To the Many-Hilbert-Space Theory of Quantum Measurements
1 To the Many-Hilbert-Space Theory of Quantum Measurements by Z.I.ISMAILOV and E.OTKUN ÇEVIK
Karadeniz Technical University, Faculty of Sciences, Department of Mathematics 61080 Trabzon, TURKEY e-mail address : [email protected]
Abstract:
In this work, a connection between some spectral properties of direct integral of operators in the direct integral of Hilbert spaces and their coordinate operators has been investigated.
Keywords:
Direct integral of Hilbert spaces and operators; spectrum and resolvent sets; compact operators;Schatten-von Neumann operator classes; power and polynomially bounded operators.
AMS Subject Classification:
It is known that the general theory of linear closed operators in Hilbert spaces and its applications to physical problems have been investigated by many researchers(for example,see ).But many physical problems require studying the theory of linear operators in direct sums or in general direct integrals of Hilbert spaces. The concepts of direct integral of Hilbert spaces and direct integral of operators as a generalization of the concept of direct sum of Hilbert spaces and direct sum of operators were introduced to mathematics and developed in 1949 by John von Neumann in his work .These subjects were incorporated in several works(see .A spectral theory of some operators on a finite sum of Hilbert spaces was investigated by N.Dunford .Note that,in terms of application ,there are some results in papers in the finite sum cases .Also in the infinite direct sum cases there is a work ,in which some spectral and compactness properties are surveyed. Furthermore,some spectral investigations of the direct integral of operators in the direct integral of Hilbert spaces have been provided by T.R.Chow ,T.R.Chow,F.Gilfeather , E.A.Azoff
L.A.Fialkow . It must be noted that the theory of direct integral of Hilbert spaces and operators on the these spaces has important role in the representation theory of locally compact groups,in the theory of decomposition rings of operators to factors,invariant measures,reduction theory,von Neumann algebras and ets.On the other hand, many physical problems of today 2 arising in the modelling of processes of multiparticle quantum mechanics , quantum field theory and in the physics of rigid bodies support to study a theory of direct integral of operators in the direct integral of Hilbert spaces (see [19] and references in it). Numereous scientific investigations have been done to explain quantum measurements .Dealing with these subjects, S.Machida and M.Namiki (see, also have offered many-Hilbert-space theory lately.Also,they find that decoherence of the wave function is only necessity to formulate quantum mechanical measurements. A direct integral space of continiously many Hilbert spaces and a continious superselection rule is starring in their theory.The direct integral space structure is assigned to the measurement devise which reflects its macroscopic features.On the other hand,observed system is left to be defined by a single Hilbert space.Furthermore,they have examined double slit experiments and negative result experiments in the framework of the many-Hilbert-space theory.Note that a direct integral space of continiously many Hilbert spaces often arises in the quantum version of Lax-Phillips theory .In this investigation the direct integral space is introduced in order to the allow the generator of motion to have a spectrum in real axis ,which is a necessary condition for the application of the Lax-Phillips theory. In second section of this paper a connection between parts of spectrum, resolvent set of direct integral of operators defined in the direct integral of Hilbert spaces and parts of the spectrum of ‘’coordinate operators ‘’has been established . Note that the another approach to analogous problem has been used in work . In this present paper sharp formulas for the connection are given. In third section the compactness properties of these operators have been examined.Finally,in fourth section in special case the analogous questions for the power and polynomially bounded operators have been provided. In the special case of direct sum of Hilbert spaces, these questions have been investigated in
Along this paper the triplet be any measure space and the Hilbert spaces are looked at will become infinite dimensional. In addition ,the space of compact operators and Schatten-von Neumann classes in any Hilbert space will be denoted by and respectively.
2. On the spectrum of direct integral of operators
In this section, the relationship between the spectrum and resolvent sets of the direct integral of operators and its coordinate operators will be investigated. Before of all prove the following result.
Theorem 2.1 .For the operator in the Hilbert space = are true , Proof . For any there exist element such that and .Then almost everywhere with respect to measure µ it is true
Since ,then there exist which satisfy the above equality and . This means that
Hence and from this it is obtained that
The proof of the second proposition is clear. Actually ,in one special case the following stronger assertions are true.
Theorem 2.3.
Assume that every one-point set is measurable and its measure is positive.
For the parts of spectrum and resolvent sets of the operator in Hilbert space = the following claims are true ; ; ; ; Proof .Here only the first and second relations of theorem will be prove. The validity of other propositions can be proved by the similar ideas. Assumed that . Then there exist such that .So for every ) and for some .Hence and from this
On the contrary,assumed that .Then at least one index is hold ,i.e. for some is true .In this case for the element and we have .Consequently, . Now we prove the second relation on the continuous spectrum.Let . In this case by the definition of continuous spectrum is a one-to-one operator, and is dense in . From this and definition of direct integral it implies that for every an operator is a one-to-one operator in , and is dense in . Since ,then or . This means that It is easy to prove the inverse implication.
On the other hand the simple calculations show that the following relations are true.
Corollary 2.4 . Under the assumptions of last theorem we have , . Corollary 2.5.
Let Λ be any countable set , = and be any measure with property for every point .In this case the formulas , , , . are true. Note that when is counting measure the analogous results have been established in work SOME COMPACTNESS PROPERTIES of DIRECT INTEGRAL of OPERATORS
In this section the compactness and spectral properties between direct integral of operators and their ‘’coordinate operators’’have been established.In general ,there is not any relation between mentioned operators in compactness means.
Example3.1 . -counting measure, . In this case for every ) ,but Example .In some cases from the relations no implies ) for every . Indeed, from the definition of direct integral of operators on the set having null µ-measure the coordinate operators may be defined by arbitrary way.
But in certain situations there are concrete results. 5
Theorem 3.3.
Let Λ be any countable set , = and be any measure with property for every point .Then (1) If ,then for every , . (2)Let Λ infinite countable set and for every .In this case if and only if This theorem is proved by analogous scheme of the proof in theorem 4.6 in . Now give one characterizating theorem on the point spectrum of compact direct integral of operators.
Theorem 3.4.
Assumed that in the Hilbert space = , and .In this case there exist countable subset of Λ such that the set is minimal and . From the definition of singular number s(.) (or characteristic numbers ) of any compact operator in any Hilbert space and Theorems 2.1 and 3.4 it is easy to prove the validity of the following result.
Theorem 3.5.
Assumed that in the Hilbert space = , and .In this case there exist countable subset of Λ such that ; (2) İf then for every , ; (3) Let .Then if and only if the series =1 =1 ( ) converges. (4) If and the series is convergent,then (5) If and sup ,then for every ( ) . (6) If , sup and for some , ,then for every . Proof.
The validity of the claims (1) and (2) are clear.Prove third assertion of theorem.If the operator ,then the series is convergent.In this case by the first proposition of this theorem and important theorem on the convergence of the rearrangement series it is obtained that the series is convergent. On the contrary, if the series is convergent ,then the series being a rearrangement of the above series,is also convergent.So . Now prove (4).If for every then from the inequality and first claim the validity of this assertion is clear.Now consider the general case.In this case the operator can be written in form . Then
On the other hand ,since and , then from the (3) of this theorem it implies that with . Therefore,by the important theorem of the operator theory . Furthermore ,by using proposition (2) of this theorem it is easy to prove the claim(5).On the other hand, the claim(6) is one of the corollary of (5).
Remark 3.6.
Note that for the some in representation in Theorem 3.4. may be hold
In these situations corresponding conditions for such index in the Theorem 3.5(3-6) may be omitted ,for example,as in the following assertion.
Theorem 3.7.
Assumed that in the Hilbert space = , and .In this case there exist countable subset of Λ such that If , , and ,then . In this section let us and is the counting measure .Here a connection of power (and polynomially ) boundedness property of the direct sum operators in the direct sum Hilbert spaces and its coordinate operators were established.In advance,give some necessary definitions for the later.
Definition 4.1 .Let ℋ be any Hilbert space. (1).An operator ℋ is called power bounded ( ℋ if there exist a constant such that for any is satisfied (3.1) (2)Operator ℋ polynomially bounded ℋ ,if there exist a constant such that for any polynomial is satisfied , (3.2) where =sup . (3) The smallest number satisfying (3.1) (resp.(3.2)) is called the power bound (resp.polynomial bound) of the operator and will be denoted by (resp. . Before of all note that the following theorem is true. Theorem 4.2. If , then for every , . This result is a one of the corollary of following equality . In general,the inverse of last assertion may be not true. Example 4.3. Let , , , It is easy to see that and , . Consequently, = Hence,for any ,but . Example 4.4. Let , and , In this case and , ,i.e.for every , ,but = Therefore, . Actually,it is true the following result . Theorem 4.5. if and only if and . Proof.If ,then from the following relation it is implies that for each .From this it is determined that for any . On the other hand,it is clear that for each Therefore, . On the contrary,if for any , and ,then from the equality it is obtained that . Now polynomially boundness property of the direct sum operators will be investigated.In advance,note that the following proposition is true. Theorem 4.6. If , and ,then for every , . Unfortunately,the inverse of last theorem may be not true in general. Example 4.7. Let , , It is known that and is a nilpotent operator with power of nilpotency 2 ,for any .In this case for any polynomial function we have . In the other words ,for every , .Unfortunately,for the polynomial we have i.e. . But in general case the following result is true. Theorem 4.8. Let , and .In order to the necessary and sufficient conditions are and . Proof. Assumed that for every polynomial p(.) .and . In this case since for every polynomial function ),then .From last relation it is obtained that .Hence . Now let us ,i.e.for any and polynomial p(.) it is valid . Then it is clear that , .In the other hand , from last equality it implies that for every is hold .Hence, .This completes the proof of the theorem. Acknowledgment
The authors are grateful to G.İsmailov(student of Trabzon Kanuni Anadolu High School) for his helping suggestion to english version and other technical discussion.
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