Functional equations of Cauchy s and d Alembert s Type on Compact Groups

Transcription

1 Proyecciones Journal of Mathematics Vol. 34, N o 3, pp , September Universidad Católica del Norte Antofagasta - Chile Functional equations of Cauchy s and d Alembert s Type on Compact roups Abdellatif Chahbi Brahim Fadli and Samir Kabbaj University of Ibn Tofail, Morroco Received : June Accepted : July 2015 Abstract Using the non-abelian Fourier transform, we find the central continuous solutions of the functional equation n 1 X f(xσ k (y)) = nf(x)f(y) x, y, where is an arbitrary compact group, n N\{0} and σ is a continuous automorphism of, such that σ n = I. We express the solutions in terms of the unitary (group) characters of. Subjclass [2010] : 39B52; 22C05; 43A30; 22E45. Keywords : Non-abelian Fourier transform, representation of a compact group.

2 298 Abdellatif Chahbi, Brahim Fadli and Samir Kabbaj 1. Introduction Let be a group, n N \{0} and σ an automorphism of, such that σ n = I, where I denotes the identity map. We consider the functional equation (1.1) f(xσ k (y)) = nf(x)f(y), x,y, where f : C is the function to determine. This equation has been solved on abelian groups. See Shin ya [[13], Corollary 3.12] and Stetkær [[17], Theorem 14.9]. The functional equation (1.1) is a generalization of Cauchy s and d Alembert s functional equations. In fact, Cauchy s functional equation (1.2) f(xy) =f(x)f(y), x,y, results from (1.1) by taking σ = I. In the particular case where n =2, (1.1) reduces to functional equation (1.3) f(xy)+f(xσ(y)) = 2f(x)f(y), x,y, When is abelian and σ = I, the functional equation (1.3) becomes d Alembert s functional equation (1.4) f(x + y)+f(x y) =2f(x)f(y), x,y. This explains the choice of the title of this paper. At present the theory of d Alembert s equation is extensively developed (cf. [2] -[11]; [14] -[20]). Because of the importance of this types of equations, it is worthwhile to provide a solution for functional equation (1.1) in the case of f being a central function and the group being compact and possibly non-abelian. Our approach uses harmonic analysis and representation theory on compact groups. The idea of using Fourier analysis for solving it goes back to [5]. Throughout the rest of this paper, B(H) is the algebra of all bounded linear operators on Hilbert space H, the group is always assumed to be compact and σ is a continuous automorphism. By solutions (resp. representations), we always mean continuous solutions (resp. continuous representations).

3 Functional equations of Cauchy s and d Alembert s Type Preliminaries In this section, we set up some notation and conventions and briefly review some fundamental facts in Fourier analysis which will be used later. Let be a compact group with the normalized Haar measure dx. Let Ĝ stand for the set of equivalence classes of continuous irreducible unitary representations of. It is known that for [π] Ĝ, π is finite dimensional. We denote its dimension by d π. Consider ε π = span{π ij : i, j =1,...,d π } the linear span of matrix-valued representative of [π]. For f L 2 (),the Fourier transform of f is defined by ˆf(π) =d π f(x)π(x) 1 dx M dπ (C), for all [π] Ĝ, where M d π (C) isthespaceofalld π d π complex matrix. As usual, left and right regular representations of in L 2 ()aredefined by (L y f)(x) =f(y 1 x), (R y f)(x) =f(xy), respectively, for all f L 2 () andx, y. A crucial property of the Fourier transform is that it converts the regular representations of into matrix multiplications. The following properties are useful. i The Fourier inversion formula is given by f(x) = X tr( ˆf(π)π(x)). [π] Ĝ ii The following identities hold: dl y f(π) = ˆf(π)π(y) 1, d R y f(π) =π(y) ˆf(π), for all x, y. A function f : C is central if f(xy) =f(yx), for all x, y. For more information about harmonic analysis we refer to [12].

4 300 Abdellatif Chahbi, Brahim Fadli and Samir Kabbaj 3. Main result In this section, we shall solve (1.1) on compact groups by using harmonic analysis of such groups. That two representations π and ρ of are equivalent is denoted π ' ρ. Lemma 3.1. Let be a compact group, and π a unitary irreducible representation of. If f : C is a central function, then ˆf(π) is a multiple of the identity operator. Proof. π(y) ˆf(π) =d π f(x)π(y)π(x 1 )dx = d π = d π = d π f(x)π((xy 1 ) 1 )dx f(xy)π(x 1 )dx f(yx)π(x 1 )dk = d π ˆf(π)π(y), for every y. By Schur s lemma, ˆf(π) is a scalar multiple of the identity operator. 2 This following lemma is inspired by the Small Dimension Lemma from [20]. Lemma 3.2. Let be a compact group and π a unitary irreducible representation of. Suppose for every x there is c x C such that (3.1) π(σ k (x)) = c x I dπ,x. Then d π =1.

5 Functional equations of Cauchy s and d Alembert s Type 301 Proof. Consider the set S = {k {1,..., n 1} π ' π σ k }. We will consider two cases, S is empty or not empty. In the first case, from (3.1) we get (3.2) π ij (x)+ π ij (σ k (x)) = 0 for i 6= j, 1 i, j d π,x. k=1 Since S = we have ε π ε π σ k for all k =1,...,n 1. Hence π ij =0 for i 6= j, so π is a diagonal matrix. Since π is irreducible this implies that d π =1. In the second case, if S 6=,then (3.3) S = {s 0, 2s 0,...,Ns 0 } and n =(N +1)s 0, where s 0 =mins and N = cards. Indeed, let k S, thereexists(q, r) N N such that k = qs 0 + r and 0 r<s 0. From π ' π σ s 0 we arrive at π σ r ' π σ qs0+r, so π σ r ' π σ k. This implies that π ' π σ r. Since 0 r<s 0 and s 0 =mins, we have r =0. Then S is contained in the set of integer multiples of s 0. An additional simple inductive argument is needed to show that S has the form S = {s 0, 2s 0,...,Ns 0 }. Furthermore, π ' π σ s 0 is equivalent to π ' π σ n s 0. From π ' π σ n s 0 we infer that n s 0 S. Since n s 0 + s 0 = n 6 S we see that n s 0 is the biggest element in S = {s 0, 2s 0,...,Ns 0 }, i.e., n s 0 = Ns 0 and n =(N +1)s 0. Let (H, h, i) denote the complex Hilbert space on which the representation π acts. Since π ' π σ s 0, there exists a unitary operator T B(H) such that π σ s 0 (x) =T π(x)t, for all x, from which induction gives the more general formula so π σ ks 0 (x) =(T k ) π(x))t k, for all x and any k = 0, 1,... H has an orthonormal basis (e 1,e 2,...,e dπ ) consisting of eigenvectors of T (by the spectral theorem for normal operators applied to T ). We write Te i = λ i e i, where λ i C for i =1;2,...,d π : Actually λ i =1,T being unitary. For any i =1;2,...,d π,k S and x we compute that E E (π σ ks 0 ) ii (x) = D(π σ ks 0 )(x)e i,e i = D(T k ) π(x))t k e i,e i E E = DT k π(x)t k e i,t k e i = Dλ k i π(x)e i,λ k i e i = λ k i λ k i hπ(x)e i,e i i = λ i 2k π ii (x) =π ii (x).

6 302 Abdellatif Chahbi, Brahim Fadli and Samir Kabbaj It is easy to see from (3.1) that (3.4) NX π ii (x)+ π ii (σ ks 0 (x)) + X π ii (σ k (x)) = c x, k=1 k S for all i =1,...,d π and x, where S is the complement of S in {1,...,n 1}. Since π ii σ ks 0 = π ii for all k =1,...,N. From (3.4) we obtain (N +1)π ii (x)+ X k S π ii (σ k (x)) = c x, for all i =1,...,d π, x. Then d π =1. Indeed, if d π > 1, then for all i =2,...,d π we have so (N +1)π ii + X π ii σ k =(N +1)π 11 + X π 11 σ k, k S k S (3.5) (N +1)(π ii π 11 )= X k S(π 11 π ii ) σ k. Since π isnotequivalenttoπ σ k for any k S, we have ε π ε π σ k for all k S. (3.5) implies that π ii = π 11 for all i =2,...,d π. But if we use Schur s orthogonality relations which say 1 d π π ij is an orthonormal basis, we get a contradiction. Then d π =1. 2 Theorem 3.3. Let be a compact group and σ is a continuous automorphism of such that σ n = I. If f is a central continuous non-zero solution of the functional equation (1.1), then there is a unitary character χ of such that f = 1 n χ σ k. Proof. Suppose that f satisfies (1.1). Rewrite (1.1) as P n 1 R σ k (y)f = nf(y)f, for all y. Taking the Fourier transform to the above equation and using the identities given in section 2, we have (3.6) π(σ k (y)) ˆf(π) =nf(y) ˆf(π)

7 Functional equations of Cauchy s and d Alembert s Type 303 Since f 6= 0, there exists [π] Ĝ with ˆf(π) 6= 0. But f is central so ˆf(π) =λ π I, and so (3.7) π(σ k (y)) = nf(y)i, for all y. By applying Lemma 3.2, we conclude from (3.7) that d π =1. From d π =1weseethatπ is a character, say π = χ where χ is a unitary character of. Then (3.7) implies 2 f = 1 n χ σ k. Corollary 3.4. Let be a compact group and σ is a continuous automorphism of such that σ n = I. If f is a central continuous non-zero solution of the functional equation (1.3), then there is a unitary character χ of such that f = χ + χ σ. 2 Acknowledgement. We are greatly indebted to the referee for valuable suggestions that led to an overall improvement of the paper. References [1] M. Akkouchi, B. Bouikhalene, and E. Elqorachi,Functional equations and K-spherical functions. eorgian.math.j, 15, pp. 1-20, (2008). [2] J. An and D. Yang, Nonabelian harmonic analysis and functional equations on compact groups. J. LieTheory, 21, pp , (2011). [3] R. Badora, On a joint generalization of Cauchy s and d Alembert s functional equations, Aequationes Math. 43, pp , (1992). [4] W. Chojnacki, roup representations of bounded cosine functions. J. Reine Angew. Math., 478, pp , (1986).

8 304 Abdellatif Chahbi, Brahim Fadli and Samir Kabbaj [5] W. Chojnacki, On some functional equation generalizing Cauchy s and d Alembert s functional equations. Colloq. Math., 55, pp , (1988). [6] W. Chojnacki, On group decompositions of bounded cosine sequences. Studia Math., 181, pp , (2007). [7] W. Chojnacki,On uniformly bounded spherical functions in Hilbert space, Aequationes Math., 81, pp , (2011). [8] T.M.K. Davison, D Alembert s functional equation on topological groups. Aequationes Math., 76, pp , (2008). [9] T.M.K. Davison, D Alembert s functional equation on topological monoids. Publ. Math. Debrecen, 75, pp , (2009). [10] E. Elqorachi, M. Akkouchi, A. Bakali, B. Bouikhalene, Badora s equation on non-abelian locally compact groups. eorgianmath.j.,11, pp , (2004). [11] P. de Place Friis, D Alembert s and Wilson s equation on Lie groups. Aequationes Math., 67, pp , (2004). [12]. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, FL, (1995). [13] H. Shin ya, Spherical matrix functions and Banach representability for lo- cally compact motion groups. Japan. J. Math.(N.S.), 28, pp , (2002). [14] H. Stetkær, D Alembert s functional equations on metabelian groups, Aequationes Math., 59, pp , (2000). [15] H. Stetkær, D Alembert s and Wilson s functional equations on step 2 nilpotent groups, Aequationes Math., 67, pp , (2004). [16] H. Stetkær, Properties of d Alembert functions, Aequationes Math., 77, pp , (2009). [17] H. Stetkær, Functional Equations on roups. World Scientific Publishing Co. (2013). [18] H. Stetkær, D Alembert s functional equation on groups, Banach Center Publ. 99, pp , (2013).

9 Functional equations of Cauchy s and d Alembert s Type 305 [19] D. Yang, Factorization of cosine functions on compact connected groups, Math.., 254, pp , (2006). [20] D. Yang, Functional Equations and Fourier Analysis, Canad. Math. Bull. 56, pp , (2013). Abdellatif Chahbi Department of Mathematics, University of Ibn tofail, Kenitra, Morroco abdellatifchahbi@gmail.com Brahim Fadli Department of Mathematics, University of Ibn tofail, Kenitra, Morroco himfadli@gmail.com and Samir Kabbaj Department of Mathematics, University of Ibn tofail, Kenitra, Morroco samkabbaj@yahoo.fr