PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 83(97) (2008), 15–25 DOI: 10.2298/PIM0897015K

COMPLEX POWERS OF OPERATORS

Marko Kostić

Communicated by Stevan Pilipović

Abstract. We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists α ∈ [0, ∞) such that the resolvent of A is bounded by O((1 + |λ|)^{α}) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (−A)^{b} is the c.i.g. of an analytic semigroup of growth order r > 0.

1. Introduction

Chronologically, the theory of fractional powers of operators dates from a paper of S. Bochner who constructed the fractional powers of −Δ in 1949. From then on, many different techniques have been established in the framework of this theory. Let us mention the papers of Balakrishnan [4], Komatsu [11], Balabane [3], Martínez, Sanz and Marco [15], Straub [20] and deLaubenfels, Yao and Wang [6]. Especially, we refer to the monograph [14] where the interested reader can find a great part of the theory of fractional powers of non-negative operators including topics related to extensions of Hirsch functional calculus, fractional powers of operators in locally convex spaces, interpolation spaces and the famous Dore-Venni theorem.

This paper is motivated by the work of Straub [20] who defined the complex powers of a closed, densely defined operator A satisfying:

(1.1) Σ(γ) := \{z ∈ ℂ : z ≠ 0, | arg(z)| ⩽ γ\} ∪ \{0\} ⊂ ρ(A), for some γ ∈ (0, \frac{π}{2} ); (1.2) ∥R(λ : A)∥ ⩽ M(1 + |λ|)^{n}, λ ∈ Σ(γ), for some M > 0 and n ∈ ℕ_{0}. For such an operator A, Straub defined in [20] the fractional powers (−A)^{b},

for all b ∈ ℂ. If A fulfills (1.1) and (1.2), then one can employ the construction given in [6] to obtain the definition of the fractional operator of −A, but only for b ∈ [0, ∞). In general, the definitions given in [6] and [20] do not coincide; see [6] and [19] for further information.

In this paper, we show how ideas developed in [20] can be applied to an essentially larger class of closed, densely defined operators. It is worth noting that the spectrum of an operator belonging this class and Σ(γ) may have non-empty intersection, for every γ ∈ (0, \frac{π}{2} ). Condition (1.2) in our analysis is replaced by ∥R(λ : A)∥ ⩽ M(1 + |λ|)^{α}, for some M > 0 and α ∈ [0, ∞). This implies that the operators J^{b} (cf. Section 3), which are fundamental in the construction of fractional powers given in [9], [20] and this paper, remain bounded for all b ∈ ℂ with Re b < −(α + 1). The fractional power (−A)^{b}, b ∈ ℂ, is defined to be the closure of J^{b}. If b ∈ (0, \frac{1}{2}), then the negative of the fractional power (−A)^{b} is the c.i.g. of an analytic semigroup of growth order r > 0. This allows one to consider the incomplete higher order abstract Cauchy problems; in this paper, it is necessary that the order of such a problem is strictly greater than two.

2000 Mathematics Subject Classification: Primary 47A99; Secondary 47D03, 47D09, 47D62.

2. Basic concepts

Throughout this paper, E denotes a complex Banach space and A a closed, densely defined operator in E. The space of all bounded linear operators from E into E is denoted by L(E); ρ(A) stands for the resolvent set of A while [D(A)] designates the Banach space D(A) equipped with the graph norm. Let a ∈ (0, 1), C ∈ (0, 1] and d ∈ (0, 1]; B_{d} := {z ∈ ℂ : |z| ⩽ d} and

P_{a,C} := {ξ + iη : ξ ∈ (0, ∞), η ∈ ℝ, ∥η| ⩽ Cξ^{a}}.

We assume that A satisfies the following conditions:

(∗) P_{a,C} ∪ B_{d} ⊂ ρ(A), (∗∗) ||R(λ : A)|| ⩽ M(1 + |λ|)^{α}, λ ∈ P_{a,C} ∪ B_{d}, for some M > 0 and α ⩾ 0. Note, if P_{a,C} ∪ {0} ⊂ ρ(A) and ∥R(λ : A)∥ = O((1 + |λ|)^{α}), λ ∈ P_{a,C} ∪ {0},

then there exists a d ∈ (0, 1] and an appropriate M > 0 such that (∗) and (∗∗) are fulfilled.

Example 2.1. (a) Let α > 0 and 0 < τ ⩽ ∞. It is said that the abstract Cauchy problem

C_{α+1}(τ) : \{ \begin{matrix}u ∈ C([0, τ) : [D(A)]) ∩ C^{1}([0, τ) : E), \\ u′(t) = Au(t) + \frac{t^{α}}{Γ(α+1)} x, 0 ⩽ t < τ, \\ u(0) = x, \end{matrix}

is well posed if it has a unique solution for every x ∈ E, cf. [1] if α ∈ ℕ and [13] if α > 0. If u(t, x) is a solution of C_{α+1}(τ), then the operators S(t)x := \frac{d}{dt}u(t, x), t ∈ [0, τ), x ∈ E, are bounded and form an α-times integrated semigroup generated by A. By [13, Theorem 2.1], the well-posedness of the problem C_{α+1}(τ) implies that for every c ∈ (0, \frac{τ}{α}), there exist constants c_{1} > 0 and M > 0 such that the exponential region E(c, c_{1}) := {ξ + iη : ξ ∈ ℝ, η ∈ ℝ, ξ ⩾ c_{1}, |η| ⩽ e^{cξ}} ⊂ ρ(A) and that ∥R(λ : A)∥ ⩽ M|λ|^{α}, λ ∈ E(c, c_{1}). If, additionally, \overline{D(A)} = E, then there exists a sufficiently large ω > 0 such that A − ω satisfies the assumptions (∗) and (∗∗) given above. Further on, if A is the densely defined generator of an α-times integrated semigroup (S(t))_{t⩾0} satisfying ∥S(t)∥ ⩽ Mt^{β}e^{ωt}, for all t ⩾ 0, where ω ⩾ 0, β ⩾ 0, α > 0, then for every a ∈ (0, 1), C ∈ (0, 1] and d ∈ (0, 1], there exists a sufficiently large ω > 0 such that A − ω satisfies (∗). Furthermore, it can be proved that A − ω satisfies (∗∗) with α − β − 1 instead of α, see [17, page 158] for this refinement.

(b) If the operator A satisfies the assumptions (1.1) and (1.2), then it can be easily proved that, for every a ∈ (0, 1), there exist C ∈ (0, 1] and b ∈ (0, 1] such that (∗) and (∗∗) are valid (with α = n). It is clear that there exist a great number of multiplication, differential and pseudo-differential operators acting on L^{p} type spaces which fulfill (∗) and (∗∗), but not (1.1). Especially, the construction given in [20] cannot be applied even if E := L^{2}(ℝ) and A is chosen to be the operator Δ^{2} − iΔ − I with maximal distributional domain. Then the spectrum of A is {ξ + iη : ξ ∈ ℝ, η ∈ ℝ, η^{2} = ξ + 1} and, for every b ∈ ℂ, our construction gives the definition of (I + iΔ − Δ^{2})^{b}.

We recall the basic fasts about semigroups of growth order r > 0, see [5], [18],

[19] and [21] for this notion. An operator family (T (t))_{t>0} in L(E) is a semigroup of growth order r if it satisfies:

(i) T (t + s) = T (t)T (s), t, s > 0, (ii) for every x ∈ E, the mapping t ↦ T (t)x is continuous, (iii) ∥t^{r}T (t)∥ = O(1), t → 0+, (iv) T (t)x = 0 for all t > 0 implies x = 0, and

(v) E_{0} = ⋃_{t>0} T (t)E is dense in E.

The infinitesimal generator of (T (t))_{t>0} is defined by

A_{0} := \{(x, y) ∈ E^{2} : lim_{t→0+} \frac{T (t)x − x}{t} = y\}.

It is a closable linear operator and its closure A = \overline{A_{0}} is called the complete infinitesimal generator (c.i.g.) of (T (t))_{t>0}. Following Tanaka [21], if the semigroup (T (t))_{t>0} of growth order r > 0 has an analytic extension to Σ_{γ} := {z ∈ ℂ : z ≠ 0, | arg(z)| < γ\}, for some γ ∈ (0, \frac{π}{2} ), denoted by the same symbol, and if additionally there exists an ω ∈ ℝ such that, for every δ ∈ (0, γ), there exists a suitable constant M_{δ} > 0 with ∥z^{r}T (z)∥ ⩽ M_{δ}e^{ω Re z}, z ∈ Σ_{δ}, then the family (T (t))_{t∈Σ_{γ}} is called an analytic semigroup of growth order r. We will use the following notations. For given a ∈ (0, 1), C ∈ (0, 1] and d ∈ (0, 1], put Γ_{1}(a, C, d) := {ξ + iη : ξ ∈ ℝ, η ∈ ℝ, η = −Cξ^{a}, ξ^{2} + η^{2} ⩾ d^{2}}. It is clear that there exists a unique ε(a, C, d) ∈ (0, d) such that (ε(a, C, d), −Cε(a, C, d)^{a}) ∈ ∂B_{d}. We define Γ_{2}(a, C, d) := {ξ + iη : ξ ∈ ℝ, η ∈ ℝ, ξ^{2} + η^{2} = d^{2}, ξ ⩽ ε(a, C, d)} and Γ_{3}(a, C, d) := {ξ+iη : ξ ∈ ℝ, η ∈ ℝ, η = Cξ^{a}, ξ^{2}+η^{2} ⩾ d^{2}}. The upwards oriented curve Γ(a, C, d) is defined by Γ(a, C, d) := Γ_{1}(a, C, d) ∪ Γ_{2}(a, C, d) ∪ Γ_{3}(a, C, d); put now H(a, C, d) := {ξ + iη : ξ ∈ (0, ∞), η ∈ ℝ, |η| ⩽ Cξ^{a}} ∪ B_{d}. For given d̃∈ (0, d] and ã ∈ (0, a], one can find a suitable constant C̃ so that Γ(ã, C̃, d̃) ⊂ H(a, C, d), where we define Γ(ã, C̃, d̃) in the same way as Γ(a, C, d).

3. Operators J^{b}, b ∈ ℂ

In order to construct the fractional powers (−A)^{b}, for every b ∈ ℂ, we define a closable linear operator J^{b}. As in [20], if α ∈ ℕ_{0} and [9], if α = −1, the construction is based on improper integrals of the form

\frac{1}{2πi} ∫_{Γ} (−λ)^{b}R(λ : A)x dλ.

Proposition 3.1. Let b ∈ ℂ satisfy Re b < −(α + 1) and let x ∈ E. Then the integral

I(b)x := \frac{1}{2πi} ∫_{Γ(a,C,d)} (−λ)^{b}R(λ : A)x dλ

exists and defines a bounded linear operator I(b) ∈ L(E). Moreover, if for some ã ∈ (0, a], C̃ ∈ (0, C] and d̃∈ (0, d]: Γ(ã, C̃, d̃) ⊂ H(a, C, d), then

I(b)x = \frac{1}{2πi} ∫_{Γ(ã,C̃,d̃)} (−λ)^{b}R(λ : A)x dλ.

Proof. The proof is essentially contained in that of Lemma 1.1 in [20]. We sketch it for the sake of completeness. Note that the function λ ↦ (−λ)^{b} (1^{b} = 1) is analytic in ℂ ∖ [0, ∞) and that

(3.1)  |(−λ)^{b}| ⩽ |λ|^{Re b}e^{π| Im b|}, λ ∈ ℂ ∖ {0}.

The integral over Γ_{2}(a, C, d) exists since Γ_{2}(a, C, d) is a finite path. The estimate (∗∗) implies

∥\frac{1}{2πi} ∫_{Γ_{3}(a,C,d)} (−λ)^{b}R(λ : A)x dλ∥

⩽ \frac{M}{2π}∫_{ε(a,C,d)}^{∞}(\sqrt{t^{2} + C^{2}t^{2a}})^{Re b}e^{π| Im b|}(1+\sqrt{t^{2} + C^{2}t^{2a}})^{α}(1 + Caε(a, C, d)^{a−1}) dt

⩽ \frac{M(1 + Caε(a, C, d))^{a−1}e^{π| Im b|}}{2π} ∫_{ε(a,C,d)}^{∞} (\sqrt{t^{2} + C^{2}t^{2a}})^{Re b}t^{α}dt.

Since (t^{2} + C^{2}t^{2a})^{Re b/2}t^{α} ∼ t^{α+Re b}, t → +∞, the integral over Γ_{3}(a, C, d) exists. Similarly, the integral over Γ_{1}(a, C, d) exists. It remains to be shown that the integral I(b) is independent of the choice of a curve Γ(a, C, d). Let R be sufficiently large and let the curve Γ_{R} = {Re^{it} : t ∈ [arctan(C̃R^{ã−1}), arctan(CR^{a−1})]} be upwards oriented. Then

∥∫_{Γ_{R}} (−λ)^{b}R(λ : A)x dλ∥ ⩽ 2πe^{π| Im b|}R^{Re b}(1 + R)^{α}R → 0, R → +∞.

The proof completes an elementary application of Cauchy’s theorem. □

If no confusion seems likely, we shall simply denote Γ(a, C, d), H(a, C, d) and ε(a, C, d) by Γ, H and ε, respectively.

Let t ∈ ℝ. Denote by ⌊t⌋ and ⌈t⌉ the largest integer ⩽ t and the smaller integer ⩾ t, respectively. Put {t} := t − ⌊t⌋. Note, if b ∈ ℂ, then Re(b − ⌊Re b + α⌋ − 2) ∈ [−(α + 2), −(α + 1)). Hence, the following definition of the operator J^{b} makes a sense.

Definition 3.1. Let b ∈ ℂ. The operator J^{b} is defined as follows: D(J^{b}) := D(A^{⌊Re b+α⌋+2}) and

J^{b}x := \{ \begin{matrix}I(b)x, −(α + 2) ⩽ Re b < −(α + 1), \\ I(b − ⌊Re b + α⌋ − 2)(−A)^{⌊Re b+α⌋+2}x, otherwise. \end{matrix}

Remark 3.1. If a densely defined operator A satisfies (1.1) and (1.2), then we have already seen that, for every a ∈ (0, 1), there exist C ∈ (0, 1] and d ∈ (0, 1] such that (∗) and (∗∗) are fulfilled. In this case, the definition of J^{b} is equivalent to the corresponding one given in Definition 1.2 of [20].

In what follows, we will use the generalized resolvent equation

(3.2)  (−λ)^{−n−1}R(λ : A)(−A)^{n+1}x = R(λ : A)x + ∑_{i=0}^{n}(−λ)^{−i−1}(−A)^{i}x,

if λ ∈ ρ(A), λ ≠ 0, n ∈ ℕ_{0}, x ∈ D(A^{n+1}), and the simple equality _{∫Γ}(−λ)^{b} dλ = 0, if Re b < −1.

Proposition 3.2. Suppose x ∈ D(A^{⌊Re b+α⌋+2}). Then

J^{b}x = \begin{cases}\frac{1}{2πi} ∫_{Γ} (−λ)^{b}R(λ : A)x dλ, & Re b < 0, \\ \frac{1}{2πi} ∫_{Γ} (−λ)^{b−⌊Re b⌋−1}R(λ : A)(−A)^{⌊Re b⌋+1}x dλ, & Re b ⩾ 0. \end{cases}

Proof. Suppose Re b < 0. If Re b ∈ [−(α+2), −(α+1)), the conclusion follows directly from the definition of J^{b}. If Re b ∉ [−(α + 2), −(α + 1)), then by Definition 3.1 and (3.2):

J^{b}x = J^{b−⌊Re b+α⌋−2}(−A)^{⌊Re b+α⌋+2}x

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b−⌊Re b+α⌋−2}R(λ : A)(−A)^{⌊Re b+α⌋+2}x dλ

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b}(R(λ : A)x + ∑_{i=0}^{⌊Re b+α⌋+1}(−λ)^{−i−1}(−A)^{i}x) dλ.

If Re b < 0 and i = 0, 1, …, ⌊Re b + α⌋ + 1, then Re b − i − 1 < −1 and the last term equals

\frac{1}{2πi}∫_{Γ} (−λ)^{b}R(λ : A)x dλ

as claimed. Suppose now Re b ⩾ 0. Then (3.2) implies

(−λ)^{b−⌊Re b⌋−1}R(λ : A)(−A)^{⌊Re b⌋+1}x = (−λ)^{b}(R(λ : A)x + ∑_{i=0}^{⌊Re b⌋}(−λ)^{−i−1}(−A)^{i}x).

By Definition 3.1,

J^{b}x = J^{b−⌊Re b+α⌋−2}(−A)^{⌊Re b+α⌋+2}x

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b−⌊Re b+α⌋−2}R(λ : A)(−A)^{⌊Re b+α⌋+2}x dλ

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b}(R(λ : A)x + ∑_{i=0}^{⌊Re b+α+1⌋}(−λ)^{−i−1}(−A)^{i}x) dλ,

and since for j = ⌊Re b + 1⌋, …, ⌊Re b + α + 1⌋, Re b − i − 1 < −1, we obtain

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b}(R(λ : A)x + ∑_{i=0}^{⌊Re b⌋}(−λ)^{−i−1}(−A)^{i}x) dλ

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b−⌊Re b⌋−1}R(λ : A)(−A)^{⌊Re b+1⌋}x dλ.

The proof is completed. □

Put C^{b} := (−A)^{⌊Re b+α⌋+2}J^{b−⌊Re b+α⌋−2}. Then, for every b ∈ ℂ, C^{b} is a closed linear operator. Proceeding similarly as in the proof of [20, Proposition 1.3], one obtains that, for every b ∈ ℂ with Re b ⩾ −(α + 1), J^{b} ⊂ C^{b} and, consequently, J^{b} is a closable operator. Clearly, J^{b} ∈ L(E) for every b ∈ ℂ with Re b < −(α + 1).

Lemma 3.1. Let b ∈ ℂ. Then J^{b} is a closable operator.

Lemma 3.2. Let b ∈ ℂ. Then

(i′) J^{b}x = J^{b+k}(−A)^{−k}x, k ∈ ℕ_{0}, x ∈ D(J^{b}), and (ii′) J^{b}x = J^{b+k}(−A)^{−k}x, if −k ∈ ℕ and x ∈ D(A^{max(−k,⌊Re b+α+2⌋)}).

Proof. (i′) If k = 0, the proof is trivial. Suppose now k = 1. If −(α + 2) ⩽ Re b < −(α + 1), then Re b + 1 < 0 and by Proposition 3.2, we obtain

J^{b+1}(−A)^{−1}x = \frac{1}{2πi} ∫_{Γ} (−λ)^{b+1}R(λ : A)(−A)^{−1}x dλ

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b+1} \frac{R(λ : A)x − (−A)^{−1}x}{−λ} dλ

= \frac{1}{2πi} ∫_{Γ} (−λ)^{b}R(λ : A)x dλ.

If Re b ∉ [−(α + 2), −(α + 1)), the assertion follows from Definition 3.1. Now (i′) follows by induction; (ii′) can be proved by the use of (i′). □

For b ∈ ℂ, denote ⟨b⟩ := max(0, ⌊Re b + α⌋ + 2). Note that ⟨b + c⟩ ⩽ ⟨b⟩ + ⟨c⟩. The expected semigroup property of the family (J^{b})_{b∈ℂ} can be proved similarly as in [20, Lemma 1.4]. More precisely, we have:

Proposition 3.3. Let b, c ∈ ℂ. Then J^{b}J^{c}x = J^{b+c}x, x ∈ D(A^{⟨b⟩+⟨c⟩}).

If k ∈ ℕ and x ∈ D(A^{k}), put ∥x∥_{k} := ∥x∥ + ∥Ax∥ + ⋯ + ∥A^{k}x∥. We prove the following lemma which naturally corresponds to [20, Lemma 1.5].

Proposition 3.4. Let b ∈ ℤ and x ∈ D(A^{⌊b+α⌋+2}). Then J^{b}x = (−A)^{b}x.

Proof. By Lemma 3.2, it is enough to prove the statement in the case b = 1. We have to prove (see Proposition 3.2) that

\frac{1}{2πi} ∫_{Γ} (−λ)^{−1}R(λ : A)(−A)^{2}x dλ = −Ax.

By the resolvent equation, if x ∈ D(A^{⌊α+1⌋}), then there exists a suitable constant M > 0 such that

∥R(λ : A)x∥ ⩽ M|λ|^{α−⌊α⌋−1}∥x∥_{⌊α+1⌋}, λ ∈ H ∪ Γ, |λ| ⩾ d.

Let R > d. Then there exists a unique number κ(R) ∈ (0, R) such that κ(R)^{2} + C^{2}κ(R)^{2a} = R^{2}. Denote Γ_{R} = {Re^{iθ} : |θ| ⩽ arctan(Cκ(R)^{a−1})}; we assume that Γ_{R} is upwards oriented. If x ∈ D(A^{⌊α⌋+3}), then A^{2}x ∈ D(A^{⌊α⌋+1}) and the previous inequality implies

∥∫_{Γ_{R}} (−λ)^{−1}R(λ : A)(−A)^{2}x dλ∥ ⩽ 2π \frac{M}{R} R^{α−⌊α⌋−1}∥x∥_{⌊α+3⌋}R → 0, R → +∞.

The rest of proof follows by an application of Cauchy’s integral formula. □

Proceeding as in [20], one can prove the next proposition.

Proposition 3.5. The next assertions are valid

(i) If b ∈ ℤ, then \overline{J^{b}} = (−A)^{b}. (ii) If Re b > α + 1, then \overline{J^{b}} = C^{b}.

4. Fractional powers and semigroups generated by them

Definition 4.1. Let b ∈ ℂ. Then the fractional power (−A)^{b} of the operator −A is defined by (−A)^{b} := \overline{J^{b}}.

The next theorem clarifies the basic structural properties of fractional powers. See [20] for the proof.

Theorem 4.1. Let b, c ∈ ℂ and k ∈ ℕ_{0}. Then

(a) D(A^{⌊Re b+α⌋+2+k}) is a core for (−A)^{b}.

(b) (−A)^{b+c} ⊂ \overline{(−A)^{b}(−A)^{c}}.

(c) (−A)^{b+c} = \overline{(−A)^{b}(−A)^{c}}, if (−A)^{b+c} = C^{b+c}.

(d) \overline{(−A)^{−b}(−A)^{b}} = I; (−A)^{−b}(−A)^{b}x = x, x ∈ D((−A)^{b}).

(e) (−A)^{b} is injective.

In the rest of this section, we consider fractional powers as generators of analytic semigroups of growth order r > 0.

Theorem 4.2. Let b ∈ (0, \frac{1}{2}). Then the operator −(−A)^{b} is the c.i.g. of an analytic semigroup (T_{b}(z))_{z∈Σ_{arctan(cos πb)}} of growth order \frac{α+1}{b}, where

(4.1)  T_{b}(z) = \frac{1}{2πi} ∫_{Γ} e^{−z(−λ)^{b}}R(λ : A) dλ, z ∈ Σ_{arctan(cos πb)}.

Proof. Our choice of b implies bπ < \frac{π}{2} . Put γ := arctan(cos πb). Then γ ∈ (0, \frac{π}{2} ) and, for every z = ξ +iη ∈ Σ_{γ}, we have ξ cos(bπ)−|η| > 0. Furthermore,

|e^{−z(−λ)^{b}}| = e^{−ξ|λ|^{b} cos(b arg(−λ))+η|λ|^{b} sin(b arg(−λ))} ⩽ e^{−(ξ cos(bπ)−|η|)|λ|^{b}}.

The convergence of the curve integral over Γ_{1} and Γ_{3} follows from the computation

∥ \frac{1}{2πi} ∫_{Γ_{1}} e^{−z(−λ)^{b}}R(λ : A) dλ∥

⩽ \frac{M}{2π} ∫_{ε}^{∞}e^{−(ξ cos(bπ)−|η|)(\sqrt{t^{2}+t^{2a}})^{b}}(1 + \sqrt{t^{2} + t^{2a}})^{α}(1 + aε^{a−1})dt

⩽ \frac{M(1 + aε^{a−1})}{2π} [ ∫_{ε}^{∞}e^{−(ξ cos(bπ)−|η|)t^{b}}(1+\sqrt{2})^{α}dt + ∫_{1}^{∞}e^{−(ξ cos(bπ)−|η|)t^{b}}(1+\sqrt{2})^{α}t^{α}dt]

⩽ \frac{M(1 + \sqrt{2})^{α}(1 + aε^{a−1})}{2π} [(1 − ε)e^{−(ξ cos(bπ)−|η|)ε^{b}} + ∫_{0}^{∞} e^{−(ξ cos(bπ)−|η|)t^{b}}t^{α}dt]

= \frac{M(1+\sqrt{2})^{α}(1+aε^{a−1})}{2π} [(1 − ε)e^{−(ξ cos(bπ)−|η|)ε^{b}}+ \frac{1}{b} Γ( \frac{α+1}{b} )(ξ cos(bπ) − |η|)^{− \frac{α+1}{b}} ],

where Γ(·) denotes the gamma function in the last estimate. The convergence of the integral over Γ_{2} is obvious and one obtains

∥\frac{1}{2πi} ∫_{Γ_{2}} e^{−z(−λ)^{b}}R(λ : A) dλ∥ ⩽ Me^{−(ξ cos(bπ)−|η|)d^{b}}(1 + d)^{α+1}.

Hence, for every δ ∈ (0, γ), we have ∥z^{\frac{α+1}{b}} T_{b}(z)∥ = O(1), z ∈ Σ_{δ}. By an elementary application of Cauchy’s formula, it follows that the integral in (4.1) does not depend on the choice of the curve Γ(a, C, d). Fix a λ_{0} ∈ ρ(A) ∖ H. Here we would like to point out that ρ(A)∖H is a nonempty set since ρ(A) is an open subset of ℂ. Using the same arguments as in [20, Propositions 2.3, 2.5, 2.6, 2.7 and 2.8], one obtains:

1. Let m ∈ {0, 1}. The improper integral

∫_{Γ} \frac{−(−λ)^{mb}e^{−z(−λ)^{b}}}{(λ − λ_{0})^{⌊b+α⌋+2}} R(λ : A) dλ

converges uniformly for z ∈ \overline{Σ_{γ}}.

2. The mapping z ↦ T_{b}(z), z ∈ Σ_{γ}, is analytic and

\frac{d^{n}}{dz^{n}} T_{b}(z) = \frac{(−1)^{n}}{2πi} ∫_{Γ} (−λ)^{nb}e^{−t(−λ)^{b}}R(λ : A) dλ, n ∈ ℕ.

3. T_{b}(z_{1} + z_{2}) = T_{b}(z_{1})T_{b}(z_{2}), z_{1}, z_{2} ∈ Σ_{γ}.

4. If x ∈ D(A^{⌊b+α⌋+2}) = D(J^{b}), then lim_{z→0, z∈Σ_{γ}} \frac{T_{b}(z)x−x}{z} = −J^{b}x. Especially, the continuity set Ω_{b} := {x ∈ E : T_{b}(t)x → x as t → 0+} is dense in E. Since the set E_{b} := ⋃_{t>0} T_{b}(t)E is dense in Ω_{b}, it follows that \overline{E_{b}} = E.

5. For every z ∈ Σ_{γ}, T_{b}(z) is an injective operator.

By the foregoing, we obtain that (T_{b}(t))_{t∈Σ_{γ}} is an analytic semigroup of growth order \frac{α+1}{b}. Denote by A_{b} the generator of (T_{b}(t))_{t>0}. By 4, we obtain that −J^{b} ⊂ A_{b}. Consequently, −(−A)^{b} ⊂ \overline{A_{b}}. Since ∫_{Γ} e^{−t(−λ)^{b}}λ^{n} dλ = 0, n ∈ ℕ_{0}, one can repeat literally the arguments given in [20, Lemma 2.10] in order to obtain that, for every x ∈ E and t > 0, T_{b}(t)x ∈ D(A^{n}) and that

(4.2)  A^{n}T_{b}(t)x = \frac{1}{2πi} ∫_{Γ} e^{−t(−λ)^{b}}λ^{n}R(λ :A)x dλ.

Particularly, E_{b} ⊂ D_{∞}(A). In order to prove that \overline{A_{b}} ⊂ −(−A)^{b}, one can proceed in the same manner as in [20]. Actually, it is sufficient to replace the natural number n in the proofs of [20, Propositions 2.11 and 2.12] with ⌊b + α⌋. □

The following theorem is the main result of this paper and possesses several natural consequences in the theory of partial differential equations.

Theorem 4.3. Suppose that a closed, densely defined operator A satisfies (∗) and (∗∗). Then, for every n ∈ ℕ with n ⩾ 3 and x ∈ D(A^{⌊ \frac{1}{n} +α⌋+2}), the abstract Cauchy problem

\{ \begin{matrix}u ∈ C((0, ∞) : [D(A)]) ∩ C^{n}((0, ∞) : E), \\ \frac{d^{n}}{dt^{n}} u(t) = (−1)^{n+1}Au(t), t > 0, \\ lim_{t→0+} u(t) = x, sup_{t>0} ∥u(t)∥ < ∞, \end{matrix}

has a solution u(t) = T_{\frac{1}{n}} (t)x, t > 0. Further on, u can be analytically extended to Σ_{arctan(cos \frac{π}{n})} and, for every δ ∈ (0, arctan(cos \frac{π}{n})) and i ∈ ℕ_{0},

sup_{z∈Σ_{δ}} ∥z^{i+nα+n} \frac{d^{i}}{dz^{i}} u(z)∥ < ∞.

Proof. One can use the assertion 2 given in the proof of the Theorem 4.2 and (4.2) to obtain that \frac{d^{n}}{dt^{n}} u(t) = (−1)^{n+1}Au(t), t > 0. By Theorem 4.2, u can be analytically extended to Σ_{arctan(cos \frac{π}{n})}. Due to the proof of Theorem 4.2 (see the assertion 4), we conclude that lim_{t→0+} u(t) = x. Let δ ∈ (0, arctan(cos \frac{π}{n})) and z ∈ Σ_{δ} be fixed. Since

\frac{d^{i}}{dz^{i}} u(z) = \frac{(−1)^{n}}{2πi} ∫_{Γ} (−λ)^{\frac{i}{n}} e^{−z(−λ)^{\frac{1}{n}}} R(λ : A)x dλ,

and ∥(−λ)^{\frac{i}{n}} R(λ : A)∥ ⩽ M(1 + |λ|)^{α+ \frac{i}{n}}, it follows (see the proofs of Theorem 4.2 and [20, Proposition 2.2]) that

∥\frac{d^{i}}{dz^{i}} u(z)∥ = O((ξ cos(\frac{π}{n}) − |η|)^{−(α+ \frac{i}{n} +1)/ \frac{i}{n}} ), z ∈ Σ_{δ}.

Hence, sup_{z∈Σ_{δ}} ∥z^{i+nα+n} \frac{d^{i}}{dz^{i}} u(z)∥ < ∞. □

The author wants to thank B. Straub and F. Periago for sending him the copies of [17]–[20] and [22]. The genesis of the paper is partially supported by Ministry of Sciences and Technology of Republic Serbia and Faculty of Technical Sciences of Novi Sad.

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Faculty of Technical Sciences (Received 14 06 2006) University of Novi Sad 21000 Novi Sad Serbia

marco.s@verat.net