PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 88(102) (2010), 67–76 DOI: 10.2298/ PIM1002067K

NOTES ON ANALYTIC CONVOLUTED C-SEMIGROUPS

Marko Kostić

Communicated by Stevan Pilipović

Abstract. We establish some new structural properties of exponentially bounded, analytic convoluted C-semigroups and state a version of Kato’s analyticity criterion for such a class of operator semigroups. Our characterizations completely cover the case of analytic fractionally integrated C-semigroups.

1. Introduction and preliminaries

An important motivational factor for the genesis of this paper presents the fact that several structural properties of exponentially bounded, analytic convoluted C-semigroups have not been fully cleared in the existing literature.

The paper is organized as follows. In Proposition 2.1 and Theorem 2.1, we refine [4, Proposition 3.7(a)], [8, Theorem 10] and transfer the assertion of [9, Theorem 5.2] to analytic convoluted C-semigroups. In Theorem 2.1, we introduce the condition (H_{1}) which holds in the case of fractionally integrated C-semigroups. In order to better explain the importance of this condition in our investigation, let us recall that the set ℘(S_{K}) consisted of all subgenerators of a (local) convoluted C-semigroup (S_{K}(t))_{t∈[0,τ)} need not be finite ([8], [10], [13]) and that, equipped with corresponding algebraic operations, ℘(S_{K}) becomes a complete lattice whose partially ordering coincides with the usual set inclusion; furthermore, ℘(S_{K}) is totally ordered iff card(℘(S_{K})) ⩽ 2 ([10], [13]), and in the case card(℘(S_{K})) < ∞, one can prove that ℘(S_{K}) is a Boolean, which implies card(℘(S_{K})) = 2^{n} for some n ∈ ℕ_{0}. In fact, the main objective in Theorem 2.1(i) is to establish the spectral characterizations of the integral generator of an analytic convoluted C-semigroup (S_{K}(t))_{t⩾0} as well as to show that such characterizations still hold for an arbitrary subgenerator of (S_{K}(t))_{t⩾0} as long as the condition (H_{1}) holds. It is an open problem whether the statements (2.6)–(2.9) quoted in the formulation of Theorem 2.1(i) remain true for an arbitrary subgenerator of (S_{K}(t))_{t⩾0} if the condition (H_{1}) is neglected. Furthermore, the condition (H_{1}) plays a crucial role in Theorem 2.2 which presents Kato’s analyticity criterion for convoluted C-semigroups. Even in the case of regularized semigroups, Theorem 2.2 and Corollary 2.2 improve the corresponding result of Zheng [14, Theorem]. It is well known that A generates an (exponentially) bounded, analytic C_{0}-semigroup of angle α ∈ (0, \frac{π}{2} ) provided that e^{±iα}A are generators of (exponentially) bounded C_{0}-semigroups (T_{±α}(t))_{t⩾0}. We transfer this assertion to analytic regularized semigroups by a slight modification of the proof of [1, Theorem 3.9.7].

2010 Mathematics Subject Classification: Primary 47D06; Secondary 47D09, 47D60, 47D62, 47D99.

Partially supported by grant 144016 of the Ministry of Science and Technological Development of the Republic of Serbia.

By E and L(E) are denoted a complex Banach space and the Banach algebra of bounded linear operators on E. For a closed linear operator A acting on E, D(A), Kern(A), R(A) and ρ(A) denote its domain, kernel, range and resolvent set, respectively. By [D(A)] is denoted the Banach space D(A) equipped with the graph norm. Given γ ∈ (0, π], put Σ_{γ} := {λ ∈ ℂ : λ ≠ 0, arg(λ) ∈ (−γ, γ)}. In what follows, we assume L(E) ∋ C is an injective operator satisfying CA ⊂ AC, τ ∈ (0, ∞], K is a complex-valued locally integrable function in [0, τ) and K is not identical to zero. Put Θ(t) := ∫_{0}^{t} K(s)ds, t ∈ [0, τ); then Θ is an absolutely continuous function in [0, τ) and Θ′(t) = K(t) for a.e. t ∈ [0, τ). We mainly use the following condition:

(P1): K is Laplace transformable, i.e., it is locally integrable on [0, ∞) and there exists β ∈ ℝ so that

K̃(λ) = L(K)(λ) := lim_{b→∞} ∫_{0}^{b} e^{−λt}K(t) dt := ∫_{0}^{∞}e^{−λt}K(t) dt

exists for all λ ∈ ℂ with Re λ > β. Put abs(K) := inf{Re λ : K̃(λ) exists}.

Definition 1.1. ([7]–[8]) Let A be a closed operator and let 0 < τ ⩽ ∞. If there exists a strongly continuous family (S_{K}(t))_{t∈[0,τ)} in L(E) such that:

(i) S_{K}(t)A ⊂ AS_{K}(t), t ∈ [0, τ),

(ii) S_{K}(t)C = CS_{K}(t), t ∈ [0, τ) and
(iii) for all x ∈ E and t ∈ [0, τ): ∫_{0}^{t} S_{K}(s)x ds ∈ D(A) and

(1.1)  A ∫_{0}^{t} S_{K}(s)x ds = S_{K}(t)x − Θ(t)Cx,

then it is said that A is a subgenerator of a (local) K-convoluted C-semigroup (S_{K}(t))_{t∈[0,τ)}. If τ = ∞, then we say that (S_{K}(t))_{t⩾0} is an exponentially bounded K-convoluted C-semigroup with a subgenerator A if, additionally, there exist M > 0 and ω ⩾ 0 such that ∥S_{K}(t)∥ ⩽ Me^{ωt}, t ⩾ 0.

The integral generator of (S_{K}(t))_{t∈[0,τ)} is defined by

Â := {(x, y) ∈ E^{2} : S_{K}(t)x − Θ(t)Cx = ∫_{0}^{t} S_{K}(s)yds, t ∈ [0, τ)},

NOTES ON ANALYTIC CONVOLUTED C-SEMIGROUPS 69

and it is a closed linear operator which is an extension of any subgenerator of (S_{K}(t))_{t∈[0,τ)}. Suppose {A, B} ⊂ ℘(S_{K}). By [10, Proposition 1.1], the following holds:

(a) C^{−1}AC = C^{−1}ÂC = Â ∈ ℘(S_{K}),

(b) A and B have the same eigenvalues,

(c) ρ_{C}(A) ⊆ ρ_{C}(B) if A ⊆ B,

(d) A = B = Â, if ρ(Â) ≠ ∅ or C = I.

The proof of the following auxiliary lemma is similar to those of [7, Theorem 2.2] and [9, Theorem 3.1, Theorem 3.3].

Lemma 1.1. Suppose K satisfies (P1) and A is a closed linear operator.

(i) Suppose M > 0, ω ⩾ 0, A is a subgenerator of an exponentially bounded, K-convoluted C-semigroup (S_{K}(t))_{t⩾0} satisfying ∥S_{K}(t)∥ ⩽ Me^{ωt}, t ⩾ 0 and ω_{1} = max(ω, abs(K)). Then {λ ∈ ℂ : Re λ > ω_{1}, K̃(λ) ≠ 0} ⊂ ρ_{C}(A) and (λ − A)^{−1}Cx = \frac{1}{K̃(λ)} ∫_{0}^{∞} e^{−λt}S_{K}(t)x dt for all x ∈ E and λ ∈ ℂ with Re λ > ω_{1} and K̃(λ) ≠ 0.

(ii) Suppose M > 0, ω ⩾ 0, (S_{K}(t))_{t⩾0} is a strongly continuous operator family, ∥S_{K}(t)∥ ⩽ Me^{ωt}, t ⩾ 0 and ω_{1} = max(ω, abs(K)). If {λ ∈ (ω_{1}, ∞) : K̃(λ) ≠ 0} ⊂ ρ_{C}(A) and (λ − A)^{−1}Cx = \frac{1}{K̃(λ)} ∫_{0}^{∞} e^{−λt}S_{K}(t)x dt, x ∈ E, λ > ω_{1}, K̃(λ) ≠ 0, then (S_{K}(t))_{t⩾0} is an exponentially bounded, K-convoluted C-semigroup with a subgenerator A.

(iii) Let A be densely defined. Then A is a subgenerator of an exponentially bounded C-semigroup (T (t))_{t⩾0} satisfying ∥T (t)∥ ⩽ Me^{ωt}, t ⩾ 0 for appropriate constants M > 0 and ω ∈ ℝ iff (ω, ∞) ⊂ ρ_{C}(A), the mapping λ ↦ (λ − A)^{−1}C, λ > ω is infinitely differentiable and

∥\frac{d^{k}}{dλ^{k}} [(λ − A)^{−1}C]∥ ⩽ \frac{Mk!}{(λ − ω)^{k+1}} , k ∈ ℕ_{0}, λ > ω.

Definition 1.2. [8] Let α ∈ (0, \frac{π}{2} ] and let (S_{K}(t))_{t⩾0} be a K-convoluted Csemigroup. Then we say that (S_{K}(t))_{t⩾0} is an analytic K-convoluted C-semigroup of angle α, if there exists an analytic function S_{K} : Σ_{α} → L(E) which satisfies

(i) S_{K}(t) = S_{K}(t), t > 0,

(ii) lim_{z→0, z∈Σ_{γ}} S_{K}(z)x = 0 for all γ ∈ (0, α) and x ∈ E.

It is said that (S_{K}(t))_{t⩾0} is an exponentially bounded, analytic K-convoluted Csemigroup, resp. bounded analytic K-convoluted C-semigroup, of angle α, if for every γ ∈ (0, α), there exist M_{γ} > 0 and ω_{γ} ⩾ 0, resp. ω_{γ} = 0, such that ∥S_{K}(z)∥ ⩽ M_{γ}e^{ω_{γ} Re z}, z ∈ Σ_{γ}.

Since no confusion seems likely, we will also denote S_{K} by S_{K}. Plugging K(t) = \frac{t^{r−1}}{Γ(r)} , t > 0 in Definition 1.1 and Definition 1.2, where r > 0 and Γ(·) denotes the Gamma function, we obtain the well-known classes of (analytic) rtimes integrated C-semigroups; an (analytic) 0-times integrated C-semigroup is defined to be an (analytic) C-semigroup (cf. [3, Definition 21.3]). The notion of (exponential) boundedness of an analytic r-times integrated C-semigroup, r ⩾ 0, is understood in the sense of Definition 1.2.

2. Analytic convoluted C-semigroups

We start this section with the following proposition.

Proposition 2.1. Suppose K satisfies (P1), α ∈ (0, \frac{π}{2} ] and A is a subgenerator of an exponentially bounded, analytic K-convoluted C-semigroup (S_{K}(t))_{t⩾0} of angle α. Suppose, further, that the condition (H) holds, where: (H) There exist functions c : (−α, α) → ℂ∖{0}, ω_{0} : (−α, α) → [0, ∞) and a family of functions (K_{θ})_{θ∈(−α,α)} satisfying (P1) so that: abs(K_{θ}) ⩽ ω_{0}(θ), \frac{abs(K)}{cos θ} ⩽ ω_{0}(θ),

Φ_{θ} =: {λ ∈ (ω_{0}(θ), ∞) : K̃(λe^{−iθ}) = 0} = {λ ∈ (ω_{0}(θ), ∞) : K̃_{θ}(λ) = 0}, (2.1)

\frac{K̃_{θ}(λ)}{K̃(λe^{−iθ})} = c(θ), λ > ω_{0}(θ), λ ∉ Φ_{θ}, θ ∈ (−α, α). (2.2)

Then, for every θ ∈ (−α, α), the operator e^{iθ}A is a subgenerator of an exponentially bounded, analytic K_{θ}-convoluted C-semigroup (c(θ)S_{K}(te^{iθ}))_{t⩾0} of angle α − |θ|. Furthermore,

(i) S_{K}(te^{iθ})A ⊂ AS_{K}(te^{iθ}), t ⩾ 0 and (ii) A ∫_{0}^{te^{iθ}} S_{K}(s)x ds = S_{K}(te^{iθ})x − \frac{1}{c(θ)} ∫_{0}^{t} K_{θ}(s) dsCx, t ⩾ 0, x ∈ E, θ ∈

(−α, α).

Proof. Let θ ∈ (−α, α) and let λ ∈ ℝ be sufficiently large with K̃_{θ}(λ) ≠ 0. Denote Γ_{θ} := {te^{−iθ} : t ⩾ 0} and notice that (c(θ)S_{K}(te^{iθ}))_{t⩾0} is a strongly continuous, exponentially bounded operator family. Clearly, K̃(λe^{−iθ}) ≠ 0 and Lemma 1.1 yields

(2.3) K̃_{θ}(λ)(λ − e^{iθ}A)^{−1}Cx = K̃_{θ}(λ)e^{−iθ}(λe^{−iθ} − A)^{−1}Cx

= e^{−iθ} \frac{K̃_{θ}(λ)}{K̃(λe^{−iθ})} ∫_{0}^{∞}e^{−λe^{−iθ}t}S_{K}(t)x dt = e^{−iθ}c(θ) ∫_{Γ_{θ}} e^{−λt}e^{iθ}S_{K}(te^{iθ})x dt

= ∫_{0}^{∞}e^{−λt}(c(θ)S_{K}(te^{iθ})x) dt, x ∈ E,

where (2.3) follows from an elementary application of the Cauchy theorem. Keeping in mind Definition 1.1 and Lemma 1.1(ii), the assertion automatically follows. □

Now we state the following generalization of [8, Theorem 10] and [9, Theorem 5.2].

Theorem 2.1. (i) Suppose K satisfies (P1), ω ⩾ max(0, abs(K)), α ∈ (0, \frac{π}{2} ], and K̃(·) can be analytically continued to a function g : ω + Σ_{\frac{π}{2} +α} → ℂ. Suppose, further, that A is a subgenerator of an analytic K-convoluted C-semigroup (S_{K}(t))_{t⩾0} of angle α and that

(2.4)  sup_{z∈Σ_{γ}} ∥e^{−ωz}S_{K}(z)∥ < ∞ for all γ ∈ (0, α).

Let us denote by Â the integral generator of (S_{K}(t))_{t⩾0} and put

(2.5)  N := \{λ ∈ ω + Σ_{\frac{π}{2} +α} : g(λ) ≠ 0\}.

Then:

N ⊂ ρ_{C}(Â), (2.6)

(2.7)  sup_{λ∈N∩ (ω+Σ_{\frac{π}{2} +γ_{1}} )} ∥(λ − ω)g(λ)(λ − Â)^{−1}C∥ < ∞ for all γ_{1} ∈ (0, α),

lim_{λ→+∞, K̃(λ)≠0} λK̃(λ)(λ − A)^{−1}Cx = 0, x ∈ E, (2.8)

(2.9) the mapping λ ↦ (λ − Â)^{−1}C, λ ∈ N is analytic.

Suppose, additionally, that the following condition holds:

(H_{1}) : (H) holds with c(·), ω_{0}(·), (K_{θ})_{θ∈(−α,α)}, and additionally, abs(K_{θ}) ⩽ ω cos θ, θ ∈ (−α, α).

Then (2.6)–(2.7) and (2.9) hold with Â replaced by A therein.

(ii) Assume α ∈ (0, \frac{π}{2} ], K satisfies (P1) and ω ⩾ max(0, abs(K)). Suppose that A is a closed linear operator with {λ ∈ ℂ : Reλ > ω, K̃(λ) ≠ 0} ⊂ ρ_{C}(A) and that the function λ ↦ K̃(λ)(λ − A)^{−1}C, Re λ > ω, K̃(λ) ≠ 0, can be analytically extended to a function q̃: ω + Σ_{\frac{π}{2} +α} → L(E) satisfying

(2.10)  sup_{λ∈ω+Σ_{\frac{π}{2} +γ}} ∥(λ − ω)q̃(λ)∥ < ∞ for all γ ∈ (0, α),

lim_{λ→+∞} λq̃(λ)x = 0, x ∈ E, if \overline{D(A)} ≠ E. (2.11)

Then the operator A is a subgenerator of an exponentially bounded, analytic Kconvoluted C-semigroup of angle α.

Proof. The proof of (i) can be obtained as follows. By Lemma 1.1(i), we have {λ ∈ ℂ : Re λ > ω, K̃(λ) ≠ 0} ⊂ ρ_{C}(A) and

K̃(λ)(λ − A)^{−1}Cx = ∫_{0}^{∞}e^{−λt}S_{K}(t)x dt, Re λ > ω, K̃(λ) ≠ 0, x ∈ E.

Put q(λ) := ∫_{0}^{∞} e^{−λt}S_{K}(t) dt, Re λ > ω. An application of [1, Theorem 2.6.1] gives that the function q(·) can be extended to an analytic function q̃: ω+Σ_{\frac{π}{2} +α} → L(E) satisfying sup_{λ∈ω+Σ_{\frac{π}{2} +γ}} ∥(λ − ω)q̃(λ)∥ < ∞ for all γ ∈ (0, α). Further on, N is an open subset of ℂ and it can be easily seen that every two point belonging to N can be connected with a C^{∞} curve lying in N; in particular, N is a connected open subset of ℂ. The function F : N → L(E) given by F (λ) := \frac{q̃(λ)}{g(λ)}, λ ∈ N is analytic and

(2.12) {λ ∈ ℂ : Re λ > ω, K̃(λ) ≠ 0} ⊂ {λ ∈ N ∩ ρ_{C}(A) : F (λ) = (λ − A)^{−1}C}.

Let us denote V = {λ ∈ N ∩ ρ_{C}(A) : F (λ) = (λ − A)^{−1}C} and suppose μ ∈ ρ_{C}(A), x ∈ D(A) and y ∈ E. Since

F (λ)(λ − A)x = (λ − A)^{−1}C(λ − A)x = Cx, λ ∈ V, (2.13)

F (λ)Cy = CF (λ)y, λ ∈ V, (2.14)

F (λ)Cy = (λ − A)^{−1}C^{2}y = (μ − A)^{−1}C^{2}y − (λ − μ)(μ − A)^{−1}CF (λ)y, λ ∈ V, (2.15)

the uniqueness theorem for analytic functions (cf. [1, Proposition A2, Proposition B.5]) implies that (2.13)–(2.15) remain true for all λ ∈ N. Suppose now that (λ − A)x = 0 for some λ ∈ N and x ∈ D(A). Owing to (2.13), one gets Cx = 0, x = 0 and λ − A is injective. By the assertion (b), we obtain that λ − Â is injective. Furthermore,

(λ − A)CF (λ)y = (λ − A)F (λ)Cy

= (λ − A)[(μ − A)^{−1}C^{2}y − (λ − μ)(μ − A)^{−1}CF (λ)y]

= C^{2}y + (λ − μ)[(μ − A)^{−1}C^{2}y − CF (λ)y − (λ − μ)(μ − A)^{−1}CF (λ)y],

and thanks to the validity of (2.15) for all λ ∈ N, one obtains that

(2.16)  (λ − A)CF (λ)y = C^{2}y, λ ∈ N.

The last equality, injectiveness of C and assertion (a) taken together imply:

λF (λ)y = C^{−1}AC[F (λ)y] + Cy = ÂF (λ)y + Cy, λ ∈ N, i.e., (2.17)

(λ − Â)F (λ)y = Cy, λ ∈ N. (2.18)

This implies R(C) ⊂ R(λ − Â), λ ∈ N, N ⊂ ρ_{C}(Â), F (λ) = (λ − Â)^{−1}C, λ ∈ N, (2.6) and (2.9). The estimate (2.7) is an immediate consequence of [1, Theorem 2.6.1]. Let x ∈ E be fixed. Then z ↦ S_{K}(z)x, z ∈ Σ_{α} is an analytic function which satisfies the condition (i) quoted in the formulation of [1, Theorem 2.6.1]. Since lim_{t↓0} S_{K}(t)x = 0, an application of [1, Theorem 2.6.4] implies that lim_{λ→+∞} λq(λ) = 0. This gives lim_{λ→+∞, K̃(λ)≠0} λK̃(λ)(λ − A)^{−1}Cx = 0, i.e., (2.8) and the first part of the proof is completed. Suppose now that (H_{1}) holds. Then abs(K_{θ}) ⩽ ω cos θ, θ ∈ (−α, α), and by Lemma 1.1(i), we have that, for every θ ∈ (−α, α), {λ ∈ ℂ : Re λ > ω cos θ, K̃_{θ}(λ) ≠ 0} ⊂ ρ_{C}(e^{iθ}A) and that:

(2.19)  K̃_{θ}(λ)e^{−iθ}(λe^{−iθ} − A)^{−1}Cx = ∫_{0}^{∞}e^{−λt}(c(θ)S_{K}(te^{iθ}))x dt,

for all x ∈ E and λ ∈ ℂ with Re λ > ω cos θ and K̃_{θ}(λ) ≠ 0. Fix a number θ ∈ (−α, α) and define G_{θ} : \{ω + te^{iφ} : t > 0, φ ∈ (−(\frac{π}{2} + θ), \frac{π}{2} − θ)\} ∩ N → ℂ

by G_{θ}(λ) := \frac{K̃_{θ}(λe^{iθ})}{g(λ)}, λ ∈ D(G_{θ}(·)). Then it is clear that D(G_{θ}(·)) is an open, connected subset of ℂ and that, owing to (2.1)–(2.2), there exists a > 0 such that Φ_{θ,a} := {te^{−iθ} ∩ N : t ⩾ a} ⊂ D(G_{θ}(·)) and that G_{θ}(λ) = c(θ), λ ∈ Φ_{θ,a}. By the uniqueness theorem for analytic functions, one obtains that G_{θ}(λ) = c(θ), λ ∈ D(G_{θ}(·)). Hence, (2.19) implies \{ω+te^{iφ} : t > 0, φ ∈ (−(\frac{π}{2} +θ), \frac{π}{2} −θ)\}∩N ⊂ ρ_{C}(A),

(2.20)  (z − A)^{−1}Cx = \frac{e^{iθ}}{g(z)} ∫_{0}^{∞}e^{−ze^{iθ}t}S_{K}(te^{iθ})x dt,

for all z ∈ \{ω+te^{iφ} : t > 0, φ ∈ (−(\frac{π}{2} +θ), \frac{π}{2} −θ)\}∩N and x ∈ E, and the mapping z ↦ (z−A)^{−1}C, z ∈ N, arg(z−ω) ∈ (−(\frac{π}{2} +θ), \frac{π}{2} −θ) is analytic. One can apply the same argument to e^{−iθ}A in order to see that \{z ∈ N : arg(z −ω) ∈ (θ− \frac{π}{2} , \frac{π}{2} +θ)\} ⊂ ρ_{C}(A) and that the mapping z ↦ (z − A)^{−1}C, z ∈ N, arg(z − ω) ∈ (θ − \frac{π}{2} , θ + \frac{π}{2} ) is analytic. Thereby, \{z ∈ N : | arg(z − ω)| < θ + \frac{π}{2} \} ⊂ ρ_{C}(A) and the mapping z ↦ (z − A)^{−1}C, z ∈ N, | arg(z − ω)| < θ + \frac{π}{2} is analytic. This completes the proof of (i). The proof of (ii) in the case \overline{D(A)} ≠ E is given in [8]. Suppose now that \overline{D(A)} = E. We will prove that (2.11) automatically holds for every x ∈ E. Arguing as in the proof of [8, Theorem 10], one obtains that there exists an analytic function S_{K} : Σ_{α} → L(E) such that sup_{z∈ω+Σ_{\frac{π}{2} +β}} ∥e^{−ωz}S_{K}(z)∥ < ∞ for all β ∈ (0, α). By [1, Proposition 2.6.3(b)] and the proof of [8, Theorem 10], it suffices to show that lim_{t↓0} S_{K}(t)x = 0. Suppose, for the time being, x ∈ D(A). Since q̃(λ)x = K̃(λ)(λ−A)^{−1}Cx, λ ∈ ℂ, Re λ > ω, K̃(λ) ≠ 0 we have that L(∫_{0}^{t} S_{K}(s)Ax ds)(λ) =

\frac{q̃(λ)}{λ}Ax = q̃(λ)x − \frac{K̃(λ)}{λ}Cx = L(S_{K}(t)x − Θ(t)Cx)(λ), λ ∈ ℂ, Re λ > ω, K̃(λ) ≠ 0 and the uniqueness theorem for Laplace transforms implies ∫_{0}^{t} S_{K}(s)Ax ds = S_{K}(t)x − Θ(t)Cx, t ⩾ 0. Therefore ∥S_{K}(t)x∥ ⩽ |Θ(t)|Cx + te^{ωt}∥Ax∥, t ⩾ 0 and lim_{t↓0} S_{K}(t)x = 0. Combined with the exponential boundedness of S_{K}(·), this indicates that lim_{t↓0} S_{K}(t)x = 0 for every x ∈ E. □

Let ∅ ≠ Ω ⊂ ρ_{C}(A) be open. By [5, Remark 2.7], we have that the continuity of mapping λ ↦ (λ − A)^{−1}C, λ ∈ Ω implies its analyticity. Furthermore, it can be simply verified that the function K(t) = \frac{t^{r−1}}{Γ(r)} , t > 0, r > 0 satisfies the condition (H_{1}) with c(θ) = e^{−irθ}, ω_{0}(θ) = 0 and K_{θ}(t) = K(t), θ ∈ (−α, α), t > 0. Keeping in mind Proposition 1.1, Theorem 2.1 and these remarks, one immediately obtains the proof of the following corollary; notice only that, in the case r = 0, the equality (2.24) follows from [1, Theorem 2.6.4] and elementary definitions.

Corollary 2.1. (i) Suppose r ⩾ 0 and α ∈ (0, \frac{π}{2} ]. Then the operator A is a subgenerator of an exponentially bounded, analytic r-times integrated C-semigroup (S_{r}(t))_{t⩾0} of angle α iff for every γ ∈ (0, α), there exist M_{γ} > 0 and ω_{γ} ⩾ 0 such that:

ω_{γ} + Σ_{\frac{π}{2} +γ} ⊂ ρ_{C}(A), (2.21)

∥(λ − A)^{−1}C∥ ⩽ M_{γ}(1 + |λ|)^{r−1}, λ ∈ ω_{γ} + Σ_{\frac{π}{2} +γ}, (2.22)

the mapping λ ↦ (λ − A)^{−1}C, λ ∈ ω_{γ} + Σ_{\frac{π}{2} +γ}is analytic (continuous) and (2.23)

lim_{λ→+∞} \frac{(λ − A)^{−1}Cx}{λ^{r−1}} = χ_{\{0\}}(r)Cx, x ∈ E, if \overline{D(A)} ≠ E. (2.24)

(ii) Let θ ∈ (−α, α) and let A be a subgenerator of an exponentially bounded, analytic r-times integrated C-semigroup (S_{r}(t))_{t⩾0} of angle α. Then e^{iθ}A is a subgenerator of an exponentially bounded, analytic r-times integrated C-semigroup (e^{−iθr}S_{r}(te^{iθ}))_{t⩾0} of angle α−|θ|, S_{r}(z)A ⊂ AS_{r}(z) and A ∫_{0}^{z} S_{r}(s)xds = S_{r}(z)x− \frac{z^{r}}{Γ(r+1)}Cx, z ∈ Σ_{α}, x ∈ E.

Now we state Kato’s analyticity criterion for convoluted C-semigroups.

Theorem 2.2. Suppose α ∈ (0, \frac{π}{2} ], K satisfies (P1), ω ⩾ max(0, abs(K)), there exists an analytic function g : ω + Σ_{\frac{π}{2} +α} → ℂ such that g(λ) = K̃(λ), λ ∈ ℂ, Re λ > ω and (H_{1}) holds. Then A is a subgenerator of an analytic K-convoluted C-semigroup (S_{K}(t))_{t⩾0} satisfying (2.4) iff:

(i.1) For every θ ∈ (−α, α), e^{iθ}A is a subgenerator of a K_{θ}-convoluted C-

semigroup (S_{θ}(t))_{t⩾0}, and

(i.2) for every β ∈ (0, α), there exists M_{β} > 0 such that

(2.25)  ∥\frac{1}{c(θ)}S_{θ}(t)∥ ⩽ M_{β}e^{ωt cos θ}, t ⩾ 0, θ ∈ (−β, β).

Proof. Suppose A is a subgenerator of an analytic K-convoluted C-semigroup (S_{K}(t))_{t⩾0} satisfying (2.4). By Proposition 1.1, we have that (i.1) and (i.2) hold with S_{θ}(t) = c(θ)S_{K}(te^{iθ}), t ⩾ 0, θ ∈ (−α, α). To prove the converse statement, notice that the argumentation given in the final part of the proof of Theorem 2.1 implies that (ω + Σ_{\frac{π}{2} +α}) ∩ N ⊂ ρ_{C}(A) and that there exists an analytic mapping G : ω + Σ_{\frac{π}{2} +α} → L(E) such that G(λ) = g(λ)(λ − A)^{−1}C, λ ∈ (ω + Σ_{\frac{π}{2} +α}) ∩ N, where N is defined by (2.5). Furthermore, for every θ ∈ (−α, α):

(2.26) G(λ) = e^{iθ} ∫_{0}^{∞}e^{−λte^{iθ}}(\frac{1}{c(θ)}S_{θ}(t)) dt if arg(λ − ω) ∈ (− (\frac{π}{2} + θ), \frac{π}{2} − θ),

(2.27) G(λ) = e^{−iθ} ∫_{0}^{∞}e^{−λte^{−iθ}}(\frac{1}{c(−θ)}S_{−θ}(t)) dt if arg(λ − ω) ∈ (θ − \frac{π}{2} , θ + \frac{π}{2} ).

Keeping in mind (i.2) as well as (2.26)–(2.27), we have that, for every β ∈ (0, α), sup_{λ∈ω+Σ_{\frac{π}{2} +β}} ∥(λ − ω)G(λ)∥ < ∞. By [1, Theorem 2.6.1], one gets the existence

of an analytic mapping S_{K} : Σ_{α} → L(E) such that sup_{z∈Σ_{β}} ∥e^{−ωz}S_{K}(z)∥ < ∞ for all β ∈ (0, α) and that G(λ) = S̃_{K}(λ) for all λ ∈ (ω, ∞). Furthermore, the uniqueness theorem for Laplace transforms implies S_{K}(z) = \frac{1}{c(arg(z))}S_{arg(z)}(|z|), z ∈ Σ_{α}, and since c(0) = 1 and K_{0} = K, it suffices to show that, for every fixed x ∈ E and β ∈ (0, α), one has lim_{z∈Σ−β, z→0} S_{K}(z)x = 0 (cf. also Lemma 1.1(ii)). To this end, notice that lim_{t↓0} S_{K}(t)x = lim_{t↓0} S_{0}(t)x = 0 and that [1, Proposition 2.6.3(b)] implies lim_{z∈Σ_{β}, z→0} e^{−ωz}S_{K}(z)x = lim_{z∈Σ_{β}, z→0} S_{K}(z)x = 0, z ∈ Σ_{α}. □

In the following corollary, we remove any density assumption from [14, Theorem]:

Corollary 2.2. Suppose r ⩾ 0, α ∈ (0, \frac{π}{2} ] and ω ∈ [0, ∞) if r > 0, resp.ω ∈ ℝ if r = 0. Then A is a subgenerator of an analytic r-times integrated C-semigroup (S_{r}(t))_{t⩾0} of angle α satisfying sup_{λ∈Σ_{β}} ∥e^{−ωz}S_{r}(z)∥ < ∞ for all β ∈ (0, α) iff the following conditions hold:

(i.1) For every θ ∈ (−α, α), e^{iθ}A is a subgenerator of an r-times integrated

C-semigroup (S_{θ}(t))_{t⩾0}, and (i.2) for every β ∈ (0, α), there exists M_{β} > 0 such that ∥S_{θ}(t)∥ ⩽ M_{β}e^{ωt cos θ},

t ⩾ 0, θ ∈ (−β, β).

Now we state the following extension of [1, Theorem 3.9.7] and [1, Corollary 3.9.9]:

Theorem 2.3. Suppose α ∈ (0, \frac{π}{2} ), A is densely defined and e^{±iα}A are subgenerators of (exponentially) bounded C-semigroups (T_{±α}(t))_{t⩾0}. Then A is a subgenerator of an (exponentially) bounded, analytic C-semigroup of angle α.

Proof. Suppose ∥T_{±α}(t)∥ ⩽ Me^{ωt}, t ⩾ 0 for appropriate constants M ⩾ 0 and ω ⩾ 0. Put μ := \frac{ω}{cos α} and A_{μ} := A − μ. Then e^{±iα}A_{μ} are subgenerators of bounded C-semigroups (S_{±α}(t) := e^{−e^{±iα}μt}T_{±α}(t))_{t⩾0} and ∥S_{±α}(t)∥ ⩽ M, t ⩾ 0. Proceeding as in the proof of [1, Theorem 3.9.7], one gets that Σ_{\frac{π}{2} +α} ⊂ ρ_{C}(A_{μ}) and that the mapping λ ↦ (λ − A_{μ})^{−1}C, λ ∈ Σ_{\frac{π}{2} +α} is analytic. Then the proof of [5, Corollary 2.8] implies that, for every n ∈ ℕ_{0} and λ ∈ Σ_{\frac{π}{2} +α}:

(2.28) R(C) ⊂ R((λ−A_{μ})^{n+1}) and \frac{d^{n}}{dλ^{n}} (λ−A_{μ})^{−1}C = (−1)^{n}n!(λ−A_{μ})^{−(n+1)}C.

Put now T_{n,k}(z) := (I − \frac{z}{n}A_{μ})^{−k}C, z ∈ \overline{Σ_{α}}, k ∈ ℕ, n ∈ ℕ. By (2.28), we obtain that, for every r ⩾ 0:

(2.29) ∥T_{n,k}(re^{±iα})∥ = ∥(I − \frac{re^{±iα}}{n} A_{μ})^{−k}C∥ = ∥\frac{n^{k}}{r^{k}} (\frac{n}{r}I − e^{±iα}A_{μ})^{−k}C∥

= ∥\frac{n^{k}}{r^{k}} \frac{(\frac{d^{k−1}}{dλ^{k−1}} (λ − e^{±iα}A_{μ})^{−1}C)_{|λ= \frac{n}{r}}}{(−1)^{k−1}(k − 1)!} ∥

= ∥\frac{n^{k}}{r^{k}} \frac{(−1)^{k−1} ∫_{0}^{∞} e^{− \frac{n}{r} t}t^{k−1}S_{±α}(t) dt}{(−1)^{k−1}(k − 1)!}∥ ⩽ M.

Arguing similarly, we get:

(2.30)  ∥T_{n,k}(z)∥ ⩽ \frac{M}{cos^{k} α}, z ∈ Σ_{α}, k ∈ ℕ, n ∈ ℕ.

Taking into account the Phragmén–Lindelöf principle (cf. for instance [1, Theorem 3.9.8]) and (2.29)–(2.30), one obtains that ∥T_{n,k}(z)∥ ⩽ M, z ∈ \overline{Σ_{α}}, k ∈ ℕ, n ∈ ℕ. In particular, ∥\frac{d^{n}}{dλ^{n}} (λI − A_{μ})^{−1}C∥ ⩽ \frac{Mn!}{λ^{n}}, λ > 0, n ∈ ℕ_{0} and Lemma 1.1(iii) implies that A_{μ} is a subgenerator of a bounded C-semigroup (T (t))_{t⩾0} such that (λ − A_{μ})^{−1}Cx = ∫_{0}^{∞} e^{−λt}T (t)x dt, λ ∈ ℂ, Re λ > 0, x ∈ E. By the Post–Widder inversion formula [1, Theorem 1.7.7], one obtains T (t)x = lim_{n→∞} T_{n,n+1}(\frac{t}{n})x, x ∈ E, t ⩾ 0 and Vitali’s theorem [1, Theorem A.5, p. 458] implies that there exists an analytic mapping T̃ : Σ_{α} → L(E) such that T̃(t) = T (t), t > 0 and that ∥T̃(z)∥ ⩽ M, z ∈ Σ_{α}. By [1, Proposition 2.6.3(b)], one yields that the mapping z ↦ T̃(z)x, z ∈ \overline{Σ_{β}} is continuous for every fixed x ∈ E and β ∈ (0, α) and the proof of theorem completes a routine argument. □

The preceding theorem has been recently generalized in [11]:

Theorem 2.4. Suppose α ∈ (0, \frac{π}{2} ), r ⩾ 0, and e^{±iα}A are subgenerators of exponentially bounded r-times integrated C semigroups (S_{r}^{±α}(t))_{t⩾0}. Then, for every ζ > 0, A is a subgenerator of an exponentially bounded, analytic (r + ζ)-times integrated C semigroup (S_{r+ζ}(t))_{t⩾0} of angle α; if A is densely defined, then A is a subgenerator of an exponentially bounded, analytic r-times integrated C semigroup (S_{r}(t))_{t⩾0} of angle α.

References

1. W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001.

2. I. Ciorănescu, G. Lumer, Problèmes d’évolution régularisés par un noyan général K(t). Formule de Duhamel, prolongements, théorèmes de génération, C. R. Acad. Sci. Paris Sér. I Math. 319 (1995), 1273–1278.

3. R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lect. Notes Math. 1570, Springer-Verlag, 1994.

4. R. deLaubenfels, Holomorphic C-existence families, Tokyo J. Math. 15 (1992), 17–38.

5. R. de Laubenfels, F. Yao, S. W. Wang, Fractional powers of operators of regularized type, J. Math. Anal. Appl. 199 (1996), 910–933.

6. T. Kato, Perturbation Theory for Linear Operators, 2nd. ed., Springer-Verlag, Berlin, 1976.

7. M. Kostić, Convoluted C-cosine functions and convoluted C-semigroups, Bull. Cl. Sci. Math. Nat. Sci. Math. 28 (2003), 75–92.

8. M. Kostić, S. Pilipović, Convoluted C-cosine functions and semigroups. Relations with hyperfunction semigroups and sines, J. Math. Anal. Appl. 338 (2008), 1224–1242.

9. M. Kostić, S. Pilipović, Global convoluted semigroups, Math. Nachr. 280 (2007), 1727–1743.

10. M. Kostić, Convoluted C-groups, Publ. Inst. Math., Nouv. Sér. 84(98) (2008), 73–95.

11. M. Kostić, Generalized Semigroups and Cosine Functions, Mathematical Institute, Belgrade, 2010.

12. F. Neubrander, Abstract elliptic operators, analytic interpolation semigroups and Laplace transforms of analytic functions, Semesterbericht Funktionalanalysis, Tübingen, Wintersemester 15 (1988/89), 163–185.

13. S. Wang, Properties of subgenerators of C-regularized semigroups, Proc. Amer. Math. Soc. 126 (1998), 453–460.

14. Q. Zheng, An analyticity criterion for regularized semigroups, Proc. Amer. Math. Soc. 130 (2002), 2271–2274.

15. Q. Zheng, Y. Lei, Exponentially bounded C-semigroups and integrated semigroups with nondensely defined generators III: Analyticity, Acta Math. Scientia 14 (1994), 107–119.

Faculty of Technical Sciences (Received 28 04 2009) University of Novi Sad 21000 Novi Sad Serbia

 marco.s@verat.net