R -bounded Representations of L 1 ( G )

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(1)Positivity 12 (2008), 151–166 c 2007 Birkh¨ auser Verlag Basel/Switzerland 1385-1292/010151-16, published online October 29, 2007 DOI 10.1007/s11117-007-2130-6. Positivity. R-bounded Representations of L1 (G) Ben de Pagter and Werner J. Ricker Dedicated to the memory of H. H. Schaefer. Abstract. We investigate R-bounded representations Ψ : L1 (G) → L (X), where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism U : G → L (X), we are then able to analyze certain classical homomorphisms U (e.g. translations in Lp (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Mathematics Subject Classification (2000). Primary 22D12, 46B20; Secondary 43A25, 46B42, 47B40, 47B60. Keywords. R-boundedness, representations, spectral measure, Banach lattice.. 1. Introduction The notion of R-boundedness for families of linear operators was formally introduced by E. Berkson and T.A. Gillespie in [3] (where it is called the R-property), although it was already implicit in earlier work of J. Bourgain, [2]. Since its conception in the mid-1990’s, R-boundedness has played an increasingly important role in various branches of functional analysis, operator theory, harmonic analysis and partial differential equations; see for example, [1], [4], [5], [6], [11], [12], [14], [24], [25] and the references therein. One of our aims is to make a study, from the viewpoint of R-boundedness, of continuous representations Ψ : L1 (G) → L (X). Here X is a Banach space and L (X) is the Banach algebra of all bounded linear operators of X into itself, equipped with the operator norm; its unit is the identity operator I on X. Moreover, for G a locally compact abelian (briefly, lca) group, the Banach algebra L1 (G) will always be equipped with convolution as its multiplication. A representation Ψ : L1 (G) → L (X) is called essential if the span of f ∈L1 (G) Ψ(f ) X is dense in X. It is shown (cf. Proposition 2.2) that an essential representation Ψ is Fourier R-bounded, meaning that (1) V1 (Ψ) = Ψ(f ) : f ∈ L1 (G) , fˆ ≤ 1 ⊆ L (X) ∞.

(2) 152. B. de Pagter and W. J. Ricker. Positivity. is R-bounded, if and only if there exists a regular R-bounded spectral measure ˆ → L (X) such that P : B(G) Ψ(f ) = (2) fˆ dP, f ∈ L1 (G) . ˆ G. ˆ denotes the dual lca group of G and fˆ the Fourier transform of f ∈ L1 (G). Here G Moreover, there then exists a special kind of group homomorphism U : G → L (X), continuous for the strong operator topology (briefly, sot) in L (X), such that f (g) U (g) dλ(g) , f ∈ L1 (G) , (3) Ψ(f ) = G. where λ is Haar measure in G. Indeed, U is precisely the Fourier-Stieltjes transform of the measure P , that is, (−g, u) dP (u) , g ∈ G, U (g) = ˆ G. ˆ its value at h ∈ G is denoted by (h, u). This can be where for each character u ∈ G, viewed as an analogue of a result due to I. Kluv´ anek, [13], Theorem 4, stating that representations of the form (3), with U weak operator continuous, have an integral ˆ representation of the form (2) relative to some regular spectral measure P on B(G) precisely when the set V1 (Ψ), defined by (1), is relatively compact for the weak operator topology in L (X). We also provide some techniques which can be used to show that certain classical homomorphisms U : G → L (X), with X a Banach function space over λ (e.g. X = Lp (G)), fail to have an R-bounded range in L (X). Consequently, the corresponding representation Ψ : L1 (G) → L (X), given by (3), cannot be R-bounded either and hence, “most” elements of {U (g) : g ∈ G} fail to be scalar-type spectral operators. The theory of spectral measures and scalar-type spectral operators and its relation to the theory of Banach lattices, of which certain aspects are treated in this note, was a favourite and well studied area of research of Prof. H.H. Schaefer. This is transparent in his well known monograph [22], especially Chapter V; see also the many relevant references in [22] to his own papers related to this topic.. 2. Representations of G and L1 (G) We investigate representations Ψ : L1 (G) → L (X), where (X, ·) is a complex Banach space, G is a lca group and L1 (G) has convolution as its multiplication. ˆ is the dual group and λ denotes Haar measure on G. The notion of Recall that G R-boundedness will play an important role as will the results of [18]..

(3) Vol. 12 (2008). R-bounded Representations of L1 (G). 153. First we recall the following definition. Given a Banach space X, a non-empty collection T ⊆ L (X) is called R-bounded if there exists M ≥ 0 such that ⎛ ⎛ 2 ⎞1/2 2 ⎞1/2 1 . 1 . n n ⎟ ⎟ ⎜ ⎜ rj (t) Tj xj dt⎠ ≤M⎝ rj (t) xj (4) ⎝ dt⎠ 0 j=1 0 j=1 n. n. ∞. for all {Tj }j=1 ⊆ T , all {xj }j=1 ⊆ X and all n ∈ N {0}. Here {rj }j=1 is the sequence of Rademacher functions on the interval [0, 1]. By Kahane’s inequality, it is possible to replace the exponent 2 in (4) by any p ∈ (0, ∞) (with a different constant, depending on p). If T ⊆ L (X) is R-bounded, then the smallest constant M ≥ 0 for which (4) holds, will be denoted by MT and is called the R-bound of T . Evidently, every R-bounded collection is uniformly bounded. For more information about R-boundedness we refer to [4], [26]. In particular, the sot closed absolute convex hull of any R-bounded collection is also R-bounded. Let Ω be a locally compact Hausdorff space and C0 (Ω) be the Banach algebra of all continuous functions on Ω which vanish at infinity (under pointwise operations), equipped with the sup-norm. A continuous linear homomorphism Φ : C0 (Ω) → L (X) is called a representation. If the span of {Φ(f ) X : f ∈ C0 (Ω)} is dense in X, then Φ is said to be essential. Let Ω∞ = Ω ∪ {∞} be the onepoint compactification of Ω. Then each f ∈ C (Ω∞ ) has a unique decomposition f = f0 + f (∞) 1 with f ∈ C0 (Ω), where 1 denotes the constant function with value 1 on Ω∞ . Moreover, f0 ∞ ≤ 2 f ∞ , f ∈ C (Ω∞ ) ,. (5). and the Borel σ-algebra B (Ω) in Ω is a σ-ideal in B (Ω∞ ). Define Φ∞ : C (Ω∞ ) → L (X) by f −→ Φ (f0 ) + f (∞) I, in which case Φ∞ (g) = Φ(g) for all g ∈ C0 (Ω). It is routine to check that Φ∞ is a linear, unital homomorphism. Moreover, via (5) it follows that Φ∞ is continuous. It is also clear from (5), that W1 (Φ∞ ) ⊆ 2W1 (Φ) + {λI : |λ| ≤ 1} ,. (6). where W1 (Φ) = {Φ (f ) : f ∈ C0 (Ω) , f ∞ ≤ 1} and W1 (Φ∞ ) is defined similarly. Proposition 2.1. Let Φ : C0 (Ω) → L (X) be an essential representation such that W1 (Φ) = {Φ(h) : h ∈ C0 (Ω) , h∞ ≤ 1} is an R-bounded subset of L (X). Then there exists a regular, R-bounded, sot σ-additive spectral measure P : B (Ω) → L (X) such that Φ(h) = h dP, h ∈ C0 (Ω) . Ω.

(4) 154. B. de Pagter and W. J. Ricker. Positivity. Proof. Let Φ∞ : C (Ω∞ ) → L (X) be defined as above. By (6) it follows that Φ∞ is R-bounded (that is, the set W1 (Φ∞ ) is R-bounded). By [18], Proposition 2.17, there exists a regular, R-bounded, sot σ-additive spectral measure P∞ : B (Ω∞ ) → L (X) such that f dP∞ , f ∈ C (Ω∞ ) . Φ∞ (f ) = Ω∞. Since B (Ω) ⊆ B (Ω∞ ), we can define P : B (Ω) → L (X) by P (A) = P∞ (A) for each A ∈ B (Ω). If we can show that P (Ω) = I, then it is evident that P is a regular, sot σ-additive spectral measure satisfying Φ(f ) = Φ∞ (f ) = f dP∞ = f dP∞ = f dP Ω∞. Ω. Ω. for all f ∈ C0 (Ω). So, it remains to show that P (Ω) = I, which clearly is equivalent to showing that P∞ ({∞}) = 0. Let B (Ω∞ ) be the space of all bounded Borel func˜ ∞ : B (Ω∞ ) → L (X) tions on Ω∞ equipped with the sup-norm. We denote by Φ the multiplicative linear extension of Φ∞ which is given by the spectral integral ˜ ∞ (f ) = Φ f dP ∞ for all f ∈ B (Ω∞ ) (see [8] or Section 2 of [18], for example). Ω∞ If f ∈ C0 (Ω), then f χ{∞} = 0 and so, ˜ ∞ f χ{∞} = 0. ˜ ∞ (f ) Φ ˜ ∞ χ{∞} = Φ Φ(f ) P∞ ({∞}) = Φ Accordingly, Φ(f ) X ⊆ (I − P∞ ({∞})) X and so, {Φ(f ) X : f ∈ C0 (Ω)} ⊆ (I − P∞ ({∞})) X. Since Φ is essential, the span of the left-hand side is dense in X and so, we may conclude that (I − P∞ ({∞})) X = X. Hence, P∞ ({∞}) = 0. ˆ For each f ∈ L1 (G), the Fourier transform Let G be a lca with dual group G. fˆ is given by ˆ f (g) (−g, u) dλ (g) , u ∈ G. fˆ (u) = G. The subalgebra. ∧ L1 (G) = fˆ : f ∈ L1 (G). ˆ (for details we refer the reader to e.g. [21]). Let is known to be dense in C0 (G) 1 (meaning also continuous) which is essenΨ : L (G) → L (X) be a representation tial (that is the span of f ∈L1 (G) Ψ(f )X is dense in X). By injectivity of the ˆ the map Φ : L1 (G)∧ → L (X) given by Fourier transform from L1 (G) into C0 (G), ∧ Φ : fˆ −→ Ψ(f ) , fˆ ∈ L1 (G) ,. (7). is well defined. Standard properties of the Fourier transform imply that Φ is linear and multiplicative..

(5) R-bounded Representations of L1 (G). Vol. 12 (2008). 155. Proposition 2.2. Let Ψ : L1 (G) → L (X) be an essential representation which is Fourier R-bounded, that is, the set V1 (Ψ) as given by (1) is R-bounded in L (X). Then there exists an R-bounded, regular, sot σ-additive spectral measure ˆ → L (X) such that P : B(G) Ψ(f ) = (8) fˆ dP, f ∈ L1 (G) . ˆ G. Conversely, given any R-bounded, regular, sot σ-additive spectral measure P : ˆ → L (X), the map Ψ defined via (8) is an essential, Fourier R-bounded B(G) representation. Proof. Since V1 (Ψ) is uniformly bounded, we have (in the notation of (7)) that sup Φ(fˆ) : f ∈ L1 (G) , fˆ ≤ 1 < ∞. ∞. 1. ∧. ˆ it follows that Φ extends uniquely to an essential By density of L (G) in C0 (G) ˆ → L (X). Since representation Φ : C0 (G) ˆ h ≤ 1 W1 (Φ) = Φ(h) : h ∈ C0 (G), ∞ is contained in the operator norm closure of V1 (Φ), it follows that the representation Φ is R-bounded. According to Proposition 2.1 there exists an R-bounded, ˆ → L (X) such that regular, sot σ-additive spectral measure P : B(G) ˆ Φ(h) = h dP, h ∈ C0 (G). (9) ˆ G. It is then clear that (8) follows from (7). ˆ → L (X) be any R-bounded, regular, sot σ-additive Conversely, let P : B(G) spectral measure and define Ψ : L1 (G) → L (X) by (8). Via the properties of spectral integration and the Fourier transform, Ψ is linear, multiplicative and also continuous, since fˆ dP ≤ 4M fˆ ≤ 4M f 1 , f ∈ L1 (G) , ˆ ∞ G ˆ < ∞. Moreover, it is clear that the correwhere M = sup P (A) : A ∈ B(G) ˆ sponding map Φ, defined by (7), extends to a representation Φ of C0 (G) given by ˆ It follows that Φ(h) = ˆ h dP for h ∈ C0 (G). G. V1 (Ψ) ⊆ W1 (Φ) ⊆ 4aco (Ran(P )) and hence, the Fourier R-boundedness of Ψ follows from that of P . It remains to show that Ψ is essential. Let Y = Ψ(f ) X : f ∈ L1 (G) and suppose that x∗ ∈ X ∗ satisfies y, x∗

(6) = 0 for all y ∈ Y . Fix x ∈ X. Then ∗ ∗ ˆ ˆ f d P x, x

(7) = f dP x, x = 0, f ∈ L1 (G) . ˆ G. ˆ G.

(8) 156. B. de Pagter and W. J. Ricker. Positivity. ˆ it follows from the density Since P x, x∗

(9) is a finite regular Borel measure on G, ∧ 1 ∗ ˆ ˆ In particular, of L (G) in C0 (G) that P x, x

(10) is the zero measure on B(G). ∗ ∗ ˆ x, x

(11) = P x, x

(12) (G) = 0. Since x ∈ X is arbitrary, we conclude that x∗ = 0 and hence the span of Y is dense in X, that is, Ψ is essential. . As a small curiosity we provide a simple application; see [23], Lemma 1, for an alternative proof. Example 2.3. Let X be a Banach space and T ∈ L (X) satisfy T n = I for some n ∈ N. Then T is a scalar-type spectral operator. For a proof, let G = Zn be ˆ = 1, ω, . . . , ω n−1 consists of the finite additive group Z (n), in which case G the n-th roots of unity (under multiplication), where ω = exp (2πi/n). Normalized ˆ Haar measure λ on G is determined by λ ({k}) = 1/n for each k ∈ G and γ ∈ G k acts on G via k −→ γ for k ∈ G. For f : G → C we have n−1 1. ˆ f (k) γ¯ k , γ ∈ G. fˆ(γ) = n k=0. From the Fourier inversion formula ˆ (γ) , k ∈ G, (k, γ) fˆ (γ) dλ f (k) = ˆ G. it is immediate that. ˆ G) ˆ f ∞ ≤ λ( fˆ. ∞. = n fˆ. ∞. , f ∈ L1 (G) .. (10). Define a representation Ψ : L1 (G) → L (X) by f −→ k∈G f (k) T k , in which case . ˆ k 1 f (k) T : f ∈ L (G) , f ≤ 1 . V1 (Ψ) = ∞. k∈G. ˆ k 1 T and so, if f ∈ L (G) satisfies But, Ψ(f ) = n2 k∈G fn(k) f ≤ 1, then (10) 2 ∞ ≤ 1. Accordingly, implies that k∈G |fn(k)| 2 V1 (Ψ) ⊆ n2 aco T k : k ∈ G and so, Ψ is a Fourier R-bounded representation. By Proposition 2.2 (or even [13], ˆ → L (X) such that Theorem 4) there is a spectral measure P : B(G) Ψ(f ) = fˆ dP, f ∈ L1 (G) . ˆ G. ˆ yields T = Ψ χ{1} = Choosing f = χ{1} , in which case fˆ(γ) = γ¯ on G, ¯ dP (γ). In particular, T is a scalar-type spectral operator. ˆγ G.

(13) Vol. 12 (2008). R-bounded Representations of L1 (G). 157. ˆ → L (X) be any regular, sot σ-additive spectral measure. For Let P : B(G) ˆ is a bounded Borel function and each g ∈ G, the function u −→ (g, u), for u ∈ G, so, UP (g) =. ˆ G. (g, u) dP (u). exists as an element of L (X). If 0 is the unit element of G, then UP (0) = Gˆ 1 dP = ˆ = I. Moreover, the multiplicative property of spectral integrals implies that P (G) UP (g1 + g2 ) = UP (g1 ) UP (g2 ), for gj ∈ G. For each x ∈ X, we see that (g, u) dP x (u) , g ∈ G, (11) UP (g) x = ˆ G. ˆ →X is the Fourier-Stieltjes transform of the regular vector measure P x : B(G) given by A −→ P (A) x. By the proof of Theorem 2 in [13] it follows that g −→ UP (g) x is continuous from G into X. Hence, the homomorphism UP : g −→ UP (g) of G into L (X) is continuous for the sot. For x ∈ X fixed, the function UP (·) x : G → X is bounded and continuous. So, if H is any σ-compact subset of G, then the set {UP (g) x : g ∈ H} is separable in X. Hence, since UP (·) x, x∗

(14) is measurable for each x∗ ∈ X ∗ , the Pettis measurability theorem ensures that the X-valued function χH UP (·) x is λ-measurable ∞ (that is, there is a sequence {sn }n=1 of X-valued, B (G)-simple functions such that sn → χH UP (·) x, as n → ∞, pointwise λ-a.e. on G). Given any f ∈ L1 (G), there exists a σ-compact set Hf ⊆ G such that f (g) = 0 λ-a.e. on GHf . Hence, the function f UP (·) x is λ-measurable and satisfies G f (g) UP (g) x dλ(g) < ∞, that is, f UP (·) x is Bochner λ-integrable. It then follows from (11) and Fubini’s theorem that f (g) UP (g) x dλ(g) , f ∈ L1 (G) , ΨP (f ) x = G. where the representation ΨP : L1 (G) → L (X) is defined by ΨP (f ) = Gˆ fˆ dP . Therefore, with the Bochner λ-integral defined relative to the sot, we have ΨP (f ) = f (g) UP (g) dλ(g) , f ∈ L1 (G) . (12) G. Moreover, if P is R-bounded, then the essential representation ΨP is also R-bounded. The following result shows that “reasonable” group homomorphisms U : G → L (X) always induce representations Ψ : L1 (G) → L (X). Lemma 2.4. Let U : G → L (X) be a homomorphism which is continuous for the sot and bounded (that is, M = supg∈G U (g) < ∞). Define ΨU : L1 (G) → L (X) by ΨU (f ) =. f (g) U (g) dλ(g) , G. f ∈ L1 (G) ,. (13). where the right-hand side of (13) is defined as a Bochner λ-integral in the sot. Then ΨU is an essential representation..

(15) 158. B. de Pagter and W. J. Ricker. Positivity. Proof. It is evident that ΨU is linear and an application of Fubini’s theorem shows that ΨU is multiplicative on the convolution algebra L1 (G) (cf. [13], Lemma 5). Moreover, it is clear that ΨU (f ) ≤ M f 1 for each f ∈ L1 (G) and so, ΨU is continuous (cf. [13], Theorem 1). It remains to show that ΨU is essential, that is, the span of Y = ΨU (f ) X : f ∈ L1 (G) is dense in X. Let x∗ ∈ X ∗ satisfy y, x∗

(16) = 0 for all y ∈ Y . If x ∈ X, then f (g) U (g) x, x∗

(17) dλ(g) = f (g) U (g) x dλ(g) , x∗ G. G. = ΨU (f ) x, x∗

(18) = 0. for all f ∈ L1 (G). Since the function g −→ U (g) x, x∗

(19) is bounded and continuous, it follows that U (g) x, x∗

(20) = 0 for all g ∈ G. In particular, x, x∗

(21) = U (e) x, x∗

(22) = 0. This holds for all x ∈ X and so, x∗ = 0. This shows that the span of Y is indeed dense in X. Remark 2.5. The converse of Lemma 2.4 is also valid. That is, every essential representation Ψ : L1 (G) → L (X) has the form (13) for some (unique) bounded homomorphism U : G → L (X) which is continuous for the sot. To see this, let D = f ∈L1 (G) Ψ (f ) X, in which case span (D) is dense in X. Fix g ∈ G. For n n x = j=1 Ψ (fj ) xj in span (D) define y = j=1 Ψ (τg fj ) xj , where τg : L1 (G) → L1 (G) is the isometric operator of translation by g. Let {uα } ⊆ L1 (G) be any approximate identity with uα 1 ≤ 1 for all α. Then the net yα =. n. Ψ (uα ∗ τg fj ) xj =. j=1. n. Ψ ((τg uα ) ∗ fj ) xj = Ψ (τg uα ) x. j=1. converges to y in X. Since yα ≤ Ψ x for all α, it follows that y ≤ Ψ x. This inequality ensures that U (g) : span (D) → X as specified by ⎞ ⎛ n n. Ψ (fj ) xj ⎠ = Ψ (τg fj ) xj U (g) ⎝ j=1. j=1. is well defined, linear and bounded. Its continuous linear extension to X is again denoted by U (g) and we have U (g) ≤ Ψ ,. g ∈ G.. Clearly, U (0) = I. To verify that U is a homomorphism it suffices to show that U (g1 + g2 ) = U (g1 ) U (g2 ) on D, for all g1 , g2 ∈ G; this follows easily from the definition of the operators U (g), g ∈ G. Next, for fixed f ∈ L1 (G) and x ∈ X, it follows from the formula U (g) (Ψ (f ) x) = Ψ (τg f ) x, together with continuity of the map g −→ τg f from G into L1 (G), that g −→ U (g) (z) is continuous from G into X for all z ∈ span (D). The uniform boundedness of {U (g) : g ∈ G} can then be invoked to deduce the sot continuity of U : G → L (X). The same argument.

(23) R-bounded Representations of L1 (G). Vol. 12 (2008). 159. as given for UP just prior to Lemma 2.4 then shows that g −→ f (g) U (g) x is Bochner λ-integrable for each f ∈ L1 (G) and x ∈ X. That is, the linear map f (g) U (g) x dλ (g) , x ∈ X, Tf : x −→ G. belongs to L (X). It remains to verify that Tf = Ψ (f ) , f ∈ L1 (G) , for which it suffices to show that Tf x = Ψ (f ) x for each x ∈ D. So, let x = Ψ (u) y for some u ∈ L1 (G) and y ∈ X. Then Tf x = G f (g) Ψ (τg u) y dλ (g). Since f has σ-compact support and g −→ τg u is bounded and continuous from G into L1 (G), it follows that g −→ f (g) τg u is Bochner λ-integrable and hence, by continuity of Ψ, that f (g) Ψ (τg u) y dλ (g) = Ψ f (g) τg u dλ (g) y = Ψ (f ∗ u) y = Ψ (f ) x. G. G. This establishes that Ψ has the form (13). Finally, for the uniqueness, suppose that V : G → L (X) is another bounded, sot-continuous homomorphism with f (g) V (g) x dλ (g) , f ∈ L1 (G) , x ∈ X. Ψ (f ) x = G. Then U (g) x, x∗

(24) = V (g) x, x∗

(25) for λ-a.e. g and hence, by continuity, for all g ∈ G, with x ∈ X and x∗ ∈ X ∗ arbitrary. It follows that V (g) = U (g) for all g ∈ G. In view of Lemma 2.4 and Remark 2.5 there arises the question of which R-bounded, sot-continuous homomorphisms U : G → L (X) are given as the Fourier-Stieltjes transforms of R-bounded spectral measures? For the case G = R, a result (due to N. Kalton and L. Weis) related to the following proposition can be found in [25] (Corollary 7.6 and Note). Proposition 2.6. Let U : G → L (X) be a bounded and sot-continuous homomorphism. The following two statements are equivalent: (i). the set. G. f (g) U (g) dλ(g) : f ∈ L1 (G) , fˆ. ∞. ≤1. (14). is R-bounded in L (X); ˆ → (ii). there exists an R-bounded, regular, sot σ-additive spectral measure P : B(G) L (X) such that U (g) = Gˆ (g, u) dP (u) for every g ∈ G. Moreover, if these conditions are satisfied, then the homomorphism U is R-bounded, that is, the set {U (g) : g ∈ G} is R-bounded in L (X)..

(26) 160. B. de Pagter and W. J. Ricker. Positivity. Proof. First assume that condition (i) is satisfied. Define the essential represen tation ΨU : L1 (G) → L (X) by ΨU (f ) = G f (g) U (g) dλ(g) for f ∈ L1 (G) (see Lemma 2.4). The set (14) is equal to V1 (ΨU ) and so, (i) states that ΨU is Fourier R-bounded. It follows from Proposition 2.2 that there exists a regular, R-bounded, ˆ → L (X) such that ΨU (f ) = ˆ fˆ dP sot σ-additive spectral measure P : B(G) G for all f ∈ L1 (G). For each x ∈ X and x∗ ∈ X ∗ we have, by Fubini’s theorem, that ∗ ∗ ∗ ˆ f (g) U (g) x, x

(27) dλ(g) = ΨU (f ) x, x

(28) = f dP x, x ˆ G G = fˆ(u) d P x, x∗

(29) (u) ˆ G = (g, u)f (g) dλ(g) d P x, x∗

(30) (u) ˆ G G f (g) (g, u) d P x, x∗

(31) (u) dλ(g) = G. ˆ G. for all f ∈ L (G). Since the functions g −→ U (g) x, x∗

(32) , Gˆ (g, u) d P x, x∗

(33) (u) are both bounded and continuous on G, this implies that U (g) x, x∗

(34) = (g, u) d P x, x∗

(35) (u) , g ∈ G. 1. ˆ G. ∗. ∗. But, x ∈ X and x ∈ X are arbitrary and so, we may conclude that U (g) = ˆ (g, u) dP (u), g ∈ G. It follows that G {U (g) : g ∈ G} ⊆ 4aco (Ran(P )) , where aco denotes the sot closure of the absolutely convex hull, and hence, that U is R-bounded. Now suppose that condition (ii) is satisfied. Define the essential representa tion ΨP : L1 (G) → L (X) by ΨP (f ) = Gˆ fˆ dP , f ∈ L1 (G). As observed in the discussion preceding Lemma 2.4, ΨP is then also given by ΨP (f ) = f (g) U (g) dλ(g), f ∈ L1 (G). By Proposition 2.2, ΨP is Fourier R-bounded, G that is, the set (14) is R-bounded. Therefore, condition (i) is satisfied. Remark 2.7. Recall that a Banach space X has property (α), [19], Definition 2.1, if there exists a constant α ≥ 0 such that 2 1 1 . n m. ε r (s) r (t) x jk j k jk dsdt 0 0 j=1 k=1 2 1 1 . m. n 2 ≤α rj (s) rk (t) xjk dsdt 0 0 j=1 k=1 for every choice of xjk ∈ X, εjk ∈ {−1, 1} and all m, n ∈ N {0}. It is shown in [19], Proposition 2.1, that every Banach space with l.u.st. and having finite.

(36) Vol. 12 (2008). R-bounded Representations of L1 (G). 161. cotype necessarily has property (α). In particular, every Banach lattice, which automatically has l.u.st., [7], Theorem 17.1, with finite cotype has property (α). As observed in [19], Remark 2.2, a Banach space with property (α) cannot contain n∞ uniformly. For a Banach space, the property of not containing n∞ uniformly is equivalent to having finite cotype, a deep result due to B. Maurey and G. Pisier, [16] (see also [7], Chapter 14). Consequently, in spaces with l.u.st., having property (α) is equivalent to finite cotype. If the Banach space X has property (α), then condition (ii) in Proposition 2.6 above may be replaced with: there exists a regular, sot σ-additive spectral measure P : B (G) → L (X) such that U (g) = Gˆ (g, u) dP (u) for every g ∈ G. Indeed, the range of P is a bounded Boolean algebra of projections in X and it follows from [17], Theorem 3.3, that such a Boolean algebra is always R-bounded (provided X has property (α)). Let (E, ·) be a (complex) Banach lattice. Recall that E satisfies a lower p-estimate, with 1 ≤ p < ∞, respectively, upper q-estimate, with 1 < q ≤ ∞, if there exists a constant K > 0 such that ⎛ ⎞ p1 . n. n p ⎝ xj xj ⎠ , ≥K j=1 j=1 respectively,. and. ⎛ ⎞ q1 n n. q ⎝ xj xj ⎠ , q < ∞, ≤K j=1 j=1 . n x j ≤ K max {xj } , q = ∞, j j=1 . for any finite disjoint system {x1 , . . . , xn } in E and all n ∈ N ([15], p. 82). Clearly, any Lp -space, with 1 ≤ p < ∞, satisfies a lower p-estimate and, for 1 < p ≤ ∞, an upper p-estimate. The following observation will be useful later. Proposition 2.8. Suppose that (E, ·) is a (complex) Banach lattice and T is a non-empty subset of L (E). (a). Suppose that E satisfies a lower p-estimate for some 1 ≤ p < 2. Assume that there exists a constant c > 0 with the property that for every n ∈ N there exist T1 , . . . , Tn ∈ T and x = 0 in E such that {T1 x, . . . , Tn x} is a disjoint system in E and Tj x ≥ c x for all j = 1, . . . , n. Then T is not R-bounded. (b). Suppose that E satisfies an upper q-estimate for some 2 < q ≤ ∞. Assume there exists a constant c > 0 with the property that for every n ∈ N, there exist R1 , . . . , Rn ∈ T and a disjoint system {y1 , . . . , yn } in E such that Rj yj = x = 0 for all j and yj ≤ c x. Then T is not R-bounded..

(37) 162. B. de Pagter and W. J. Ricker. Positivity. Proof. (a). Suppose that T is R-bounded. Then there exists a constant MT ≥ 0 such that ⎛ ⎛ 2 ⎞ 12 2 ⎞ 12 1 . 1 . n n ⎟ ⎟ ⎜ ⎜ rj (t) Sj xj dt⎠ ≤ MT ⎝ rj (t) xj (15) ⎝ dt⎠ 0 j=1 0 j=1 for all choices of S1 , . . . , Sn ∈ T and x1 , . . . , xn ∈ X and all n ∈ N. By hypothesis, given n ∈ N there exist x = 0 in E and T1 , . . . , Tn ∈ T such that {T1 x, . . . , Tn x} is a disjoint system in E and Tj x ≥ c x for all j = 1, . . . , n. Using the disjointness of the elements T1 x, . . . , Tn x, we find that . . . n n n rj (t) Tj x = |rj (t) Tj x| = |Tj x| j=1 j=1 j=1 for all t ∈ [0, 1]. Since E satisfies a lower p-estimate, we also have (for some K > 0) that ⎛ ⎞ p1 ⎞ p1 ⎛ n n n. . 1 p p ⎝ |Tj x| Tj x ⎠ ≥ K ⎝ cp x ⎠ = Kcn p x . ≥K j=1 j=1 j=1 Hence,. ⎛ ⎜ ⎝. 0. 1. 2 ⎞ 12 . ⎟ n 1 p r (t) T x j j dt⎠ ≥ Kcn x . j=1. On the other hand, it follows from (15) that ⎛ ⎛ 2 ⎞ 12 2 ⎞ 12 1 . 1 . n n ⎟ ⎟ 1 ⎜ ⎜ 2 rj (t) Tj x dt⎠ ≤ MT ⎝ rj (t) x ⎝ dt⎠ = MT n x . 0 j=1 0 j=1 1. 1. This implies that Kcn p x ≤ MT n 2 x. Since 1 ≤ p < 2 and n ∈ N is arbitrary, this is a contradiction. Therefore, we may conclude that T is not R-bounded. (b). Assume that T is R-bounded and let the constant MT be as in (15). n n Given n ∈ N, choose {Rj }j=1 ⊆ T and {yj }j=1 ⊆ E as in statement (b) of the proposition and set x = Rj yj (same value for all j). Evidently, 2 1 n dt = nx2 . Furthermore, using the disjointness of the sysj=1 rj (t)Rj yj 0 2 2 1 n n n tem {yj }j=1 , we also have 0 j=1 rj (t) yj dt = j=1 |yj | . Since E satisfies an upper q-estimate, there exists a constant K > 0 such that (for 2 < q < ∞) ⎞ q1 ⎛ n n. . 1 1 q ⎝ n 2 x ≤ MT |yj | yj ⎠ ≤ MT K x n q , ≤ MT K j=1 j=1.

(38) Vol. 12 (2008) 1. R-bounded Representations of L1 (G). 163. 1. that is, n 2 ≤ MT Kn q . Since 2 < q < ∞ and n ∈ N is arbitrary, this is a 1 contradiction. In the case q = ∞ we find similarly that n 2 ≤ MT K, which also leads to a contradiction. Hence, T cannot be R-bounded. The following result illustrates the use of Proposition 2.8. Proposition 2.9. Let E be a Banach lattice which satisfies either a lower p-estimate for some 1 ≤ p < 2 or an upper q-estimate for some 2 < q ≤ ∞ and H be a bounded, multiplicative group in L (E). Suppose that for every n ∈ N there exist T1 , . . . , Tn ∈ H and x = 0 in E with the property that {T1 x, T2 x, . . . , Tn x} is a disjoint system in E. Then H is not R-bounded. Proof. Let M = sup {T : T ∈ H}. Suppose that E satisfies a lower p-estimate for some 1 ≤ p < 2. Given n ∈ N, let T1 , . . . , Tn ∈ H and 0= x ∈ Ebe as in the statement of the proposition. For 1 ≤ j ≤ n, we have x = Tj−1 Tj x ≤ M Tj x and so, Tj x ≥ c x (where c = M −1 ). The conclusion follows from Proposition 2.8(a). Now suppose that E satisfies an upper q-estimate for some 2 < q ≤ ∞. Given n ∈ N, let T1 , . . . , Tn ∈ H and 0 = x ∈ E be as in the statement of the proposition. Define Rj = Tj−1 and yj = Tj x for 1 ≤ j ≤ n. Then Rj yj = x = 0 for all j and {y1 , . . . , yn } is a disjoint system. Since yj ≤ M x for j = 1, . . . , n, we may apply Proposition 2.8(b) to conclude that H fails to be R-bounded. A homomorphism U : Z → L (X) is necessarily of the form m −→ T m , for m ∈ Z, where T = U (1). In particular, U is a bounded homomorphism if and only if supm∈Z T m < ∞. In this case the isomorphism T : X → X is called power bounded. The next corollary follows immediately from Proposition 2.9. Corollary 2.10. Let E be a Banach lattice which satisfies either a lower p-estimate for some 1 ≤ p < 2 or an upper q-estimate for some 2 < q ≤ ∞. Let T ∈ L (E) be a power bounded operator such that, for k1 , . . . , kn ∈ Z every n ∈ N, there exist and x = 0 in E with the property that T k1 x, T k2 x, . . . , T kn x is a disjoint system in E. Then the homomorphism m −→ T m , for m ∈ Z, of Z in E fails to be R-bounded. As another consequence of Proposition 2.9 we mention the following result. For G = H = R, see [14], Example 2.12. Proposition 2.11. Let G be a lca group and p ∈ [1, 2) ∪ (2, ∞]. For any h ∈ G, the translation operator τh : Lp (G) → Lp (G) is defined by (τh f ) (g) = f (g − h) for λ-a.e. g ∈ G, and all f ∈ Lp (G). If H is an infinite subgroup of G, then {τh : h ∈ H} is not R-bounded. Proof. For every p ∈ [1, 2) ∪ (2, ∞] the Banach lattice E = Lp (G) satisfies the hypothesis of Proposition 2.9 and H = {τh : h ∈ H} is surely a bounded, multiplicative group in L (E). Given n ∈ N, there exist mutually different elements h1 , . . . , hn ∈ H. Hence, there exists a compact neighbourhood K of the unit.

(39) 164. B. de Pagter and W. J. Ricker. Positivity. element (so λ (K) > 0) such that K + h1 , . . . , K + hn are pairwise disjoint. Accordingly, χK+hj : 1 ≤ j ≤ n is a disjoint family in E. Since χK+hj = τhj (χK ) for j = 1, . . . , n, it follows from Proposition 2.9 that H is not R-bounded. Corollary 2.12. Let G be a lca group and p ∈ [1, 2) ∪ (2, ∞]. If h ∈ G has infinite order, then the bounded homomorphism m −→ τhm of Z in Lp (G) is not R-bounded. Corollary 2.13. If G is a lca group, p ∈ [1, 2) ∪ (2, ∞] and h ∈ G has infinite order, then the translation operator τh : Lp (G) → Lp (G) fails to be a scalar-type spectral operator. Proof. Let T = {z ∈ C : |z| = 1} denote the circle group, in which case the spectrum σ (τh ) ⊆ T, as τh is a surjective isometry. So, if τh is scalar-type spectral operator, then there exists a sot σ-additive spectral measure P : B (T) → L (X) such that τh = T λ dP (λ). If p = ∞, then Lp (G) is a Grothendieck space with the Dunford-Pettis property. Accordingly, Ran(P ) is finite, [20], and hence, is Rbounded. If p ∈ [1, 2) ∪ (2, ∞), then Lp (G) has property (α) and so, by [17], Theorem 3.3, the bounded Boolean algebra Ran(P ) is R-bounded. Since τhm = T λm dP (λ) for all m ∈ Z, it follows that m f dP (λ) : f ∈ C (T) , f ∞ ≤ 1 ⊆ 4aco (Ran(P )) . {τh : m ∈ Z} ⊆ T. Accordingly, {τhm : m ∈ Z} is R-bounded in L (X) which contradicts Corollary 2.12. . The above corollary, for p ∈ [1, 2) ∪ (2, ∞), was obtained by a quite different method of proof in [10]. The result for p = ∞ occurs in [9]. Remark 2.14. Suppose that G is an infinite lca group and let p ∈ [1, 2) ∪ (2, ∞). Consider the (strongly operator continuous and bounded) canonical homomorphism h −→ τh of G in Lp (G). It follows easily from Corollary 2.10 and Proposition 2.11 that {τh : h ∈ G} is not R-bounded. This implies that there is no sot ˆ → L (X) such that τh = ˆ (h, u) dP (u) for σ-additive spectral measure P : B(G) G all h ∈ G. Indeed, if such a spectral measure P would exist, then Ran(P ) is Rbounded, which implies that {τh : h ∈ G} is R-bounded (see the proof of Corollary 2.13). Accordingly, the canonical homomorphism can never be the Fourier-Stieltjes ˆ We point out that if G has the transform of any regular spectral measure on G. property that every element h ∈ G has finite order, then each individual operator τhis scalar-type spectral (see Example 2.3). This is, for instance, the case if ∞ G = n=2 Zn ..

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(41) 166. B. de Pagter and W. J. Ricker. Positivity. [20] W. J. Ricker, Spectral operators of scalar-type in Grothendieck spaces with the Dunford-Pettis property, Bull. London Math. Soc., 17 (1985), 268–270. [21] W. Rudin, Fourier Analysis on Groups, Wiley Classics Library Edition, John Wiley & Sons, New York (1990). [22] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Heidelberg (1974). [23] J. G. Stampfli, Roots of scalar operators, Proc. Amer. Math. Soc., 13 (1962), 796–798. [24] L. Weis, A new approach to maximal Lp -regularity, In: G. Lumer, L. Weis (eds.) Evolution Equations and their Applications in Physical and Life Sciences, pp. 195–214, Marcel Dekker, New York (2001). [25] L. Weis, The H ∞ -holomorphic functional calculus for sectorial operators – a survey, In: Erik Koelink, Jan van Neerven, Ben de Pagter, Guido Sweers (eds.) Partial Differential Equations and Functional Analysis (the Philippe Clément Festschrift), pp. 263–294, Operator Theory Adv. Appl. (Vol. 168), Birkh¨ auser (2006). [26] H. Witvliet, Unconditional Schauder Decompositions and Multiplier Theorems, Ph.D. thesis, Delft University of Technology (2000). Ben de Pagter Delft Institute of Applied Mathematics Faculty EEMCS Delft University of Technology P.O. Box 5031, 2600 GA Delft The Netherlands e-mail: b.depagter@ewi.tudelft.nl Werner J. Ricker Math.-Geogr.Fakult¨ at Katholische Universit¨ at Eichst¨ att-Ingolstadt D-85072 Eichst¨ att Germany e-mail: werner.ricker@ku-eichstaett.de Received 15 January 2007; accepted 7 June 2007. To access this journal online: www.birkhauser.ch/pos.

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