On the Operator-valued $\mu $-cosine functions

arXiv:1701.07229v1 [math.RT] 25 Jan 2017

ON THE OPERATOR-VALUED µ-COSINE FUNCTIONS BOUIKHALENE BELAID AND ELQORACHI ELHOUCIEN Abstract. Let (G, +) be a topological abelian group with a neutral element e and let µ : G −→ C be a continuous character of G. Let (H, h·, ·i) be a complex Hilbert space and let B(H) be the algebra of all linear continuous operators of H into itself. A continuous mapping Φ : G −→ B(H) will be called an operator-valued µ-cosine function if it satisfies both the µ-cosine equation Φ(x + y) + µ(y)Φ(x − y) = 2Φ(x)Φ(y), x, y ∈ G and the condition Φ(e) = I, where I is the identity of B(H). We show that any hermitian operator-valued µ-cosine functions has the form Γ(x) + µ(x)Γ(−x) 2 where Γ : G −→ B(H) is a continuous multiplicative operator. As an application, positive definite kernel theory and W. Chojnacki’s results on the uniformly bounded normal cosine operator are used to give explicit formula of solution of the cosine equation. Φ(x) =

1. Introduction 1.1. The µ-cosine equation, also called the pre-d’Alembert equation, on abelian group G is the equation (1.1)

f (x + y) + µ(y)f (x − y) = 2f (x)f (y), x, y ∈ G

where f : G −→ C is the unknoun. Davison [8] gave solution of (1.1) in terms of traces of certain representations of G on C2 . In [22] Stetkær proves that a non-zero solution of (1.1) has the form (1.2)

f (x) =

χ(x) + µ(x)χ(−x) , x ∈ G, 2

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] and by Aczél and Dhombres [3] have references and detailed discussions. 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, multiplicative function. 2010 MSC: 47D09; 22D10; 39B42. 1

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B. BOUIKHALENE AND E. 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 fˇ by 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, where Φ∗ (x) is the adjoint operator of Φ(x). A continuous mapping Γ : 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

and the conditions Φ(e) = 1. The scalar case of (1.4) is given by the equation (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 Kisy´ nski [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 Mn (C) of complex n × n matrices. By using positive definite kernels and linear algebra theory we find that any hermitian continuous solution where Γ : G −→ B(H) is a continuous of (1.4) has the form Φ(x) = Γ(x)+µ(x)Γ(−x) 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, x1 , ..., xn ∈ G and arbitrary complex numbers c1 , c2 , ..., cn we have n X n X ci cj K(xi , xj ) ≥ 0. (1.6) i=1 j=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 VK := span{K(x, ·) : x ∈ G}. Then i) VK ⊂ C(G), ii) VK is equipped with the inner product n n X X αi βj K(xi , xj ) hf, giK = i=1 j=1

where Pn Pm f = i=1 αi K(xi , .), g = i=1 βj K(xj , .), x1 , ..., xsup(m,n) ∈ G and α1 , ..., αn , β1 , ..., βm ∈ C.

ON THE OPERATOR-VALUED µ-COSINE

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Let HK be the completion of VK . Then (HK ,h., .iK ) is a Hilbert space of continuous functions on G. The function K is the reproducing kernel of the Hilbert space HK . Theorem 1.3. Let K be a positive definite kernel on G. Then there exists a Hilbert space (HK ,h., .iK ) and a continuous mapping T : G −→ HK , x 7−→ K(x, ·), such that 1) K(x, y) = hT (x), T (y)iK for all x, y ∈ G. 2) span{T (x) : x ∈ G} is dense in HK . Moreover, the pair (HK , T ) is unique in the following way : if another pair (L, U ) satisfies (1) and (2), there exists a unique unitary isomorphism Ψ : HK −→ L such that U = Ψ ◦ T . For all f ∈ C(G) and for all x, y ∈ G we define 1 (1.7) Kf (x, y) := {f (−y + x) + µ(x)f (−y − x)}. 2 and f (x) = fµ+ (x) + fµ− (x), x ∈ G,

(1.8) where fµ+ (x) =

f (x)+µ(x)f (−x) 2

and fµ− (x) =

f (x)−µ(x)f (−x) 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 −1 SΦ(y)S −1 .

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From which we get that SΦ(x + y)S −1 + µ(y)SΦ(x − y)S −1 = 2SΦ(x)S −1 SΦ(y)S −1 for all x, y ∈ G. Furthermore we have SΦ(e)S −1 = I. Proposition 2.2. Let M : G −→ B(H) be a multiplicative operator. Then M (x) + µ(x)M (−x) , x∈G 2 is an operator-valued µ-cosine functions. Φ(x) =

Proof. Since M (x + y) = M (x)M (y) for all x, y ∈ G and M (e) = I, we get by easy computations that Φ is an operator-valued µ-cosine functions. 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−→ Kf (x, y) is continuous. ii) Kf (−x, y) = µ(−x)Kf (x, y) for all x, y ∈ G. iii) Kf (0, −y) = f (y) for all y ∈ G. 4i) Kf (x, 0) = fµ+ (x) for all x ∈ G. 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, x 7−→ 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) Kf (x, y) = hΦ(x)ξ, Φ(y)ξi for all x, y ∈ G. 4i) Kf is a positive definite kernel. 5i) Kf (x, .) = 12 {(R−x fˇ) + µ(x)(Rx fˇ)} 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 Kf (x, y)

= =

1 {f (−y + x) + µ(x)f (−y − x)} 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

= =


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4i) For all n ∈ N, x1 , ..., xn ∈ G and arbitrary complex numbers c1 , c2 , ..., cn we have n X n n X n X X ci cj hΦ(xi )ξ, Φ(xj )ξi ci cj Kf (xi , xj ) = i=1 j=1 n X

i=1 j=1

= h

ci Φ(xi )ξ,

i=1 n X

= k

n X

cj Φ(xj )ξi

j=1

ci Φ(xi )k2 ≥ 0.

i=1

5i) For all x, y ∈ G we have Kf (x, y)

= = =

1 {f (−y + x) + µ(x)f (−y − x)} 2 1 {(Rx f )(−y) + µ(x)(R−x f )(−y)} 2 1 ˇ f )(y).} {(Rˇx f )(y) + µ(x)(R−x 2

Since (Rˇx f ) = R−x fˇ for all x ∈ G it follows that Kf (x, y) = 21 {R−x fˇ(y) + µ(x)(R− fˇ)(y)} for all x, y ∈ G. So that we have Kf (x, .) = 21 {(R−x fˇ)+µ(x)(Rx fˇ)} for all x ∈ G. 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 Φ(x) =

M (x) + µ(x)M (−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 Kϕ (x, y) = hΦ(x)ξ, Φ(y)ξi. So that Kϕ is a positive definite kernel. According to Theorem 1.3 there exists a Hilbert space (Hϕ , h·, ·iϕ ) and a mapping T : G −→ Hϕ , x 7−→ Kϕ (·, x) such that Kϕ (x, y)

= =

hKϕ (x, ·), Kϕ (y, ·)i hΦ(x)ξ, Φ(y)ξi

and a unique unitary isomorphism ψ : Hϕ −→ H such tat 1 ˇ ϕ)]. ψ[(Rˇx ϕ) + µ(x)(R−x 2 Since Φ(e) = I we get by setting x = e in (3.1) that ξ = ψ(ϕ). ˇ From which we get ˇ ϕ) = Rx ϕˇ for all x ∈ G. that ϕˇ = ψ −1 (ξ). We show that (Rˇx ϕ) = R−x ϕˇ and (R−x So that for all x ∈ G and ξ ∈ H we have (3.1)

Φ(x)ξ = ψ(Kϕ (x, .)) =

Φ(x)ξ =

1 ψ[(R− x + µ(x)Rx )ψ −1 (ξ)]. 2

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Hence Φ(x) = ψ ◦ R(x) ◦ ψ −1 for all x ∈ G. x Since R(x) = R−x +µ(x)R for all x ∈ G we get that 2 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 Φ(x) =

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 ◦ Re ◦ ψ −1 = I. for all x ∈ G where M : G −→ H is Finally we have that Φ(x) = M(x)+µ(x)M(−x) 2 a multiplicative operator. This ends the proof of theorem. 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 −→ Mn (C) be a continuous hermitian solution of (1.4). Then there exists A ∈ GL(n, C) such that E(x) + µ(x)E(−x) −1 A , x∈G 2 where E : G −→ Mn (C) has the form   γ1 0 ... 0  0 γ2 ... 0     0 0 ... 0     ... ... γi ...  0 0 ... γn

(3.2)

Φ(x) = A

where γ1 , ..., γn ∈ M (G) and γi 6= γj for all i, j ∈ {1, ..., n} such that i 6= 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   ω1 0 ... 0  0 ω2 ... 0    −1 0 ... 0  Φ(x) = A  0 A .  ... ... 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 ωi (x + y) + µ(y)ωi (x − y) = 2ωi (x)ωi (y) for all i ∈ {1, ..., n}. According to [22] there exists i (−x) for all x ∈ G. So γi ∈ M(G) for all i ∈ {1, ..., n}. such thatω(x) = γi (x)+µ(x)γ 2  γ1 0 ... 0  0 γ2 ... 0    −1  0 0 ... 0  and γi ∈ M(G) such Φ(x) = A E(x)+µ(x)E(−x) A where E = 2    ... ... γi ...  0 0 ... γn that γi 6= γj for i 6= j. This ends the proof of corollary


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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 cosine functions. Then there is an invertible S ∈ B(H) such that Ψ(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 Φ(x) = S

M (x) + M (−x) −1 S , x ∈ G. 2

Proof. By using Proposition 4.1 we get that Ψ(x) = SΦ(x)S −1 is a solution of (1.5) such that Ψ(x)∗ = Ψ(x−1 ) for all x ∈ G. According to Theorem 3.1 we get the remainder. References [1] Akkouchi M., Bakali A., Khalil I, A class of functional equations on a locally compact group, J. Lond. Math. Soc. (2) 57 (1998), 694-705. [2] J. Acz´ el, Vorlesungen u ¨ber Funktionalgleichungen und ihre Anwendungen, Birkh¨ auser 1961. [3] J. Acz´ el and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. [4] J. A. Baker and K. R. Davidson, Cosine, exponential and quadratic functions, Glasnik Mathemati˘ cki. 16 (36) (1981), 269-274. [5] B. Bouikhalene, E. Elqorachi and A. Bakali, On generalized Gajda’s functional equation of DAlembert type, Adv. Pure Appl. Math. 3 (2012), 293-313 DOI 10.1515/apam-2012-0008. [6] W. Chojnacki, Fonctions cosinus hilbertiennes bornes dans les groupes commutatifs localement compacts, Compos. Math. 57 (1986), 15-60. [7] W. Chojnacki, On uniformly bounded spherical functions in Hilbert space. Aequationes Math. 81 (2011), no. 1-2, 135-154. [8] T. M. K. Davison, d’Alembert’s functional equation on topological monoids. Publ. Math. Debrecen 75 1/2 (2009), 41-66. [9] H. O. Fattorini, Uniformly bounded cosine functions in Hilbert space, Indiana Univ. Math. J. 20 (1970), 411-425. [10] J. Kisy´ nski, On operator-valued solutions of d’Alembert’s functional equation I, Colloq Math. 32 (1971), 107-114. [11] J. Kisy´ nski, On operator-valued solutions of d’Alembert’s functional equation II, Studia Math. 42 (1972), 43-66. [12] S. Kurepa, A cosine functional equation in Banach algebra, Acta Sci. Math. (Szeged) 23 (1962), 255-267. [13] S. Kurepa, Uniformly bounded cosine function in a Banach space, Math. Balkanica 2 (1972), 109-115. [14] M. A. Mckiernan, The matrix equation a(x ◦ y) = a(x) + a(x)a(y) + a(y), Aequationes Math. 15 (1977), 213-223.

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Bouikhalene Belaid Departement of Mathematics and Informatics Polydisciplinary Faculty, Sultan Moulay Slimane university, Beni Mellal, Morocco. E-mail : [email protected]. Elqorachi Elhoucien, Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco, E-mail: [email protected]