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ON THE OPERATOR-VALUED µ-COSINE FUNCTIONS 7 BOUIKHALENEBELAIDANDELQORACHIELHOUCIEN 1 0 2 Abstract. Let(G,+)beatopologicalabeliangroupwithaneutralelemente andletµ:G−→CbeacontinuouscharacterofG. Let(H,h·,·i)beacomplex n Hilbert space and let B(H) be the algebra of all linear continuous operators a of H into itself. A continuous mapping Φ : G −→ B(H) will be called an J operator-valuedµ-cosinefunctionifitsatisfiesboththeµ-cosineequation 5 Φ(x+y)+µ(y)Φ(x−y)=2Φ(x)Φ(y), x,y∈G 2 and the condition Φ(e) = I, where I is the identity of B(H). We show that ] anyhermitianoperator-valuedµ-cosinefunctions hastheform T Γ(x)+µ(x)Γ(−x) Φ(x)= R 2 . where Γ : G −→ B(H) is a continuous multiplicative operator. As an ap- h plication, positive definite kernel theory and W. Chojnacki’s results on the t a uniformlyboundednormalcosineoperatorareusedtogiveexplicitformulaof m solutionofthecosineequation. [ 1 v 9 1. Introduction 2 2 1.1. The µ-cosine equation, also called the pre-d’Alembert equation, on abelian 7 group G is the equation 0 . 1 (1.1) f(x+y)+µ(y)f(x−y)=2f(x)f(y), x,y ∈G 0 7 where f : G −→ C is the unknoun. Davison [8] gave solution of (1.1) in terms of 1 tracesofcertainrepresentationsofGonC2. In[22]Stetkærprovesthatanon-zero v: solution of (1.1) has the form i X χ(x)+µ(x)χ(−x) (1.2) f(x)= ,x∈G, r 2 a where χ is a character of G. In the case where µ = 1, equation (1.1) becomes the classic cosine functional equation (also called the d’Alembert functional equation) (1.3) f(x+y)+f(x−y)=2f(x)f(y), x,y ∈G. Several mathematicians studied the equation (1.3). The monographs by Aczél [2] andbyAczélandDhombres[3]havereferencesanddetaileddiscussions. The main purpose of this work is to extend equation (1.1) to functions taking values in the algebra B(H) of bounded operators on a Hilbert space H. Keywords: cosine equation; locally compact group; unitary representation; character, multi- plicativefunction. 2010MSC:47D09;22D10;39B42. 1 2 B.BOUIKHALENEANDE.ELQORACHI 1.2. Throughout this paper, G will be a topological abelian group with the unit element e. The space of continuous complex-valued functions is denoted by C(G) and the set of all continuous homomorphisms γ : G −→ C\{0} by M(G). Let µ : G −→ C∗ be a continuous character of the group G i.e. µ ∈ M(G) such that µ(e)=1. For all f ∈C(G) we define the function fˇby fˇ(x)=f(−x). Let (H,h,i) be a Hilbert space over C and let B(H) be the algebra of all linear continuous operators of H into itself with the usual operator norm denoted k.k. A mapping Φ:G−→B(H) is said to be hermitian if it satisfies Φ∗(x)=Φ(−x) for all x∈G, whereΦ∗(x)istheadjointoperatorofΦ(x). AcontinuousmappingΓ:G−→B(H) is said to be a multiplicative operator if Γ(x+y) = Γ(x)Γ(y) for all x,y ∈ G and Γ(e) = I. Also we say that a continuous mapping Φ : G −→ B(H) is an operator valued µ-cosine function if it satisfies both the µ-cosine functional equation (1.4) Φ(x+y)+µ(y)Φ(x−y)=2Φ(x)Φ(y), x,y ∈G andtheconditionsΦ(e)=1. Thescalarcaseof(1.4)isgivenbytheequation(1.1). For µ=1 we obtain the cosine functional equation (1.5) Φ(x+y)+Φ(x−y)=2Φ(x)Φ(y), x,y ∈G. Several variants of (1.5) has been studied by Kisyn´ski [10] and [11], Székelyhidi [23], Chojnacki [6] and [7], Stetkær [19], [20] and [21]. 1.3. The main purpose of this work is to solve the equation (1.4), where the unknown Φ is an hermitian continuous functions on G taking its values in B(H) or in the algebra M (C) of complex n×n matrices. By using positive definite n kernels and linear algebra theory we find that any hermitian continuous solution of (1.4) has the form Φ(x)= Γ(x)+µ(x)Γ(−x) where Γ:G−→B(H) is a continuous 2 multiplicative operator. 1.4. Notation and preliminary. Definition 1.1. A continuous function K : G×G −→ C is said to be a positive definite kernel on G if for all n∈N, x ,...,x ∈G and arbitrary complex numbers 1 n c ,c ,...,c we have 1 2 n n n (1.6) c c K(x ,x )≥0. XX i j i j i=1j=1 We provide some known results on positive definite kernel theory. For more details we refer to [15]. Proposition 1.2. Let K be a positive definite kernel on G and let V :=span{K(x,·):x∈G}. K Then i) V ⊂C(G), K ii) V is equipped with the inner product K n n hf,gi = α β K(x ,x ) K XX i j i j i=1j=1 where f = n α K(x ,.), g = m β K(x ,.), x ,..P.,xi=1 i ∈iG and αP,i.=..1,αj,β ,j...,β ∈C. 1 sup(m,n) 1 n 1 m ON THE OPERATOR-VALUED µ-COSINE 3 LetH bethecompletion ofV . Then (H ,h.,.i ) is aHilbert spaceofcontinuous K K K K functions on G. The function K is the reproducing kernel of the Hilbert space H . K Theorem1.3. LetK beapositivedefinitekernelonG. ThenthereexistsaHilbert space (H ,h.,.i ) and a continuous mapping K K T :G−→H ,x7−→K(x,·), K such that 1) K(x,y)=hT(x),T(y)i for all x,y ∈G. K 2) span{T(x):x∈G} is dense in H . K Moreover, the pair (H ,T) is unique in the following way : if another pair (L,U) K satisfies (1) and (2), there exists a uniqueunitary isomorphism Ψ:H −→L such K that U =Ψ◦T. For all f ∈C(G) and for all x,y ∈G we define 1 (1.7) K (x,y):= {f(−y+x)+µ(x)f(−y−x)}. f 2 and (1.8) f(x)=f+(x)+f−(x), x∈G, µ µ where f+(x)= f(x)+µ(x)f(−x) and f−(x)= f(x)−µ(x)f(−x) µ 2 µ 2 2. General properties Proposition 2.1. Let Φ:G−→B(H) be a solution of (1.4). Then i) Φ(−x)=µ(−x)Φ(x) for all x∈G. ii) Φ(x)Φ(y)=Φ(y)Φ(x) for all x,y ∈G. iii) For all invertible operator S ∈ B(H) we have SΦ(x)S−1 for all x ∈ G is a solution of (1.4). Proof. i) For all x,y ∈G we have 2Φ(x)Φ(y) = Φ(x+y)+µ(y)Φ(x−y) = µ(y)(Φ(x−y)+µ(−y)Φ(x+y)) = 2µ(y)Φ(x)Φ(−y). From which we get that (2.1) Φ(x)Φ(y)=µ(y)Φ(x)Φ(−y). Setting x=e in (2.1) and Φ(e)=I we get that Φ(y)=µ(y)Φ(−y) for all y ∈G. ii) For all x,y ∈G we have 2Φ(y)Φ(x) = Φ(y+x)+µ(x)Φ(y−x) = Φ(x+y)+µ(x)µ(y−x)Φ(x−y) = Φ(x+y)+µ(y)Φ(x−y) = 2Φ(x)Φ(y). From which we get that Φ(x)Φ(y)=Φ(y)Φ(x) for all x,y ∈G. iii) For all x,y ∈G we have SΦ(x+y)S−1+µ(y)SΦ(x−y)S−1 = S(Φ(x+y)+µ(y)Φ(x−y))S−1 = S2Φ(x)Φ(y)S−1 = 2SΦ(x)S−1SΦ(y)S−1. 4 B.BOUIKHALENEANDE.ELQORACHI From which we get that SΦ(x+y)S−1+µ(y)SΦ(x−y)S−1 =2SΦ(x)S−1SΦ(y)S−1 for all x,y ∈G. Furthermore we have SΦ(e)S−1 =I. (cid:3) Proposition 2.2. Let M :G−→B(H) be a multiplicative operator. Then M(x)+µ(x)M(−x) Φ(x)= , x∈G 2 is an operator-valued µ-cosine functions. Proof. Since M(x+y)=M(x)M(y) for all x,y ∈G and M(e)=I, we getby easy computations that Φ is an operator-valuedµ-cosine functions. (cid:3) By easy computations we get the following proposition Proposition 2.3. For all f ∈C(G) we have the following statements i) the mapping (x,y)7−→K (x,y) is continuous. f ii) K (−x,y)=µ(−x)K (x,y) for all x,y ∈G. f f iii) K (0,−y)=f(y) for all y ∈G. f 4i) K (x,0)=f+(x) for all x∈G. f µ We need the following proposition in the main result Proposition 2.4. Let Φ:G−→B(H) be be a solution of (1.4) such that Φ(x)∗ = Φ(−x) for all x∈G and let f :G−→C, x7−→hΦ(x)ξ,ξi for ξ ∈H. Then i) f(−x)=µ(−x)f(x) for all x∈G. ii) f(x)=f(−x) for all x∈G. iii) K (x,y)=hΦ(x)ξ,Φ(y)ξi for all x,y ∈G. f 4i) K is a positive definite kernel. f 5i) Kf(x,.) = 12{(R−xfˇ)+µ(x)(Rxfˇ)} for all x ∈ G where R is the right regular representation of G. Proof. i) Since Φ(−x)=µ(−x)Φ(x) for all x∈G we get that f(−x)=hΦ(−x)ξ,ξi=hµ(−x)Φ(x)ξ,ξi =µ(−x)hΦ(x)ξ,ξi =µ(−x)f(x). ii) for all x∈G we have f(x)=hΦ(−x)ξ,ξi=hξ,Φ(x)ξi=hΦ(x)∗ξ,ξi=hΦ(−x)ξ,ξi=f(−x). iii) For all x,y ∈G we have 1 K (x,y) = {f(−y+x)+µ(x)f(−y−x)} f 2 1 = {(hΦ(−y+x)+µ(x)Φ(−y+x)ξ,ξi)} 2 = hΦ(−y)Φ(x)ξ,ξi = hΦ(y)∗Φ(x)ξ,ξi = hΦ(x)ξ,Φ(y)ξi ON THE OPERATOR-VALUED µ-COSINE 5 4i) For all n ∈ N, x ,...,x ∈ G and arbitrary complex numbers c ,c ,...,c we 1 n 1 2 n have n n n n c c K (x ,x ) = c c hΦ(x )ξ,Φ(x )ξi XX i j f i j XX i j i j i=1j=1 i=1j=1 n n = h c Φ(x )ξ, c Φ(x )ξi X i i X j j i=1 j=1 n = k c Φ(x )k2 ≥0. X i i i=1 5i) For all x,y ∈G we have 1 K (x,y) = {f(−y+x)+µ(x)f(−y−x)} f 2 1 = {(Rxf)(−y)+µ(x)(R−xf)(−y)} 2 1 = {(Rˇxf)(y)+µ(x)(R−ˇxf)(y).} 2 Since (Rˇxf) = R−xfˇ for all x ∈ G it follows that Kf(x,y) = 21{R−xfˇ(y) + µ(x)(R−fˇ)(y)}forallx,y ∈G. SothatwehaveKf(x,.)= 21{(R−xfˇ)+µ(x)(Rxfˇ)} for all x∈G. (cid:3) 3. Main Result In the next theorem we solve the equation (1.4). Theorem 3.1. Let Φ : G −→ B(H) be an hermitian operator-valued µ-cosine functions. Then there exists a multiplicative operator M :G−→B(H) such that M(x)+µ(x)M(−x) Φ(x)= , x∈G. 2 Proof. Let ξ ∈ H. By the same way as in the proof of Theorem 2.2 in [1], we can suppose that the vector ξ is cyclic Let now ϕ(x) = hΦ(x)ξ,ξi for all x ∈ G. For all x,y ∈ G we have K (x,y) = ϕ hΦ(x)ξ,Φ(y)ξi. So that K is a positive definite kernel. According to Theorem ϕ 1.3 there exists a Hilbert space (H ,h·,·i ) and a mapping T : G −→ H , x 7−→ ϕ ϕ ϕ K (·,x) such that ϕ K (x,y) = hK (x,·),K (y,·)i ϕ ϕ ϕ = hΦ(x)ξ,Φ(y)ξi and a unique unitary isomorphism ψ :H −→H such tat ϕ 1 (3.1) Φ(x)ξ =ψ(Kϕ(x,.))= ψ[(Rˇxϕ)+µ(x)(R−ˇxϕ)]. 2 Since Φ(e)=I we get by setting x=e in (3.1) that ξ =ψ(ϕˇ). From which we get that ϕˇ=ψ−1(ξ). We show that (Rˇxϕ)=R−xϕˇ and (R−ˇxϕ)=Rxϕˇ for all x∈G. So that for all x∈G and ξ ∈H we have 1 Φ(x)ξ = ψ[(R−x+µ(x)Rx)ψ−1(ξ)]. 2 6 B.BOUIKHALENEANDE.ELQORACHI Hence Φ(x)=ψ◦R(x)◦ψ−1 for all x∈G. Since R(x)= R−x+µ(x)Rx for all x∈G we get that 2 Φ(x)= ψ◦R−x◦ψ−1+µ(x)ψ◦Rx◦ψ−1. 2 Setting M(x)=ψ◦R−x◦ψ−1 for all x∈G. We have for all x,y ∈G that M(x+y) = ψ◦R−x−y◦ψ−1 = ψ◦R−x◦R−y ◦ψ−1 = ψ◦R−x◦ψ◦ψ−1◦R−y ◦ψ−1 = M(x)M(y). and that M(e)=ψ◦R ◦ψ−1 =I. e Finally we have that Φ(x) = M(x)+µ(x)M(−x) for all x ∈ G where M : G −→ H is 2 a multiplicative operator. This ends the proof of theorem. (cid:3) In the next corollary we determine solutions of (1.4) taking their values in the complex n×n matrices Corollary 3.2. Let Φ:G−→M (C) be a continuous hermitian solution of (1.4). n Then there exists A∈GL(n,C) such that E(x)+µ(x)E(−x) (3.2) Φ(x)=A A−1, x∈G 2 where E :G−→M (C) has the form n γ 0 ... 0 1   0 γ ... 0 2 0 0 ... 0   ... ... γi ...   0 0 ... γn where γ ,...,γ ∈M(G) and γ 6=γ for all i,j ∈{1,...,n} such that i6=j 1 n i j Proof. since Φ(x)Φ(y) = Φ(y)Φ(x) for all x,y ∈ G and Φ(x)∗ = Φ(−x) for all x ∈ G it follows that Φ(x) for all x ∈ G can be diagonalized simultaneously. So there exists A∈GL(n,C) such that ω 0 ... 0 1   0 ω ... 0 2 Φ(x)=A0 0 ... 0 A−1.   ... ... ωi ...   0 0 ... ωn Since Φ(x+y)+µ(y)Φ(x−y)=2Φ(x)Φ(y) for all x,y ∈G it follows that ω (x+ i y)+µ(y)ω (x−y)=2ω (x)ω (y)foralli∈{1,...,n}. Accordingto[22]thereexists i i i γ ∈M(G) forall i∈{1,...,n}. suchthatω(x)= γi(x)+µ(x)γi(−x) forall x∈G. So i 2 γ 0 ... 0 1   0 γ ... 0 2 Φ(x) = AE(x)+µ(2x)E(−x)A−1 where E = 0 0 ... 0 and γi ∈ M(G) such ... ... γi ...   0 0 ... γn that γ 6=γ for i6=j. This ends the proof of corollary (cid:3) i j ON THE OPERATOR-VALUED µ-COSINE 7 4. applications Throughout this section we adhere to the terminology used in [7]. Let G be a locally compact commutative group and let µ = 1. A mapping Φ : G −→ B(H) will be said to be uniformly bounded if sup{kΦ(x)k : x ∈ G}B(H) < +∞. The hermitian operator-valued cosine functions is denoted by ∗-operator-valued cosine functions in [7]. According to Theorem 1 in [7] we get the following proposition Proposition 4.1. Let Φ : G −→ B(H) be a uniformly bounded operator-valued cosinefunctions. ThenthereisaninvertibleS ∈B(H)suchthatΨ(x)=SΦ(x)S−1 for all x∈G is an hermitian operator-valued cosine functions In the next theorem we use our study to solve the equation (1.5) Theorem 4.2. Let Φ:G−→B(H) be a uniformly bounded operator-valued cosine functions. Then there is an invertible S ∈ B(H) and a multiplicative operator M :G−→B(H) such that M(x)+M(−x) Φ(x)=S S−1, x∈G. 2 Proof. ByusingProposition4.1wegetthatΨ(x)=SΦ(x)S−1 isasolutionof(1.5) such that Ψ(x)∗ = Ψ(x−1) for all x ∈ G. According to Theorem 3.1 we get the remainder. (cid:3) References [1] AkkouchiM.,BakaliA.,KhalilI,Aclassoffunctionalequationsonalocallycompactgroup, J. Lond. Math. Soc. (2) 57(1998), 694-705. [2] J.Aczél,Vorlesungenu¨berFunktionalgleichungen undihreAnwendungen, Birkhäuser1961. 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Stetkær, D’Alembert’s And Wilson’s functional equations for vector and 2×2 matrix valuedfunctions,Math.Scand.87(2000), 115-132. [20] H.Stetkær, Functional equations andmatrix-valuedsphericalfunctions, Aequationes Math. 69(2005), 271-292. [21] H.Stetkær,Onoperator-valuedsphericalfunctions,J.Funct. Anal.224(2005), 338-351. [22] H.Stetkær, d’Alembert’s functional equation on groups. Recent developments in functional equations and inequalities, 173-191, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw,2013. [23] L. Székelyhidi, Functional equations on abelian groups, Acta Math. Acad. Sci. Hung. 73 (1981), 235-243. Bouikhalene Belaid Departement of Mathematics and Informatics PolydisciplinaryFaculty, Sultan Moulay Slimane university, Beni Mellal, Morocco. E-mail : [email protected]. ElqorachiElhoucien, Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco, E-mail: [email protected]