unitarymap U : K → K ′ such that U F ( X ) J φ = F ′ ( X ) J ′ φ for all X ∈ Σ, φ ∈ V and U W = W ′ U ; inparticular, U J φ = J ′ φ for all φ ∈ V . Theorem 5.5.
Two dilations M and M ′ of E are unitarily equivalent if and only if thedecompositions of E associated to M and M ′ are the same, that is, E ( k ) X ( φ, ψ ) = h J k φ | F ( X ) J k ψ i = h J ′ k φ | F ′ ( X ) J ′ k ψ i for all k ∈ { , , , } , X ∈ Σ and φ, ψ ∈ V . Moreover, then the relevant U is unique and U F ( X ) = F ′ ( X ) U for all X ∈ Σ .Proof. Assume first that the decompositions of E associated to M and M ′ are the same. Sinceboth ( K k , F k , I k , J k ) and ( K ′ k , F ′ k , I ′ k , J ′ k ) are dilations of the positive SFM E ( k ) , it follows fromTheorem 3.6 of [3] that there is a unique unitary map U k : K k → K ′ k such that U k F k ( X ) J k φ = F ′ k ( X ) J ′ k φ for all X ∈ Σ, φ ∈ V , and U k F k ( X ) = F ′ k ( X ) U k for all X ∈ Σ. Then U := P k =0 U k P k has the desired properties.Suppose conversely that M and M ′ are unitarily equivalent. Since U F ( X ) W k F ( Y ) J φ = U W k F ( X ∩ Y ) J φ = W ′ k U F ( X ∩ Y ) J φ = W ′ k F ′ ( X ∩ Y ) J ′ φ = F ′ ( X ) W ′ k F ′ ( Y ) U J φ = F ′ ( X ) U W k F ( Y ) J φ , it follows from 5.1(3) that
U F ( X ) = F ′ ( X ) U . As U W = W ′ U implying U P k = P ′ k U , one sees that h P ′ k J ′ φ | F ′ ( X ) P ′ k J ′ ψ i = h P ′ k U J φ | F ′ ( X ) P ′ k U J ψ i = h U P k J φ | F ′ ( X ) U P k J ψ i = h P k J φ | F ( X ) P k J ψ i , i.e., the associated decompositions coincide. Since U W k F ( X ) J φ = F ′ ( X ) W ′ k J ′ φ , the unique-ness of U is clear. (cid:3) Remark 5.6.
Since k J φ k K = P k =0 k J k φ k K k = P k =0 E ( k )Ω ( φ, φ ), we see that J : V → K isinjective if and only if(5.7) X k =0 E ( k )Ω ( φ, φ ) > φ ∈ V \ { } . This situation can always be achieved by writing E = ( E + ǫE ) − ǫE , where ǫ > E : Σ → L ( H ) is a semispectral measure, which automatically satisfies (5.7). Remark 5.8.
In analogy with the case of positive SFMs treated in [3], it is possible to describe aconcrete representation of the dilation (
K, F, W, J ) associated with any decomposition ( E ( k ) ) k =0 of a SFM E into positive parts.Let L (Ω , µ ; ℓ ( Z )) h L (Ω , µ ; ℓ ( Z + ) ) be the usual Bochner space of ℓ ( Z )-valued func-tions f = ( f (0) , . . . , f (3) ), where f ( k ) = ( f ( k ) j ) ∞ j =1 ∈ L (Ω , µ ; ℓ ( Z + )). Given a measurable n ( · ) = ( n ( · ) , . . . , n ( · )) : Ω → ( N ∪ {∞} ) , we denote by L n ( · ) (Ω , µ ; ℓ ( Z )) the closed sub-space consisting of the functions f such that for a.e. ω ∈ Ω, all j and k , there holds f ( k ) j ( ω ) = 0if j > n k ( ω ). This is analogous to the “direct integral” Hilbert space of a measurable familyof ℓ spaces of variable dimension considered in Section 5 of [3]; extending the notation usedthere we could write L n ( · ) (Ω , µ ; ℓ ( Z )) = Z ⊕ Ω ( ℓ ) n ( ω ) dµ ( ω ) . Let then E = P k =0 i k E ( k ) be a SFM. By the construction of Section 4 (or Theorem 4.5 of [3]),the positive SFMs E ( k ) have representations E ( k ) X ( φ, ψ ) = Z X X j ∈ Z + h φ | d ( k ) j ( ω ) ih d ( k ) j ( ω ) | ψ i dµ ( ω ) , where d ( k ) j ( ω ) = D − g ( k ) j ( ω ), and the g ( k ) j ( ω ) are as the g j ( ω ) in Corollary 4.4. We now fix aspecific n ( · ) by setting n k ( ω ) := sup { j ∈ Z + : d ( k ) j ( ω ) = 0 } (with sup ∅ := 0), and define K := L n ( · ) (Ω , µ ; ℓ ( Z )) , F ( X ) f := 1 X f, W f := ( f (0) , if (1) , − f (2) , − if (3) ) , ( J φ )( ω ) := ( h d (0) j ( ω ) | φ i , . . . , h d (3) j ( ω ) | φ i ) ∞ j =1 . The conditions 5.1(1) and 5.1(2) of a dilation follow from simple algebra. The density re-quirement 5.1(3) is a consequence of the fact that the component dilations ( K k , F k , I k , J k ), k = 0 , . . . ,
3, are dilations of the positive parts E ( k ) of E by Theorem 5.1 of [3]. References [1] G. Da Prato, J. Zabczyk,
Stochastic equations in infinite dimensions . Encyclopedia of Mathematics andits Applications, 44. Cambridge University Press, Cambridge, 1992.[2] N. Dunford, J. T. Schwartz,
Linear Operators. I. General Theory . With the assistance of W. G. Bade andR. G. Bartle. Pure and Applied Mathematics, 7. Interscience Publishers, New York–London, 1958.[3] T. Hyt¨onen, J.-P. Pellonp¨a¨a, K. Ylinen, Positive sesquilinear form measures and generalized eigenvalueexpansions. J. Math. Anal. Appl., in press; arXiv:math.FA/0703589v1[4] K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors.
Bull. Acad. Polon. Sci. S´er. Sci. Math.Astronom. Phys. (1965), 397–403. ONPOSITIVE SESQUILINEAR FORM MEASURES 15 [5] V. Paulsen,
Completely bounded maps and operator algebras . Cambridge Studies in Advanced Mathematics78. Cambridge University Press, Cambridge, U.K., 2002.[6] J.-P. Pellonp¨a¨a,
Covariant Phase Observables in Quantum Mechanics
J. Phys. A (2001), 7901–7916.[8] J. Peˇrina, Coherence of Light . D. Reidel Publishing Company, Dordrecht, 1985.[9] S. Stratila, L. Zsido,
Lectures on von Neumann algebras . Editura Academiei, Bucuresti, Romania; AbacusPress, Turnbridge Wells, Kent, England, 1979.[10] M. Takesaki,
Theory of operator algebras I . Springer-Verlag, New York Heidelberg Berlin, 1979.[11] B. T. H. Varcoe, S. Brattke, H. Walther, The creation and detection of arbitrary photon number statesusing cavity QED.
New J. Phys.6