JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 215, 423᎐442Ž. 1997 ARTICLE NO. AY975643
Quasi *-Algebras and Multiplication of Distributions
Agnese Russo
Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Palermo, Via Archirafi 34, I-90123, Palermo, Italy
View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector Camillo Trapani
Istituto di Fisica dell’Uni¨ersita,` Via Archirafi 36, I-90123, Palermo, Italy
Submitted by Joseph A. Ball
Received April 8, 1996
A self-adjoint operator A in L2 Ž.⍀, defines in a natural way a space of X test functions SAŽ.⍀ and a corresponding space of distributions SA Ž.⍀ . These are considered as quasi *-algebras and the problem of multiplying distributions is studied in terms of multiplication operators defined on a rigged Hilbert space. ᮊ 1997 Academic Press
1. INTRODUCTION
Partial algebraic structuresŽ e.g., linear spaces with a non-everywhere defined multiplication. have received a certain attention in the last decade, mainly because of the fact that a lot of instances arising in quantum physical applications fit in a quite reasonable way into themwx 1᎐7 . But also pure mathematicians are familiar with a series of relevant examples where the multiplication is defined only for certain couples of elements: think of L p-spaces or of spaces of distributions. Actually, the problem of the multiplication of distributions, within the framework of the so-called duality method wx8, Sect. II.5 , is the main topic of this paper. After the celebrated result of L. Schwartzwx 9 on the impossibility of multiplying two Dirac delta measures massed at the same point, several possible approaches to the multiplication of distributions have been proposedŽ seewx 8 and references therein. .
423
0022-247Xr97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 424 RUSSO AND TRAPANI
In this paper we will extend the duality method to a more general situation and the multiplication of distribution is studied in the more abstract setting of composition of continuous linear maps acting in locally convex spaces. As we shall see, partial algebraic structures will play a relevant role in the whole discussion. In Section 2 we introduce distributions associated to aŽ possibly un- bounded. self-adjoint operator A in L2 Ž.Ž⍀, where ⍀ is a locally compact Hausdorff measure space with a positive Borel measure . as ϱ continuous linear functionals on the space SAŽ.⍀ of all C -vectors for A. We then study the structure of the corresponding space of distributions, X denoted with SA Ž.Ž⍀ and called, shortly, A-distributions. . In particular we X give conditions on A for SA Ž.⍀ to be a quasi *-algebra in the following sense.
Let A be a linear space and A0 a *-algebra contained in A. We say that Ais a quasi *-algebra with distinguished *-algebra A00Ž.or, simply, over A ifŽ. i the right and left multiplications of an element of A and an element of A0 are always defined and linear; andŽ. ii an involution * Ž which extends .Ž. the involution of A0 is defined in A with the property AB * s B*A* whenever the multiplication is defined.
A quasi *-algebra Ž.A, A0 is said to have a unit މ if there exists an element މ g A0 such that Aމ s މ A s A, ᭙A g A. A quasi *-algebra Ž.A, A0 is said to be topological if a locally convex topology is defined in A such thatŽ. a the involution is continuous and the multiplications are separately continuous; andŽ. b A0 is dense in Awx . The most typical instance of a topological quasi *-algebra is provided by the completion of a topological *-algebra whose multiplication is not jointly continuous Ž L p-spaces are examples of this kind, since they can be viewed as the completion of the space of continuous functions, with compact supportwx 10. . In Subsection 2.1, we show that the space S ЈŽޒ n. is a topological quasi *-algebra over S Žޒ n. with respect to the natural multiplication and then, in Subsection 2.2 we extend, under certain conditions for A, this result to spaces of A-distributions. X In the case where Ž SAAŽ.⍀ , S Ž..⍀ is a quasi *-algebra one can consider XŽ. Ž. XŽ. each ⌽ g SA ⍀ as a multiplication operator L⌽ from SAA⍀ into S ⍀ . We start from this idea to reformulate in a different way the problem of multiplying distributions: we just look for conditions under which the product of the corresponding multiplication operators exists and is still an operator of multiplication by a distribution. In order to do this, we first discuss, more abstractly, the problem of the multiplication in spaces of continuous linear maps acting in rigged Hilbert spaces. To be clearer we give some basic definitions. MULTIPLICATION OF DISTRIBUTIONS 425
Let D be a dense subspace of Hilbert space H. Let us endow D with a locally convex topology t, stronger than that induced on D by the Hilbert norm and let DЈwxtЈ be its topological conjugate dual endowed with the strong dual topology tЈ defined by the set of seminorms
⌽¬55⌽M[sup Ž.⌽, ,1Ž. gM where M runs in the family of bounded subsets of Dwxt . In this way we get the familiar triplet
D ; H ; DЈ called a rigged Hilbert space. Given a rigged Hilbert space, D ; H ; DЈ, we denote with L Ž.D, DЈ the set of all continuous linear maps from Dwxt into DЈ wtЈ x. The space † LŽ.D,DЈcarries a natural involution A ª A defined by
† Ž.Af,g sŽ.Ag, f ᭙f , g g D.
Furthermore, we denote by L qŽ.D the *-algebra of all closable operators in H with the properties DŽ.A s D, D ŽA* .= D and both A and A* leave D invariantŽ. * denotes here the usual Hilbert adjoint . The involu- qŽ. q qŽ. tion in L D is defined by A ª A s A*uD . The space L D is not, in general a subset of L Ž.D, DЈ but, for instance, when t is the so-called graph-topology wx11 defined by L qŽ.D on D then L qŽ.D ; L ŽD, DЈ . † and ŽŽL D, DЈ ., L qŽ..D is a quasi *-algebra. In this case, A s Aq ᭙AgLqŽ.ŽDfor this reason we denote withq both involutions. . As mentioned above we are interested in refining the multiplication in LŽ.D,DЈto allow that a larger number of pairs of elements can be multiplied. This is done by factorizing the operators through some inter- mediate spaces between D and DЈ which we call interspaces. In this way, under certain conditions on the family of interspaces, L Ž.D, DЈ becomes apartial *-algebra wx4. A partial *-algebra is a vector space A with involution A ª A*w i.e., Ž.AqB*sA*qB*; A s A**x and a subset ⌫ ; A = A such thatŽ. i Ž.A,Bg⌫implies ŽB*, A* .g ⌫;ii Ž.Ž.Ž.A,Band A, C g ⌫ and g ރ imply Ž.A, B q C g ⌫; and Ž.Ž. iii if A, B g ⌫, then there exists an element AB g A and for this multiplication the distributive property holds in the following sense: if Ž.A, B g ⌫ and Ž.A, C g ⌫ then
AB q AC s ABŽ.qC.
Furthermore Ž.AB * s B*A*. The product is not required to be associative. 426 RUSSO AND TRAPANI
The partial *-algebra A is said to have a unit if there exists an element މ Ž.Ž.necessarily unique such that މ* s މ, މ, A g ⌫, މ A s Aމ s A, ᭙A g A. If Ž.A, B g ⌫ then we say that A is a left multiplier of B wand write AgLBŽ.xwor B is a right multiplier of ABgRAŽ.x. For S ; A we put LSsFÄ LAŽ.:AgS4; the set R S is defined in an analogous way. The set M S s L S l R S is called the set of universal multipliers of S. Coming back to the multiplication of distributions, the outcome of Section 3 is that, even when two distributions can be multiplied, their product is not, in general a distribution, but a more general objectŽ a necessary and sufficient condition is given in the case of tempered distri- butions. . Finally we show that there is no way, also in this set-up, to define the square of a Dirac ␦-measure, according to the already quoted Schwartz’s result.
2. QUASI *-ALGEBRAS OF DISTRIBUTIONS
Let Ž.⍀, be a measure space with a positive Borel measure on the locally compact Hausdorff space ⍀. When ⍀ is an open subset of ޒ n we take as the Lebesgue measure. With L2 Ž.⍀, we denote, as usual, the space of all measurable functions f on ⍀ such that
2 2 55f2sHfxŽ. d-ϱ ⍀ modulo the subspace of all -almost everywhere zero functions. The Hilbert space structure of L2 Ž.⍀, , defined by scalar product
Žf , g .s Hfxgxd Ž.Ž., ⍀ will play a fundamental role in what follows. To begin with, let us recall that given a self-adjoint operator A in Hilbert space H it is possible to construct a chain of Hilbert spaces. More precisely, let DŽ.A denote the domain of A. Then DŽ.A can be made into a Hilbert space defining the scalar productŽ. ,1 by
Ž.Ž.Žf , g 1s f , g q Af , Ag ., ᭙f , g g D Ž.A .
The completeness of DŽ.A with respect to the norm defined by this new scalar product is ensured by the closedness of A. The Hilbert space obtained in this way will be denoted by H11. The identity map i: f g H ¬ f gHis then a continuous embedding with dense range. MULTIPLICATION OF DISTRIBUTIONS 427
Let now Hy11denote the conjugate dual of H with respect to the scalar product of H; i.e., H1 consists of all continuous conjugate linear functionals on H1.
An element F g Hy1 satisfies the inequality
FfŽ.F55Ffy11 55,᭙fgH1, 55 <Ž.< where, as usual F y1s sup5f5F1 Ff. By Riesz’s lemma, there exists a unique element F g H1 such that Ž. Ž . Ffs f,F 1. This correspondence defines a surjective isometric opera- tor from Hy11into H . This makes of Hy1a Hilbert space too. If g g H, we can identify g with the continuous conjugate linear Ž. Ž.Ž. Ž .᭙ functional jg gHy11defined by jg f s g,f, fgH. As is known, j is one-to-one and linear; therefore H can be identified with a subspace of
Hy1. So, in conclusion,
H1 ¨ H ¨ Hy1 , where ¨ denotes a continuous embedding. The same construction can be made starting from any power An of A. All these powers are indeed self-adjoint operators in H. This construction leads, at the end, to the chain of Hilbert spaces иии иии H21; H ; H s H 0; Hy1; Hy2, n where Hn denotes the Hilbert space obtained defining on DŽA .the Ž. Ž.Žnn.Ž᭙ n. scalar product f, g n s f, g q Af,Ag, f,ggDA and Hyn its conjugate dual. The space ϱ ϱ n D Ž.A s FD ŽA . ns0 plays an important role. First, it is dense in H Ž.and in any other of the Hn’s and is a Frechet` space if endowed with the projective topology defined by Ä4 all spaces Hnngޚ. This topology, denoted as tA, can be, for instance, described by the seminorms
ϱ q 2 nr2 gDŽ.A¬55n[ ŽމqA . , ngގ.
This family of seminorms is non-decreasing, i.e., 55 nnF 55 q1, ᭙ g DϱŽ.Aand is equivalent to the family of seminorms
ϱ n gDŽ.A¬55n[ 5A5, ngގ which is non-decreasing when A G މ. 428 RUSSO AND TRAPANI
The conjugate dual, D ϱϱŽ.A ,of D Ž.A is endowed with the strong dual X y topology, tA which can be viewed as the inductive limit of the spaces Ä4 Hnngޚ. In this case, the strong dual topology can be defined, instead of the set of seminormsŽ. 1 , by the equivalent family of seminorms
fAŽ.⌽, fgF,2Ž. where F is the class of all the positive bounded and continuous functions Ž. ޒ fx on q, which decrease faster than any inverse power of x, such that
k sup x fxŽ.-ϱ, ᭙kgގ. ޒ xgq In what follows we will investigate chains of the type described above 2 when H s L Ž.⍀, . 2.1. S ЈŽޒ n. as a Quasi *-Algebra. Before going forth we discuss a n 2 n n special example. Let S Žޒ . ; L Žޒ .Ž; S Ј ޒ .be the rigged Hilbert space constituted by Schwartz’s space of a rapidly decreasing Cϱ-function, by the space of square integrable functions on ޒ n, and by the space of n ϱ temperedŽ. conjugate distributions. As shown inwx 12 , S Žޒ .s D Ž.J ϱ n where J is the closure of the operator J000defined on C Žޒ . by Jf 1Ž2.Ž.ϱn s2 < kl n 55k,lssup n 55 nkkF C 5555, ᭙,gSŽ.ޒ,3 Ž. since J G މ. n n PROPOSITION 2.1. Let S ЈŽޒ .Žbe the conjugate . dual of S Žޒ .. If we n define the multiplication of an element ⌽ g S ЈŽޒ . and an element g MULTIPLICATION OF DISTRIBUTIONS 429 S Žޒ n. by n Ž.Ž.Ž.⌽, s ⌽, [ ⌽, * , ᭙ g S Ž.ޒ , n X n then ŽŽS Ј ޒ .wtJ x, S Žޒ .. is a topological quasi *-algebra. n n Proof. First, we prove that ⌽ g S ЈŽޒ .Ž, ᭙ g S ޒ .. Indeed, using the continuity of ⌽ andŽ. 3 , we can find a bounded set M, n g ގ and positive constants C12, C such that Ž.Ž.⌽, s⌽,* FC1555⌽M* 5n n FC25555kk, ᭙,gSŽ.ޒ. Therefore, ⌽ is a continuous linear functional on S Žޒ n.. Now, we define an involutionq in S ЈŽޒ n. which extends the involution of S Žޒ n.. This can be done by setting n n ⌽ g S ЈŽ.ޒ ¬ ⌽qg S Ј Ž.ޒ , where n Ž.Ž.⌽q, [ ⌽, *, ᭙gS Ž.ޒ . The involution defined in this way, satisfies the equality q nn Ž.⌽sqq⌽,᭙⌽gSЈŽ.ޒ , ᭙gS Ž.ޒ . Furthermore, it is continuous, since, if M is a bounded subset of S Žޒ n., there exists C ) 0 and n g ގ such that qq 55⌽Mssup Ž.⌽ , s sup Ž.⌽, * F C55 n. gM gM For each fixed in S Žޒ n., the map n n ⌽ g S ЈŽ.ޒ ¬ ⌽ g S Ј Ž.ޒ is continuous. Indeed, let M be a bounded subset of S Žޒ n.. We have 55⌽Ms 55⌽M. The set M is still bounded in S Žޒ n.since it is the continuous image of a bounded set. Finally, it is well known that as a consequence of the reflexivity of SŽ.ޒnn, this latter is dense in S ЈŽ.ޒ wx13 . 2.2. A-Distributions. All these facts suggest considering the space DϱŽ.Aas a kind of space of test functions and itsŽ. conjugate dual as a space of distributions. Let A be self-adjoint in L2 Ž.⍀, . The space 430 RUSSO AND TRAPANI ϱ DŽ.A, endowed with the topology tAA, is denoted by S Ž.⍀ . From the Žn.Žn. previous example it follows that SJ ޒ s S ޒ the usual Schwartz space ϱ of rapidly decreasing C -functions. As shown above, SAŽ.⍀ and its conju- X gate dual SA Ž.⍀ are the extreme spaces of a chain of Hilbert spaces Ä4 Žn. Hnngޚ with Hns D A . For this special chain we adopt a different nŽ. notation; we put HnAs H ⍀ , n g ޚ. It is worth mentioning that in the nn case of the Example of Subsection 2.1, the corresponding spaces HJ Žޒ . are nothing but the well known global Sobelev type spacesŽ see, e.g.,wx 14. . We will now discuss the problem of defining a multiplication in the rigged Hilbert space 2 X SAAŽ.⍀ ; L Ž⍀, .; S Ž.⍀ . X We refer to elements of SA Ž.⍀ as A-distributions. ϱŽ. Ž. Ž In general, D A s SA ⍀ is not *-invariant i.e., invariant under conjugation.Ž . For this to happen we need first that if f g DA.then f*gDAŽ.with f * Ž.x s fxŽ..If Af * s "Ž.Af *, this property is true for the powers An of A, too. From now on, we assume that A always commutes with the conjugation. This fact also implies that the map Ž. Ž. fgSAA⍀¬f*gS⍀is continuous. Remark 2.2. The space DϱŽ.A can be constructed also under the lighter assumption that A is a closed and symmetric operator. Even in this case, in fact DϱŽ.A is a dense domain in Hilbert space and is left invariant by all powers of A wx11 . But, as is known, the assumption that A commutes with the conjugation forces it to have a self-adjoint extensionwx 15, No. 123 . For this reason we will confine ourselves to the case of self-adjoint A. For the reasons discussed above, from now on we will always consider self-adjoint operators in L2 Ž.⍀, with the following properties: Ž.iiffgDA Ž.then f * g DA Ž.and Af * s " ŽAf .*; Ž.ii A G މ. ϱ ϱ There is no loss of generality in the assumptionŽ. ii since D Ž.A s D ŽŽމ 2 1 2 q A . r .. We will now study the algebraic structure of SAŽ.⍀ . First, it is easy to prove that SAŽ.⍀ is a partial *-algebra considering ⌫ s Ä4Žf , g .g SAA Ž.⍀ : fg g S Ž.⍀ . ŽXŽn.Žn.. As we saw, for A s J, SJJޒ , S ޒ is a topological quasi *-algebra. X What can we say about ŽSAAŽ.⍀ , S Ž..⍀ ? First of all we define an XŽ.⍀ Ž. involution on SA s Dyϱ A . MULTIPLICATION OF DISTRIBUTIONS 431 PROPOSITION 2.3. Let us define q X q X : F g SAAŽ.⍀ ¬ F g S Ž.⍀ withŽ. Fq, f s Ž.F, f *.Then Ž. q XŽ. i FgSA⍀. Ž.ii qis continuous. Proof. Ž.i Since F is continuous, there exists C ) 0 and n g ގ such that q n Ž.Ž.Ž.F,fsF,f*sF,f*FCAf5555*sCfn since F is continuous. Ž.ii Let sup Ž.Fq, s sup Ž.F, * s sup Ž.F, gM gM gM* where M * is bounded, since so M is. Let M SAAŽ.⍀ denote the following subset of S Ž.⍀ : MSAAŽ.⍀sÄ4fgS Ž.Ž⍀:f,g .g⌫,᭙ggSAŽ.⍀. Ž. Ž. It is easy to see that if f g M SAA⍀ , then f * g M S ⍀ and as a simple consequence of the associativity of theŽ. ordinary multiplication of func- tions, it turns out that M SAŽ.⍀ is a *-algebra. XŽ. Ž. Now let F g SAA⍀ and f g M S ⍀ . Then we can define a conjugate linear functional Ff on SAŽ.⍀ by the formula Ž.Ž.Ff , g s F, f *g XŽ. and whenever Ff g SA ⍀ we put fF [ Ff and call it the product of F and f. Ž. Ž. PROPOSITION 2.4. If the map g g SAA⍀ ¬ fg g S ⍀ is continuous for Ž. XŽ. Ž XŽ. Ž.. each f g M SAAA⍀ , then Ff g S ⍀ and thus S ⍀ , M SA⍀ is a quasi *-algebra. Proof. By the continuity of F, there exists C ) 0 and n g ގ such that n Ž.ŽFf , g s F, f * g .F CA Ž. f*g . Ž. Ž. Ž. If the map g g SAAA⍀ ¬ fg g S ⍀ , f g M S ⍀ , is continuous in Ž. SA ⍀, then there exists CЈ ) 0 and l g ގ such that nl AfŽ.*g FCЈ55Ag 432 RUSSO AND TRAPANI and thus finally, for a suitable CЉ ) 0 l Ž.Ff , g F CЉ 55Ag . Ž. Ž. Ž. PROPOSITION 2.5. If g g SAAA⍀ ¬ fg g S ⍀ , f g M S ⍀ , is contin- uous, then the map X X F g SAAŽ.⍀ ¬ Ff g S Ž.⍀ , f g M S AŽ.⍀ , X X is continuous. Therefore, if M SAAŽ.⍀ is dense in S Ž.⍀ wtAx, then XX ŽSAAŽ.⍀wtx,MS AŽ..⍀ is a topological quasi *-algebra. Proof. Let N be bounded in SAŽ.⍀ ; we get sup Ž.Ff , g s sup ŽF, f *g .s sup Ž.F, . ggN ggN gf*N Now f *N is bounded, since g ¬ f *g is continuous. In many situations the continuity condition given in the above two propositions is automatically fulfilled. n PROPOSITION 2.6. If ⍀ is an open subset of ޒ with finite Lebesgue Ž. Ž. measure <<⍀ , then the map g g SAA⍀ ¬ fg g S ⍀ is continuous for each Ž. fgMSA⍀. Ž. Ž. Proof. We will show that the map g g SAA⍀ ¬ fg g S ⍀ is closed Ž. for each f g M SA ⍀ . The statement then follows from the closed graph theoremŽ. for Frechet` spaces . Ž. Ž. Let gnAª g in S ⍀ and fg nAª h in S ⍀ . Then necessarily, 5gny g50 and so it is possible to find a subsequence Ä4g which converges 2 ª n k to g almost everywhere. Therefore fg fg almost everywhere and, since n k ª <<⍀-ϱ, also in measure. Now since also 5fgnny h 52 ª 0, fg ª h in measure. Hence fg s h almost everywhere. We will now revisit some well-known examples in the light of the above discussion. 2 Ž. Ž.2Ž. EXAMPLE 2.7. If A is bounded in L ⍀, , then SA ⍀ s L ⍀, Ž. ϱŽ.2Ž. and M SA ⍀ s L ⍀, l L ⍀, wx10 . In this case, as is clear, the 2 topology tA coincides with the norm topology of L Ž.⍀, and therefore XŽ. 2Ž.Ž . SA ⍀sL⍀,up to an isomorphism . This example shows that the unboundedness of A is crucial, in order to get significant spaces of A-distributions. MULTIPLICATION OF DISTRIBUTIONS 433 2 EXAMPLE 2.8. Let ⍀ s ޒ; as is known, any f g L Ž.ޒ admits a deriva- tive Df, in the sense of distributions, and in general Df g DЈŽ.ޒ . We put 2 2 DŽ.P s Ä4f g L Ž.ޒ : Df g L Ž.ޒ . By the definition itself DŽ.P coincides with the Sobolev space W 1, 2 Ž.ޒ . For f g DŽ.P we define, as usual, Pf syiDf. Then P is a self-adjoint operatorwx 16, Chap. VII, Sect. 34 in L2Ž.ޒ with domain W 1, 2 Ž.ޒ . We will now describe the space S Ž.ޒ D ŽP k .W k ,2 Ž.ޒ . Ps FFD s kG1 kG1 As a consequence of Sobolev’s lemmawx 17, Vol. II, Theorem IX.24 , any Ž. ϱ fgSP ޒis a C -function. But something more can be said making use of the Fourier transform. Let us consider 2 2 DŽ.Q s Ä4f g L Ž.ޒ : xf g L Ž.⍀ and define Ž.Ž.Qf x s xf Ž. x , ᭙f g D Ž.Q . 2 2 As is easy to see Q is self-adjoint. Now, if U: L Ž.ޒ ¬ L Ž.ޒ denotes the 2 2 operator which maps f g L Ž.ޒ into its Fourier transform fˆg L Ž.ޒ ,we have, from well-known properties of the Fourier transform k 2 k 2 k U DŽ.Q s Ä4f g L Ž.ޒ : D f g L Ž.ޒ s D ŽP . and k k k Ž.U*P UfsQf, ᭙fgDŽ.Q . Therefore k 2 k ,2 DŽ.Q sÄ4fgL Ž.⍀ : fˆgW Ž.ޒ and ϱ k,2 D Ž.Q s UWž/F Ž.ޒ . kG1 Because of these properties it is enough to consider only DϱŽ.Q to get ϱ information also on D Ž.P and conversely. First we notice that, if f g DϱŽ.Q, then a simple application of the Schwarz inequality implies that 434 RUSSO AND TRAPANI 1Ž. Ž. fgLޒand so fˆg Cϱ ޒ , the space of continuous functions vanishing at Ž.ϱ Ž. infinity. Thus fˆg Cϱ ޒ l C ޒ . In a similar way, one can prove that k Ž.ϱ Ž. DfˆgCϱ ޒlC ޒfor any k g ގ. Now, it is knownwx 18, Corollary VIII.9 that W 1, 2 Ž.ޒ is an algebra. The k,2 same, of course, holds true for W Ž.ޒ for any k g ގ. In conclusion, SP Ž.ޒ is an algebra. By taking Fourier transforms it follows also that Ž. SQ ޒis a con¨olution algebra. It is not, however, an algebra with respect to the ordinary multiplication of functions. Indeed, it is easy to see that the function 1 ¡ , x g Ž.0,1 fxŽ.s~'x ¢0, elsewhere Ž. 2 Ž. is in SQQޒ but f f S ޒ . Remark 2.9. It is worth mentioning that the construction outlined in this section allows us to get spaces of A-distributions whose elements are not distributions in the usual sense. Indeed, starting from any self-adjoint operator A in L2 Ž.⍀, , we can define, for any ) 1, a new operator ϱ Ak 0 A s Ý ks0Ž.k! on the dense set DAŽ.of all analytic vectors for A. Let A be the 0 self-adjoint operator obtained by closing A . Ž. A straightforward estimation shows that if f g DA then there exists a constant C ) 0 such that k 55Af FCkŽ.!.Ž. 4 To be more definite, let us now consider the case where ⍀ s ޒ and A s J as in Subsection 2.1, or A s P s iD as in Example 2.8 and construct the corresponding spaces S Ž.ޒ , S XŽ.ޒ . As well as S Ž.ޒ S Ž.ޒ S Ž.ޒ AA Js ; P one finds S Ž.ޒ S Ž.ޒ and therefore S XŽ.ޒ S XŽ.ޒ . Applying Ž. 4 for JP; PJ; AsPsiD, we get the existence of a constant C ) 0 such that k 55Df FCkŽ.!, ᭙fgDPŽ.. For this reason each f S Ž.ޒ can be viewed as an element of a modified g J Gevrey type spaceŽ see, e.g.,wx 19 .GmodŽ.ޒ consisting of all functions ϱ r,2Ž.ޒ fgFrs0 W for which there exists a constant C ) 0 such that kk1 55Df FCkqŽ.! Žwe note that the definition of the Gevrey spaces makes use on the l.h.s. of the Lϱ-norm instead of the L2-one. . The corresponding space of -ultra- MULTIPLICATION OF DISTRIBUTIONS 435 distributions is then a subset of S XŽ.ޒ . We hope to discuss in more detail J examples of this kind in a further paper. 3. MULTIPLICATION OF DISTRIBUTIONS Throughout this section we will consider only spaces of A-distributions for which the assumptions of Proposition 2.5 are fulfilled. Accordingly, X Ž SAAŽ.⍀,MS Ž..⍀ is a Ž topological . quasi *-algebra and each element F X of SAFŽ.⍀ can be viewed as an operator L of multiplication on M SAŽ.⍀ defined by X LFA: g M S Ž.⍀ ª F g S A Ž.⍀ . X This is a continuous linear map of M SAAŽ.⍀ into S Ž.⍀ . Indeed, if M is bounded in SAAwxt , by the continuity of F, there exists C ) 0 and n g ގ such that sup Ž.F, s sup ŽF, * .F C sup 55* n. gM gM gM Then, making use of the continuity of the multiplication, we can find a new constant CЈ ) 0 and m g ގ such that sup Ž.F, F CЈ55* mms CЈ 55 . gM Ž. Ž. Ž. XŽ. Moreover, if M SAA⍀иS ⍀is dense in SA⍀ , the map j: F g SA⍀ ŽŽ.XŽ.. ªLFAAgLMS⍀,S⍀is injective; indeed, LFFs 0 implies L s 0, Ž. ᭙gMSA ⍀and so Ž.Ž.F, s F, * s 0, ᭙ g MSAAŽ.Ž.⍀ , g S ⍀ . At this point, if M SAAŽ.⍀иS Ž.⍀ is dense in SAŽ.Ž⍀ or if <<⍀ - ϱ, the Ž. Ž .. function u, with uxs1, ᭙ x g ⍀, belongs to SA ⍀ we get F s 0. XŽ. Clearly, for F, G g SA ⍀ and g ރ we have LFqGFGs L q L , LFFFFs L , Ž.L * s L *. Therefore, we can get information about multiplication of two distribu- tions F, G from the multiplication of the corresponding operators LF and X LGA. Then, the problem is referred to L ŽŽ.M S ⍀ , SAŽ..⍀ . It is worth mentioning that if M SAAŽ.⍀ is dense in S Ž.⍀ then X LŽŽ.MSAA⍀,SŽ..⍀ can be identified, in a natural way, with X LŽŽ.SAA⍀,SŽ..⍀. 436 RUSSO AND TRAPANI The problem of the multiplication of operators acting in a rigged Hilbert space D ; H ; DЈ has been often considered in the literaturewx 6, 20, 21 . We will now discuss the possibility of refining the lattice of multipliers of L Ž.ŽD, DЈ it con- sists, up to now, of two elements only: L Ž.D, DЈ and L qŽ..D in order to allow the possibility that a larger number of pairs of elements in L Ž.D, DЈ may be multiplied. This problem of refinement of the multiplication in quasi *-algebras has been discussed from a general point of view inwx 21 where the notion of multiplication framework was introduced and in the case of L Ž.D, DЈ inwx 22 . In both cases one has to consider refinements of rigged Hilbert spaces in the sense ofwx 23 . In what follows we reformulate in the language ofwx 21 some results of wx 22 . Let E be a locally convex space satisfying D ¨ E ¨ DЈ,5Ž. where, as before, ¨ denotes continuous embeddings with dense range. In the language ofwx 21 , a subspace of DЈ satisfyingŽ. 5 was called an interspace. We will maintain this terminology. PROPOSITION 3.1. Let E be an interspace and EЈ its dual. If E ¨ EЈ then E ; H. Proof. By the assumption, E is a non-degenerate pre-Hilbert space with respect to the form defining the duality and extending the scalar product of D. Since for each g g E, the linear functional f ¬ Ž.f, g is Ä4 continuous on E and from the embedding E ¨ EЈ,if p␣ is a directed family of seminorms defining the topology of E we get Ž.f , g F Cp␣ Ž.Ž. f p␣ g , ᭙f , g g E. From this it follows that the norm-topology on E defined by the scalar product is weaker than the initial topology of E. Let E h denote the conjugate dual of E with respect to the norm. Then we have D ¨ E ¨ E h ¨ EЈ ¨ DЈ. Dis norm-dense in E. Then D and E have the sameŽ. up to isomorphism completion. So, in conclusion, E ; H , E h. MULTIPLICATION OF DISTRIBUTIONS 437 Let E, F be interspaces. We denote with L Ž.E, F the space of all continuous linear maps from E into F and define CŽ.E,FsÄ4XgL ŽD,DЈ .:XsX˜˜Dfor some X g L Ž.E , F . DEFINITION 3.2. The product X и Y of two elements of L Ž.D, DЈ is defined in L Ž.D, DЈ if there are three interspaces E, F, G such that XgCŽ.F,Gand Y g C Ž.E, F . In this case the multiplication X и Y is defined by X и Y s Ž.XY˜˜ u D or, equivalently, by X и Yf s XYf˜ , f g D, where the X˜˜Ž.resp., Y denote the extension of X Ž.Žresp., Y to E resp., F .. This definition, however, may depend on the particular choice of the interspaces E, F wx22 . We say that a rigged Hilbert space D ¨ H ¨ DЈ is of regular type if the topology t of D is the projective limit of the locally convex topologies of a family L of interspaces. In this case, the definition of multiplication of a pair X, Y of elements of L Ž.D, DЈ can be simplified as follows as it can be shown by using easy duality properties. PROPOSITION 3.3. Let D ¨ H ¨ DЈ be a rigged Hilbert space of regular type and X, Y g L Ž.D, DЈ . Then the product X и Y is well-defined if, and only if, there exists an interspace F such that Y: D ª F continuously and Xq: D ª F Ј continuously. We will now show that the above proposition allows us to get partly two well-known results on the multiplication of distributions. n ϱ n EXAMPLE 3.4. Let OM denote, as usual, the set of C -functions on ޒ which together with their derivatives are polynomially boundedŽ see, e.g., wx17, Vol. I, Sect. V.3.Ž . Then it is well known that u⌽ exists in S Ј ޒ n.for n Ž n. each u g OM and for each ⌽ g S Ј ޒ . This fact can partly be obtained from Proposition 3.3 in the following way. First, observe that the product n n uwith g S Žޒ .Žalways exists in S ޒ .and that the map n n Lu: g S Ž.ޒ ª u g S Ž.ޒ is continuous. Furthermore, as shown in Subsection 2.1, for any ⌽ g n nn SЈŽޒ.Ž, the map L⌽ is continuous from S ޒ .Žinto S Ј ޒ .. Therefore Ž n. Proposition 3.3 applies with F s S ޒ and so L⌽ и Lu is well-defined in n n L ŽŽS ޒ .Ž, S Ј ޒ ..; however, Proposition 3.3 does not imply that L⌽ и Lu s L⌽ u. 438 RUSSO AND TRAPANI k,2 EXAMPLE 3.5. As is known, if u, ¨ g W Ž.ޒ , k g ގ, then u¨ g k,2 n W Ž.ޒ. Also the fact that u and ¨ are multiplicable elements of S ЈŽޒ . k,2 can be obtained from Proposition 3.3. Indeed, if ¨ g W Ž.ޒ then the Žޒ n. corresponding multiplication operator L¨ maps continuously S into k,2Ž. qqk,2Ž. W ޒ. Now since Luus L *the same holds true for Lu. But W ޒ ; Wyk,2Ž.Žޒ up to isomorphisms. ; therefore Proposition 3.3 applies and и Lu L¨ is then well-defined, in the sense discussed above. Also in this case и we cannot conclude directly that Lu L¨ s Lu¨ . To get this, as we shall see below, we need further information. In order to make of L Ž.D, DЈ a partial *-algebra with respect to the multiplication defined above we need confine ourselves to families of interspaces that make possible the product to be independent of the choice of the interspaces used to factorize it. In order to get this the conditionŽ. iii below is essentialwx 22, Proposition 3.2 . A family L of interspaces is called a multiplication framework wx21 if Ž.i DgL Ž.ii ᭙E g L , its dual EЈ also belongs to L Ž.iii ᭙E, F g L , E l F g L and D is dense in E l F with the projective topology. For instance, if L0 is a chain of interspaces, the chain L , consisting of D, DЈ, the elements of L0 and their duals, is a multiplication framework. PROPOSITION 3.6. Let L be a multiplication framework in L Ž.D, DЈ Ž.Ž.Ž. and let ⌫L be the set of pairs X, Y g L D, DЈ = L D, DЈ such that there exist E, F, G g L such that X g CŽ.F, G and Y g C Ž.E, F . Then ŽŽ . . LD,DЈ,⌫L is a partial *-algebra. X We now apply these ideas to the case of L ŽŽ.M SAA⍀ , S Ž..⍀ . First we need to choose the multiplication framework L. A natural n choice is, clearly, to take it as the chain HA Ž.⍀ defined in Subsection 2.2; r yr in this case L ŽHAAŽ.⍀ , H Ž..⍀ is a Banach space with respect to its natural norm 55Xr , yr ssup 5X 5yr . 55rF1 XŽ. DEFINITION 3.7. Let ⌽ g SA ⍀ and l, m g ޚ. We say that ⌽ is of Ž. ŽlŽ. mŽ.. type l, m if L⌽ g C HAA⍀ , H ⍀ . If r g ގ, we say that the A-distribution ⌽ is of order r if L⌽ g Ž r Ž. yrŽ.. Ž sŽ. ysŽ.. CHAA⍀,H ⍀ but L⌽f C HAA⍀ , H ⍀ for s - r. X We denote with SA, rŽ.⍀ the set of all A-distributions of order r. X Ž. XŽ. Clearly SA, rA⍀ is a linear space and if ⌽ g S ,r⍀ , then also ⌽* g X SA,rŽ.⍀. MULTIPLICATION OF DISTRIBUTIONS 439 X PROPOSITION 3.8. If M SAAŽ.⍀иS Ž.⍀ is dense in SAA Ž.⍀ , then S ,rŽ.⍀ is a Banach space with respect to the norm X 55⌽rrs 5L⌽ 5,yr, ᭙⌽gSA,rŽ.⍀ .6Ž. XŽ. Ž rŽ. yrŽ.. Proof. The map ⌽ g SA, r ⍀ ª L⌽ g C HAA⍀ , H ⍀ is linear and Ž.Ž..Ž. injective; indeed, if L⌽⌽s 0 then L , s ⌽ , s ⌽, * s 0, Ž. Ž. Ž. ᭙gMSAA⍀,᭙gS⍀; by the hypothesis, we have ⌽ s 0. Thus 6 X defines actually a norm. It remains to prove that SA, rŽ.⍀ is complete. Let X ⌽nAbe a Cauchy sequence in S ,rŽ.⍀ , i.e., 55⌽nmy⌽rª0.Ž. 7 But 5555⌽ ⌽ L L and since L ŽH rŽ.⍀ , H yrŽ..⍀ is a nmy rrs ⌽⌽nmy ,yr AA Banach space, there exists L L ŽH rŽ.⍀ , H yrŽ..⍀ such that L L. g AA ⌽nª Since A G މ we can write yr yr 55L⌽⌽r,yrs 5ALA5 and then, for f as inŽ. 2 , we get, for a certain C ) 0, r fAŽ.⌽FCA55y ⌽. X Then, the topology of SA, rŽ.⍀ , defined byŽ. 6 is stronger than the X topology induced by the strong dual topology of SA Ž.⍀ . Then Ž. 7 implies XŽ. XŽ. the convergence in SAA⍀ , that is, there exists ⌿ g S ⍀ such that ŽŽ.ryr Ž.. XŽ. LsL⌿ . Since L g L HAA⍀ , H ⍀ then ⌿ g S A,r⍀ . The following lemma is a particular case of a statement shown inw 6, Lemma 5.2x . ŽŽ. XŽ.. LEMMA 3.9. Let L g L SAA⍀ , S ⍀ . Then there exists r g ގ such ŽrŽ. yrŽ.. that L g L HAA⍀ , H ⍀ . From this statement we get immediately the following generalization of a well known property of tempered distributionsŽ see, e.g.,w 24, Corollary 1, Sect. 5.2x. . XŽ. PROPOSITION 3.10. Each ⌽ g SA ⍀ is of finite order. Taking into account the discussion developed up to now, we get XŽ. PROPOSITION 3.11. Let ⌽, ⌿ g SA ⍀ . If ⌽ is of order r and ⌿ is of Ž. ŽrŽ. yrŽ.. type l, m with l F r and m G r then L⌽⌿и L exists in L HAA⍀ , H ⍀ . 440 RUSSO AND TRAPANI Of course, this does not guarantee that L⌽⌿и L is also an operator of multiplication by a distribution. In the case where A s J we have ŽŽn.Žn.. PROPOSITION 3.12. Let X g L S ޒ , S Ј ޒ . Then X s LV for some n V g S ЈŽޒ . if, and only if, the following two conditions are fulfilled n Ž.i X Ž .s X , ᭙, g S Žޒ .; Ž.ii there exists C ) 0 and n g ގ such that n Ž.X,FC55*n,᭙,gSŽ.ޒ. Proof. The necessity is obvious, so we will only prove the sufficiency. n n Let Ž.x be a fixed regularizing function in ޒ ŽŽ.i.e., x G 0onޒ; Ž. ϱŽ n. Ä n 4 Ž. . xgC0 ޒ ; supp ; x g ޒ : < n VŽ. s lim ŽX␣⑀, ., g S Žޒ .. ⑀ª0 We have n V⑀⑀Ž.slim ŽX␣ , .F C lim 55␣ nnsC 55 , ᭙gSŽ.ޒ . ⑀ª0⑀ª0 Žn.Ž.Žn. Therefore, V g S Ј ޒ . Now, making use of i , we get, for , g S ޒ , L , V , * lim Ž.Ž.X␣ , * X, . Ž.Vs Ž.⑀s s ⑀ª0 This equation shows that the definition of V is actually independent of and that X is of the desired form. ŽŽ. XŽ..Ž Ž. Remark 3.13. Let X g L SAA⍀ , S ⍀ with M SA⍀ dense in XŽ.. XŽ. SAVA⍀. Then if X s L for some V g S ⍀ the following slight modifi- cations ofŽ. i and Ž ii . of Proposition 3.12 hold: Ž. Ž . Ž . Ž . iЈ X s X , ᭙ g M SAA⍀ , g S ⍀ ; Ž.iiЈ there exists C ) 0 and n g ގ such that Ž.X,FC55*n,᭙gMSAAŽ.⍀,gS Ž.⍀. As for the sufficiency part, the argument used above has no easy general- ization. The existence of a well-behaved net like the ␣⑀ introduced there is still unclear when we consider the more general case. MULTIPLICATION OF DISTRIBUTIONS 441 To end this paper, we will show how the results obtained so far allow us to proveŽ. in a different way the well known result of L. Schwartzwx 9 on the non-existence of ␦ 2. n PROPOSITION 3.14. The distribution ␦ g S ЈŽޒ . cannot be multiplied by itself in the sense of Definition 3.2. Proof. It is easily seen that n L␦ s Ž.0 ␦ , ᭙ g S Ž.ޒ . Ä Ž n. 4 Let M␦ s q ␦; g S ޒ , g ރ be endowed with any topology Ž n. such that u S Žޒ n. is not weaker than the topology of S ޒ . Then M␦ is n the minimal interspace containing L␦ S Žޒ .. By Proposition 3.3, L␦␦и L is n X well-defined if, and only if, L␦ maps also S Žޒ .into M␦ Žthe dual of . X M␦ wx . This implies that M␦␦; M and thus, by Proposition 3.1, M␦; L2Žޒn.. This, in turn, implies that ␦ is a function and this is a contradic- tion. This result, as well as other impossibility results of the same kind, depends strictly on the framework where the problem of multiplying two distributions is considered. Apart from the duality method, several other approaches have been developed in the literatureŽ regularizing procedures, methods of complex or harmonic analysis, etc.,wx 8. . For this reason these impossibility results do not prevent us from trying to develop methods that allow also a definition of ␦ 2 Žfor recent results in this direction seewx 25 and references therein. . ACKNOWLEDGMENTS The authors thank the referee for many valuable suggestions that really improved the paper and, in particular, for having drawn their attention to the example discussed in Remark 2.9. We are also indebted to Professor K. D. Kursten¨ for remarking some inaccuracies in an earlier version of the paper. REFERENCES 1. G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. Karl-Marx-Uni¨. Leipzig Math.-Natur. Reihe 6 Ž.1981 , 572᎐595. 2. G. Lassner, Algebras of unbounded operators and quantum dynamics, Phys. A 124 Ž.1984 , 471᎐480. 3. J.-P. Antoine and W. Karwowski, Partial *-algebras of closed linear operators in Hilbert space, Publ. Res. Inst. Math. Sci. 21 Ž.1985 , 205᎐236. 4. J.-P. Antoine, A. Inoue, and C. Trapani, Partial *-algebras of closable operators. I. The basic theory and the Abelian case, Publ. Res. Inst. Math. Sci. 26 Ž.1990 , 359᎐395. 442 RUSSO AND TRAPANI 5. J.-P. Antoine, A. Inoue, and C. Trapani, Partial *-algebras of closable operators. II. States and representations of partial *-algebras, Publ. Res. Inst. Math. Sci. 27 Ž.1991 , 399᎐430. 6. F. Bagarello and C. Trapani, States and representations of CQ*-algebras, Ann. Inst. H. Poincare´ 61 Ž.1994 , 103᎐133. 7. C. Trapani, Quasi *-algebras of operators and their applications, Re¨. Math. Phys. 7 Ž.1995 , 1303᎐1332. 8. M. Oberguggenberger, Multiplication of distributions and applications to partial differen- tial equations, in ‘‘Pitman Research Notes in Mathematics Series,’’ Vol. 259, Longman, Harlow, 1992. 9. L. Schwartz, Sur l’impossibilite´ de la multiplication des distributions, C. R. Acad. Sci. Paris 239 Ž.1954 , 847᎐848. 10. F. Bagarello and C. Trapani, L p-spaces as Quasi *-algebras, J. Math. Anal. Appl. 197 Ž.1996 , 810᎐824. 11. K. Schmudgen,¨ ‘‘Unbounded Operator Algebras and Representations Theory,’’ Birkhauser,¨ Basel, 1990. 12. R. T. Powers, Selfadjoint algebras of unbounded operators, Comm. Math. Phys. 21 Ž.1971 , 85᎐124. 13. H. H. Schaefer, ‘‘Topological Vector Spaces,’’ Springer-Verlag, New York, 1971. 14. M. Shubin, ‘‘Pseudodifferential Operators and Spectral Theory,’’ Springer-Verlag, BerlinrHeidelberg, 1987. 15. F. Riesz and B. Sz. Nagy, ‘‘Lec¸ons d’Analyse Fonctionelle,’’ Gauthier᎐Villars, Paris, 1972. 16. H. Triebel, ‘‘Hohere¨ Analysis,’’ VEB, Berlin, 1972. 17. M. Reed and B. Simon, ‘‘Methods of Modern Mathematical Physics,’’ Vols. I, II, Academic Press, New York, 1980. 18. H. Brezis, ‘‘Analyse Fonctionelle,’’ Masson, Paris, 1983. 19. L. Rodino, ‘‘Linear Differential Operators in Gevrey Spaces,’’ World Scientific, Singa- pore, 1993. 20. J.-P. Antoine and A. Grossmann, Partial inner product spaces, I, J. Funct. Anal. 23 Ž.1976 , 369᎐378; IIŽ. 1976 , 379᎐391. 21. G. Epifanio and C. Trapani, Partial *-algebras and Volterra convolution of distribution kernels, J. Math. Phys. 32 Ž.1991 , 1096᎐1101. 22. K. D. Kursten,¨ The completion of the maximal Op *-algebra on a Frechet domain, Publ. Res. Inst. Math. Sci. 22 Ž.1986 , 151᎐175. 23. J.-P. Antoine and W. Karwowski, Interpolation theory and refinement of nested Hilbert spaces, J. Math. Phys. 22 Ž.1981 , 2489᎐2496. 24. V. S. Vladimirov, ‘‘Le Distribuzioni in Fisica Matematica,’’ Moscow, 1981. 25. F. Bagarello, Multiplication of distributions in one dimension: Possible approaches and applications to ␦-function and its derivative, J. Math. Anal. Appl. 196 Ž.1995 , 885᎐901.