MULTIPLIERS ON SPACES OF FUNCTIONS ON A LOCALLY COMPACT ABELIAN GROUP WITH VALUES IN A HILBERT SPACE. Violeta Petkova

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2 Serdica Math. J. 32 (2006), MULTIPLIERS ON SPACES OF FUNCTIONS ON A LOCALLY COMPACT ABELIAN ROUP WITH VALUES IN A HILBERT SPACE Violeta Petkova Communicated by S. L. Troyansky We prove a representation theorem for bounded operators commuting with translations on L 2 ω(, H), where is a locally compact abelian group, H is a Hilbert space and ω is a weight on. Moreover, in the particular case when = R, we characterize completely the spectrum of the shift operator S 1,ω on L 2 ω (R, H). 1. Introduction. Let be a locally compact abelian group. Denote by Ĝ the dual group of. The groups and Ĝ are equipped with the Haar measure. Let H be a separable Hilbert space and denote by u, v the scalar product of two elements u and v in H. Let ω be a weight on i.e. ω is a continuous, positive, measurable function on such that ω(x + y) 0 < sup < +, y. x ω(x) 2000 Mathematics Subject Classification: Primary 43A22, 43A25. Key words: multipliers, translations, spaces of vector-valued functions.

3 216 Violeta Petkova For 1 p < +, we denote by L p ω(, H) the space of the functions f on with values in H such that x f(x) R + is a function in L p ω(), where L p ω() is the set of the measurable functions g on such that g(x) p ω(x) p dx < +. Let C c () be the space of the continuous functions from into C with compact support. Denote by C c (, H) the space of the functions f on with values in H such that f(.) C c (). For a, define by the formula S a,ω : L p ω(, H) L p ω(, H) (S a,ω f)(x) = f(x a), f L p ω(, H), a.e. and let S a,ω : L p ω () Lp ω () be the operator defined by the formula Notice that we have (S a,ω g)(x) = g(x a), g L p ω (), a.e. S a,ω = S a,ω = sup x ω(x + a), a, ω(x) and consequently ρ(s a,ω ) = ρ(s a,ω ), a. Here ρ(s a,ω ) (resp. ρ(s a,ω )) denotes the spectral radius of S a,ω (resp. S a,ω ). Denote by C c () H the closed vector space generated by functions fu : x f(x)u H with f C c () and u H. The space C c () H is dense in L p ω(, H), for 1 p < +. We say that M is a multiplier on L p ω(, H) if M is a bounded operator from L p ω(, H) into L p ω(, H) such that MS a = S a M, a.

4 Multipliers on spaces of functions Define M p ω the algebra of the multipliers on L p ω(, H). We denote by F (resp. F) the usual Fourier transformation from L 2 (, H) (resp. L 2 ()) into L2 (Ĝ, H) (resp. L2 (Ĝ)). We have the following representation theorem for the multipliers on L p (, H). Theorem 1 ([2]). For every M multiplier on L p (, H), 1 p < +, there exists a measurable function Φ M : Ĝ L(H), which is essentially bounded for the operator norm of L(H) such that F(Mf)(χ) = Φ M (χ)[f(f)(χ)], a.e. on Ĝ, for every f L p (, H) L 2 (, H). Moreover, ess sup Φ M (χ) M. χ The proof of this theorem is based on the well-known result about the multipliers on L p (). Indeed, for every bounded operator M commuting with the translations on L p () there exists a function h L (Ĝ) (see [3]) such that (1.1) Mf = h ˆf, f Cc () and h M. This paper is motivated by a recent result generalizing the representation (1.1) for a more general class of Banach spaces of functions on. The spaces L p ω() are included in this class. Denote by p ω the set of the continuous morphisms θ from into C such that (1.2) f(x)θ 1 (x)dx M f L(L p ω ()), where M f is the operator of convolution by f on L p ω(). Define that p ω = We have the following proposition. p+ ω = { θ, θ p ω}. p+ ω It was proved in [6] that the set is not empty, log-convex and compact for the topology of the uniform convergence on every compact set of. It is clear p+ ω Ĝ. Let be the set of the continuous morphisms from into C.

5 218 Violeta Petkova Proposition 1 (see [6], [7]). group, we have If is either a discrete group or a compact p ω = {θ θ 1 (x) ρ(s x,ω ), x } and p ω is isomorphic to the joint spectrum of {S x,ω } x. The same result holds for = R. Also in [6], was proved the following result, which we will use. Theorem 2 ([6], [7]). Fix θ p ω. For every bounded operator M commuting with the translations on L p ω(), we have (Mg)θ 1 L 2 (), g C c (). There exists a function h M,θ L (Ĝ) such that (Mg)θ 1 = h M,θ (gθ 1 ), g C c () and h M,θ C ω M, where C ω is a constant independent of M. The main result in this paper is the following. Theorem 3. For M M p ω and θ p ω, we have: 1) (Mg)θ 1 L 2 (, H), g C c () H. 2) There exists Φ θ L (Ĝ, L(H)) such that F((Mg)θ 1 )(χ) = Φ θ (χ)[f(gθ 1 )(χ)], g C c () H, a.e. Moreover, ess sup χ Φ θ (χ) C ω M. 2. Proof of Theorem 3. Since Theorem 2 plays an important role in the proof of Theorem 3, for the convenience of the reader we give a sketch of it s proof. The full proof is exposed in [7] and [6]. P r o o f o f T h e o r e m 2. First, every multiplier M in L p ω() is the limit for the strong operators topology of a net (M φα ) where φ α C c () and M φα C ω M. This may be proved using the fact that the restriction of every multiplier on C c () is a convolution with a quasimeasure (see [2]). Fix M M p ω and let (M φα ) be a net satisfying the above property. Fix θ p ω. From the definition of Ĝ, it follows that φ α θ 1 (χ) M φα C ω M, χ p ω.

6 Multipliers on spaces of functions If we replace (φ α ) by a suitable subnet, we obtain that ( φ α θ 1 ) converges to a function h M,θ L (Ĝ) for the weak topology σ(l (Ĝ), L1 (Ĝ)). This implies that for each f C c (), the net ( ) ( ) ( F((M φα f)θ 1 ) = F((φ α f)θ 1 ) = φα θ 1 fθ ) 1 converges to h M,θ fθ 1 with respect to the weak topology of L2 (Ĝ). Consequently, lim α (M φα f)θ 1 = F 1 (h M,θ fθ 1 ), f C c () with respect to the weak topology of L 2 (). On the other hand, since L p ω() L 1 loc () and the inclusion is continuous, for g C c(), we get ( ) lim g(y)θ 1 (y) M φα f(y) Mf(y) dy = 0. α We conclude that for every f C c () the functions (Mf)θ 1 and F 1 (h M,θ fθ 1 ) define the same linear functional on C c () and so (Mf)θ 1 (x) = F 1 (h M,θ fθ 1 )(x), for almost every x. We conclude that (Mf)θ 1 L 2 () and (Mg)θ 1 = h M,θ(gθ 1 ), g C c (). In order to proof Theorem 3, we need the following lemma. Lemma 1. for almost every χ Ĝ. Let g L 2 (, H) and v H. Then we have F( g(.), v )(χ) = F(g)(χ), v, P r o o f. Let g L 2 (, H) and (g n ) n N C c (, H) be a sequence converging to g in L 2 (, H). Then, we have F( g(.), v ) = lim F( g n(.), v ), n + with respect to the norm of L2 (Ĝ). For fixed χ Ĝ and v H, the map C c (Ĝ, H) h h(χ), v C is a continuous linear form. For given φ C c (Ĝ), the integral g n (x)φ(χ)χ 1 (x)dx

7 220 Violeta Petkova is a convergent Bochner integral with values in C c (Ĝ, H). Indeed, we have sup g n (x)φ(χ)χ 1 (x) dx χ φ g n (x) dx < +. Since the Bochner integral commutes with continuous linear maps, we have for almost every χ Ĝ, φ(χ)f( g n (.), v )(χ) = φ(χ) F(g n )(χ), v. Since lim F(g n) = F(g) with respect to the norm of L2 (Ĝ, H), if we replace n + (g n ) n N by a suitable subsequence, we get and hence We conclude that for almost every χ Ĝ. lim F(g n)(χ) F(g)(χ) = 0, a.e. n + lim F(g n)(χ), v = F(g)(χ), v, a.e. n + F( g(.), v )(χ) = F(g)(χ), v, P r o o f o f T h e o r e m 3. Fix M M p ω and fix u and v H. Introduce the operator M u,v defined for f L p ω() by the formula (2.1) M u,v (f)(x) = M(fu)(x), v, a.e. Notice that M u,v (f) L p ω() for every f L p ω(). L p ω(, H), we have M(fu)(x), v p ω(x) p dx Moreover, notice that It is clear that M(fu)(x) p v p ω(x) p dx < +. M u,v M u v. M(S a (fu))(x), v = M(fu)(x a), v, a.e. Indeed, since M(fu)

8 Multipliers on spaces of functions hence M u,v is a multiplier on L p ω(). From Theorem 2, we obtain for every θ p ω, (2.2) (M u,v f)θ 1 L 2 (), f C c () and there exists Φ θ,u,v L (Ĝ) such that (2.3) F((M u,v f)θ 1 )(χ) = Φ θ,u,v (χ)f(fθ 1 )(χ), a.e. Let O be a countable orthonormal basis of H and let F be the set of finite linear combinations of elements of O. We have Φ θ,u,v (χ) C ω M u,v, χ Ĝ\N u,v, where N u,v is a set of measure zero. Without loss of generality, we can modify Φ θ,u,v on N = (u,v) F F N u,v in order to obtain For fixed χ Ĝ\N Φ θ,u,v (χ) C ω M u,v C ω M u v, u, v F, a.e. F F (u, v) Φ θ,u,v (χ) C is a sesquilinear and continuous form on F F and since F is dense in H, we conclude that there exists an unique map H H (u, v) Φ θ,u,v (χ) C such that Φ θ,u,v (χ) = Φ θ,u,v (χ), u, v F. Consequently, there exists an unique map Φ θ : Ĝ L(H) such that It is clear that Φ θ (χ)[u], v = Φ θ,u,v (χ), u, v H. Φ θ (χ) = sup Φ θ (χ)[u], v C ω M, a.e. u =1, v =1 Fix θ p ω and f C c (). For every χ Ĝ, we have fθ 1 (χ)u H. Next for almost every χ Ĝ, we obtain Φ θ (χ)[ fθ 1 (χ)u], v = Φ θ (χ)[u], v fθ 1 (χ)

9 222 Violeta Petkova Consequently, = Φ θ,u,v (χ) fθ 1 (χ) = F((M u,v f)θ 1 )(χ) = F( M[fu], v θ 1 )(χ). (2.4) F 1 ( Φ θ (.)[ fθ 1 (.)u], v )(x) = M[fu](x), v θ 1 (x), for almost every x. Now, consider the function Ψ θ on Ĝ defined for almost every χ Ĝ by the formula Ψ θ (χ) = Φ θ (χ)[ fθ 1 (χ)u] and observe that Ψ θ L2 (Ĝ, H). Indeed, we have C 2 ω M 2 Φ θ (χ)[ fθ 1 (χ)u] 2 dχ Φ θ (χ) 2 fθ 1 (χ)u 2 dχ fθ 1 (χ) 2 u 2 dχ < +. This makes possible to apply Lemma 1, and we get F 1 ( Φ θ (.)[ fθ 1 u(.)], v )(x) = F 1 (Φ θ (.)[ fθ 1 (.)u])(x), v, for almost every x. It follows from (2.4) that we have M[fu](x)θ 1 (x) = F 1 (Φ θ (.)[ fθ 1 (.)u])(x), for almost every x and this yields Moreover, we obtain M[fu]θ 1 L 2 (, H). F(M[fu]θ 1 )(χ) = Φ θ (χ)[ fθ 1 (χ)u], for almost every χ Ĝ.

10 Multipliers on spaces of functions The case = R. In [4] we have established a more complete version of Theorem 2 concerning multipliers on weighted spaces on R. Let w be a weight on R and denote by S ω the operator S 1,ω on L 2 ω(r, H). Define and I ω = [ ln ρ(s 1 ω ), ln ρ(s ω )] Ω ω = {z C, Im z I ω }. For f L 2 ω(r, H) denote by (f) a the function We have the following theorem. (f) a (x) = f(x)e ax, a I ω. Theorem 4 ([4]). Let ω be a weight on R and let M be a multiplier on L 2 ω(r). i) We have (Mf) a L 2 (R), f Cc (R) and there exists h a L (R) such that (Mf) a (x) = h a (x)(f) a (x), a I ω, f Cc (R), a.e. and h a C ω M. ii) If Ω ω, then there exists h H ( Ω ω ), such that for every f C c (R), Mf(z) = h(z) ˆf(z), z Ω ω, where Mf(x + ia) = (Mf) a (x), x + ia Ω ω. Using the same methods as those exposed in Section 2 combined with Theorem 4, we obtain the following interesting version of Theorem 3 in the particular case = R. Theorem 5. Let ω be a weight on R. Let M be a multiplier on L 2 ω(r, H). Then i) We have (Mf) a L 2 (R, H), f C c (R) H and there exists Φ a L (R, L(H)) such that (Mf) a (x) = Φ a (x)[ (f) a (x)], a I ω, f C c (R) H, a.e.

11 224 Violeta Petkova and ess sup Φ a (x) C ω M. x R ii) If Ω ω, then there exists such that for every f C c (R) H, where For every u, v H the function is in H ( Ω ω ). Φ : Ω ω L(H) Mf(z) = Φ(z)[ ˆf(z)], z Ω ω, Mf(x + ia) = (Mf) a (x), x + ia Ω ω. z Φ(z)[u], v Since the proof of Theorem 5 is very similar to that of Theorem 3, we omit the details. Notice that following the results of [4] and [6], if = R, the set p ω given by (1.2), that we use in Theorem 3 is isomorphic to the strip Ω ω and p+ ω the set is isomorphic to the segment I ω. Applying Theorem 5, we get the following proposition. Proposition 2. spec(s ω ) = Let ω be a weight on R. We have { 1 } z C, ρ(sω 1 ) z ρ(s ω). P r o o f. Let α / spec(s ω ). Then M = (S ω αi) 1 is a multiplier. Applying Theorem 5, we get that for every a I ω, there exists Φ a L (R, H) such that (Mf) a (x) = Φ a (x)[(f) a (x)], a I ω, f C c (R) H, a.e. and ess sup Φ a (x) C ω M. x R

12 Multipliers on spaces of functions Replacing f by (S ω αi) 1 g in the above formula, we obtain [ ) ] (g) a (x) = Φ a (x) F (((S ω αi)g) a (x), g C c (R) H, a I ω, a.e. We have ) F (((S ω αi)g) a (x) = (g(t 1) αg(t))e at e itx dt Consequently, we get and hence = (g) a (x)(e ix e a α), g C c (R) H, a I ω, a.e. Φ a (x)[ (g) a (x)] = Φ a (x) 1 (g) e ix e a a (x), a.e. α 1 e ix e a α, a.e This shows that e a α, for every a I ω and from the definition of I ω it follows that { 1 } α / z C, ρ(sω 1 ) z ρ(s ω). We deduce that { 1 } z C, ρ(sω 1 ) z ρ(s ω) spec(s ω ) and this completes the proof. R E F E R E N C E S [1]. I. audry. Quasimeasures and operators commuting with convolution. Pacific J. Math. 18 (1966), [2]. I. audry, B. R. F. Jefferies, W. J. Ricker. Vector-valued multipliers: convolution with operator-valued measures. Dissertations Math. 385 (2000). [3] R. Larsen. The Multiplier Problem. Springer-Verlag, Berlin, 1969.

13 226 Violeta Petkova [4] V. Petkova. Symbole d un multiplicateur sur L 2 ω(r). Bull. Sci. Math. 128 (2004), [5] V. Petkova. Multipliers and Toeplitz operators on Banach spaces of sequences (submitted). [6] V. Petkova. Multipliers on Banach spaces of functions on locally compact abelian group (submitted). [7] V. Petkova. Multiplicateurs sur les espaces de Banach de fonctions sur un groupe localement compact abélien. Thése de l Université Bordeaux 1, (2006), ( Violeta Petkova Laboratoire Bordelais d Analyse et éométrie U.M.R Université Bordeaux 1 351, cours de la Libération Talence, France petkova@math.u-bordeaux1.fr Received January 18, 2006 Revised May 12, 2006