CARLEMAN INTEGRAL OPERATORS AS MULTIPLICATION OPERATORS AND PERTURBATION THEORY

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1 Kragujevac Joural of Mathematics Volume 41(1) (2017), Pages CARLEMAN INTEGRAL OPERATORS AS MULTIPLICATION OPERATORS AND PERTURBATION THEORY S. M. BAHRI 1 Abstract. I this paper we itroduce a multiplicatio operatio that allows us to give to the Carlema itegral operator of secod class [3, 8] the form of a multiplicatio operator. Also we establish the formaly theory of perturbatio of such operators. 1. Itroductio It is well kow that the multiplicatio operators [1,2] possess a very rich structure theory, such that these operators played a importat role i the study of operators o Hilbert Spaces. I this paper, we itroduce a multiplicatio operatio that allows us to give to the Carlema itegral operator of secod class [3, 8] the form of a multiplicatio operator. I what follows, we shall assume that the reader is familiar with the fudametal results ad the stadard otatio of the Itegral operators theory [8 12]. Let X be a arbitrary set, µ a σ fiite measure o X (µ is defied o a σ algebra of subsets of X, we do t idicate this σ algebra), L 2 (X, µ) the Hilbert space of square itegrable fuctios with respect to µ. Istead of writig µ measurable, µ almost everywhere ad (dµ (x)) we write measurable, a.e. ad dx. A liear operator A : D (A) L 2 (X, µ), where the domai D (A) is a dese liear maifold i L 2 (X, µ), is said to be itegral if there exists a measurable fuctio K o X X, a kerel, such that, for every f D (A), (1.1) Af (x) = K (x, y) f (y) dy a.e. X Key words ad phrases. Carlema kerel, defect idices, itegral operator, formal series Mathematics Subject Classificatio. Primary: 45P05, 47B25. Secodary: 13F25. Received: May 7, Accepted: August 10,

2 72 S. M. BAHRI A kerel K o X X is said to be Carlema if K (x, y) L 2 (X, µ) for almost every fixed x, that is to say K (x, y) 2 dy < a.e. X A itegral operator A with a kerel K is called Carlema operator if K is a Carlema kerel. Every Carlema kerel K defies a Carlema fuctio k from X to L 2 (X, µ) by k (x) = K (x, ) for all x i X for which K (x, ) L 2 (X, µ). Now we cosider the Carlema itegral operator (1.1) of secod class [3,8] geerated by the followig symmetric kerel K (x, y) = a ψ (x) ψ (y), =0 where the overbar deotes the complex cojugatio, (ψ (x)) =0 is a orthoormal sequece i L 2 (X, µ) such that ψ (x) 2 < a.e., =0 ad (a ) =0 is a real umber sequece verifyig a 2 ψ (x) 2 < a.e. =0 We call (ψ (x)) =0 a Carlema sequece. Moreover, we assume that there exists a umeric sequece (γ ) =0 such that (1.2) γ ψ (x) = 0 a.e., ad (1.3) =0 γ a λ =0 2 <. With the coditios (1.2) ad (1.3), the symmetric operator A = (A ) admits the defect idices (1, 1) (see [3 6]), ad its adjoit operator is give by A f (x) = a (f, ψ ) ψ (x), D (A ) = =0 f L 2 (X, µ) : } a (f, ψ ) ψ (x) L 2 (X, µ). Moreover, we have ϕ λ (x) = γ ψ a λ (x) N λ, λ C, λ a, = 1, 2,..., =0 ϕ a (x) = ψ (x), =0

3 CARLEMAN INTEGRAL OPERATORS 73 N λ beig the defect space associated to λ (see [3, 4]). 2. Positio Operator Let ψ = (ψ ) =0 be a fixed Carlema sequece i L2 (X, µ). It is clear from the foregoig that ψ is ot a complete sequece i L 2 (X, µ). We set L ψ the closure of the liear spa of the sequece (ψ (x)) =0 L ψ = spa ψ : N}. We start this sectio by defiig some formaly spaces Formal Elemets. Defiitio 2.1 ([7]). We call formal elemet ay expressio of the form (2.1) f = N a ψ, where the coefficiets a ( N) are scalars. The sequece (a ) is said to geerate the formal elemet f. Defiitio 2.2. We say that f is the zero formal elemet ad we ote f = 0 if a = 0 for all N. We say that two formal elemets f = N a ψ ad g = N b ψ are equal if a = b for all N. If ϕ is a scalar fuctio defied for each a, we set ( ) ϕ a ψ = ϕ (a ) ψ, or i aother form, For example let ϕ (a 1, a 2,..., a,... ) = (ϕ (a 1 ), ϕ (a 2 ),..., ϕ (a ),... ). ϕ (x) = 1 x, x 0. If a 0 for all N, the the formal elemet ( ) ϕ a ψ = 1 a ψ is called iverse of the formal elemetf = a ψ. Furthermore, we defie the cojugate of a formal elemet f by f = a ψ. Deotes by F ψ the set of all formal elemets (2.1). O F ψ, we defie the followig algebraic operatios:

4 74 S. M. BAHRI (a) the sum (b) ad the product + : F ψ F ψ F ψ ( a ψ ) + ( b ψ ) = (a + b ) ψ, : C F ψ F ψ λ ( a ψ ) = (λ a ) ψ. Hece, we obtai a complex vector space structure for F ψ Bouded Formal Elemets. Defiitio 2.3. A formal elemet f = Na ψ is bouded if its sequece (a ) is bouded. We deote by B ψ the set of all bouded formal elemets. It s clear that B ψ is a subspace of F ψ. We claim that: (a) L ψ is a subspace of B ψ. (b) Furthermore we have the strict iclusios: L ψ B ψ F ψ. We defie a liear form, o F ψ by settig (2.2) a ψ, b ψ = a b, with the series covergig o the right side. Propositio 2.1. The form (2.2) verifies the properties of scalar product. Proof. Ideed, let f = a ψ, g = b ψ, f 1 = a 1 ψ ad f 2 = a 2 ψ, i F ψ. We have the: (a) f, g = a b = b a = g, f, (b) ( ) λf, g = λ a ψ, a ψ = (λa ) ψ, = (λa ) b = λ a ψ, b ψ = λ f, g, b ψ

5 CARLEMAN INTEGRAL OPERATORS 75 (c) f 1 + f 2, g = = ( ) a 1 + a 2 ψ, b ψ ( ) a 1 + a 2 b = a 1 b + a 2 b = f 1, g + f 2, g, (d) f, f = a 2 0 ad f, f > 0, if f 0. Remark 2.1. O L ψ, the scalar product.,. coicides with the scalar product (.,.) of L 2 (X, µ) The Multiplicatio Operatio. Here, we itroduce the crucial tool of our work. Defiitio 2.4. We call multiplicatio with respect to the Carlema sequece (ψ ), the operatio deoted ad defied by f g = f, ψ g, ψ ψ = a b ψ, for all (f, g) F 2 ψ. Defiitio 2.5. We call positio operator i L ψ ay uitary selfadjoit operator satisfyig U (f g) = Uf Ug, for all f, g L ψ. The term positio operator comes from the fact (as it will be show i the followig theorem) that for the elemets of the sequece ψ = (ψ ), the operator U acts as operator of chage of positio of these elemets Mai Results. Theorem 2.1. A liear operator defied o L ψ is a positio operator if ad oly if there exists a ivolutio j (i.e. j 2 = Id)of the set N such that for all N (2.3) Uψ = ψ j(). Proof. (a) It is easy to see that if (2.3) holds, the U is a positio operator. (b) Let U be a positio operator. Accordig to Propositio 2.1 we ca write: Uψ = k α,k ψ k, with k α,k 2 = 1 (2.4) sice Uψ L ψ. O the other had, we have α,k ψ k = k k α 2,k ψ k

6 76 S. M. BAHRI (2.5) as Uψ = U (ψ ψ ) = Uψ Uψ. The equalities (2.4) lead to the resolutio of the system: α,k 2 = 1, α,k 2 = α k N.,k, We get the for all N, there exists oly oe k N : Let us ow cosider the followig applicatio j : N N, j () = k. It s clear that j is ijective. Now let m N. Sice U 2 = I, the Hece Remark 2.2. α,k = 1, α,k = 0, U (Uψ m ) = Uψ j(m) = ψ j(j(m)) = ψ m. j(j(m)) = m. Fially j is well defied as ivolutio. for all k k. (a) We emphasize that ivolutio j depeds of the operator U, i.e. j = j U. We the write Uψ = ψ j() = ψ ju () ad ( ) Uf = U a ψ = a ψ j() = f U. (b) The positio operator U ca be exteded over F ψ as follows. If f = a ψ F ψ, the Uf = f U = a ψ ju (). 3. Carlema Operator i F ψ 3.1. Case of Defect Idices (1, 1). Let α = α pψ p F ψ, we itroduce the operator A α defied i L ψ by A α f = α f = α, ψ f, ψ ψ.

7 CARLEMAN INTEGRAL OPERATORS 77 It is clear that A α is a Carlema operator iduced by the kerel K (x, y) = α ψ (x) ψ (y), with domai D( A α ) = f L ψ : α (f, ψ ) 2 < }. Moreover, if α = α, the A α is selfadjoit. Now let Θ = γ ψ F ψ ad Θ / L ψ (i.e., γ 2 = ). We itroduce the followig set (3.1) H Θ = f + µθ : f L ψ, µ C}, which verify the followig properties. Propositio 3.1. (a) H Θ is a subset of F ψ. (b) H θ = L ψ Cθ, i.e. direct sum of L ψ with Cθ = µθ : µ C}. Proof. The first property is easy to establish. We show the uiqueess for the secod. Let g 1 = f 1 + µ 1 θ ad g 2 = f 2 + µ 2 θ two formal elemets i H θ. The g 1 = g 2 f 1 f 2 = (µ 2 µ 1 ) θ. This last equality is verified oly if µ 2 = µ 1. Therefore f 1 = f 2. Deote by Q the projector of H Θ o L ψ, that is to say: if g H Θ, g = f + µθ with f L ψ ad µ C the Qg = f. We defie the operator B α by It is clear that, B α f = Q (α f), f L ψ. D (B α ) = f L ψ : α f H Θ }. Theorem 3.1. B α is a desely defied ad closed operator.

8 78 S. M. BAHRI Proof. (a) Sice spa ψ : N} D (B α ) ad that (ψ ) is complete i L ψ, the D (B α ) = L ψ. (b) Let (f ), be a sequece of elemets i D (B α ). Checkig f f, (covergece i the L 2 sese). B α f g, We have the with The This implies that: B α f = Q (α f ), α f = g + µθ, g L ψ. g = α f µ Θ L ψ, g, ψ m = α m f, ψ m µ γ m ψ m, for all m N. Or, whe teds to, we have: g g ad f f. Therefore, there exist µ C such that: lim µ = µ. Ad as Q is a closed operator, the we ca write Fially f D (B α ) ad g = B α f. α f H Θ ad g = Q (α f). It follows from this theorem that the adjoit operator B α exists ad B α = B α. Let us deote by A α, the operator adjoit of B α A α = B α. I the case α = α, the operator A α is symmetric ad we have the followig results. Theorem 3.2. A α admits defect idices (1, 1) if ad oly if ϕ λ = (α λ) 1 Θ L ψ. I this case ϕ λ N λ (defect space associated with λ, [3]).

9 CARLEMAN INTEGRAL OPERATORS 79 Proof. We kow (see [3]) that A α has the defect idices (1, 1) if ad oly if, its defect subspaces N λ ad N λ are uidimesioal. We have N λ = ker (A α λi) = ker (B α λi). So it suffices to solve the system Bαϕλ = λϕλ, ϕ λ L ψ, i.e., Q (α ϕ λ ) = λϕ λ, ϕ λ L ψ, (α ϕ λ ) = λϕ λ + µθ, µ C, ϕ λ L ψ, (α λ) ϕ λ = Θ, ϕ λ L ψ, ϕ λ = (α λ) 1 Θ, ϕ λ L ψ Case of Defect Idices (m, m). I this sectio we give the geeralizatio for the case of defect idices (m, m), where m > 1. Let Θ 1, Θ 2,..., Θ m, (where m N) formal elemets ot belogig to L ψ ad let } m H Θ = f + µ k Θ k, f L ψ, µ k C, k = 1,..., m. k=1 We cosider the operator B α defied by We assume that α = α ad we set B α f = Q (α f), for f D (B α ), D (B α ) = f L ψ : α f H Θ }. A α = B α. By aalogy to the case of defect idices (1, 1) we also have the followig. Theorem 3.3. The operator B α is desely defied ad closed. Theorem 3.4. The operator A α admits defect idices (m, m) if ad oly if ϕ (k) λ = (α λ) Θ k L ψ, k = 1,..., m. I this case the fuctios ϕ (k) λ (k = 1,..., m) are liearly idepedet ad geerate the defect space N λ.

10 80 S. M. BAHRI Refereces [1] M. B. Abrahamse, Multiplicatio operators, Lecture Notes i Mathematics 693 (1993), [2] N. I. Akhiezer ad I. M. Glazma, Theory of Liear Operators i Hilbert Space, Dover Books o Mathematics, Dover Publicatios, New York, [3] S. M. Bahri, O the extesio of a certai class of Carlema operators, Z. Aal. Awed. 26 (2007), [4] S. M. Bahri, Spectral properties of a certai class of Carlema operators, Arch. Math. (Bro) 43 (2007), [5] S. M. Bahri, O covex hull of orthogoal scalar spectral fuctios of a Carlema operator, Bol. Soc. Paraa. Mat. (3) 26(1 2) (2008), [6] S. M. Bahri, O the localizatio of the spectrum for quasi-selfadjoit extesios of a Carlema operator, Math. Bohem. 137 (2012), [7] M. K. Belbahri, Series de Lauret formelles, Office des Publicatios Uiversitaires, Alger, [8] T. Carlema, Sur les Equatios Itégrales Sigulières à Noyau Réel et Symétrique, Uppsala Almquwist Wiksells Boktryckeri, Uppsala, [9] V. B. Korotkov, Itegral Operators with Carlema Kerels (i Russia), Nauka, Novosibirsk, [10] G. I. Tagroski, O Carlema itegral operators, Proc. Amer. Math. Soc. 18 (1967), [11] J. Weidma, Carlema operators, Mauscripts Math. 2 (1970), [12] J. Weidma, Liear Operators i Hilbert Spaces, Sprig Verlag, New-York Heidelberg Berli, Laboratory of Pure ad Applied Mathematics, Abdelhamid Ib Badis Uiversity, BP 227, Mostagaem 27000, Algeria address: sidimohamed.bahri@uiv-mosta.dz