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By Seymour Lipschutz

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H. Hardy, is adapted from [Ka, Thm. 2] where the discrete case is proved. 3 is a special case of a theorem due to F. Neubrander [Nb1] who gives a different proof. , in [Wi, Thm. 5b]. 1 is proved in part in [GWV]; the rest appears in [Na]. The identity s(A) = ω1 (T) is due to F. Neubrander [Nb2]. 4 is due to W. Arendt [Ar2]. It was the first example of a positive semigroup on a rearrangement invariant Banach function space whose spectral bound and uniform growth bound do not coincide. Earlier, it was shown in [GVW] that the spectral bound and uniform growth bound do not coincide for the translation semigroup T defined by (T (t)f )(s) = f (s + t) in the Banach function space 2 Lp (IR+ ) ∩ Lq (IR+ , et dt).

In this step we prove that 2π φ(−2πm)T (2πm) = I, m∈ZZ the convergence being in the operator norm. By the estimate of φ we have φ(−s − 2πm)T (s + 2πm) ≤ m∈ZZ |φ(−s − 2πm)| ω(s + 2πm) m∈ZZ (1 + (s + 2πm)2 )−1 . 6) m∈ZZ This shows that the series m∈ZZ φ(−s−2πm)T (s+2πm) converges absolutely with respect to the operator norm of L(X). 4) as follows: ∞ e−iks φ(−s)T (s) ds Pk = −∞ 2π(m+1) e−iks φ(−s)T (s) ds = m∈ZZ 2πm  2π e−iks  = 0  φ(−s − 2πm)T (s + 2πm) ds. m∈ZZ Observe that for all x ∈ X the 2π-periodic continuous function φ(−s − 2πm)T (s + 2πm)x, ξx (s) := 2π s ∈ IR, m∈ZZ is continuous for every x ∈ X and that Pk x is the k-th Fourier coefficient of ξx .

1 can be replaced by mere boundedness. This is the content of the following result, usually referred to as Gearhart’s theorem. 4. Let T be a C0 -semigroup on a Hilbert space H, with generator A. Then the following assertions are equivalent: (i) 1 ∈ (T (2π)); (ii) iZZ ⊂ (A) and supk∈ZZ R(ik, A) < ∞. Proof: (i)⇒(ii): By the spectral inclusion theorem we have iZZ ⊂ (A). 2), 2π R(ik, A)x = (I − T (2π))−1 e−iks T (s)x ds, 0 ∀x ∈ X, Spectral mapping theorems 35 it is evident that supk∈ZZ R(ik, A) < ∞.

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