Operator theoretic properties of semigroups in terms of their generators

S. Blunck , L. Weis

Abstract

Let (T_{t}) be a C_{0} semigroup with generator A on a Banach space X and let A be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ, A) of A belongs to A if and only if the integrated semigroup S_{t} := ∫_{0}^{t} T_{s} ds belongs to A. For analytic semigroups S_{t} ∈ A implies T_{t} ∈ A, and in this case we give precise estimates for the growth of the A-norm of T_{t} (e.g. the trace of T_{t}) in terms of the resolvent growth and the imbedding D(A) ↪ X. ^{1}

0 Introduction

In this paper we study how operator theoretic properties of the generator and the resolvent of a C_{0} semigroup on a Banach space X are reflected in the properties of the semigroup. Often operator theoretic properties of an operator T can be checked conveniently by showing that T belongs to a suitable operator ideal. If T belongs to the ideal of compact or strictly singular operators we know that its spectrum σ(T) consists of a series of eigenvalues with possible limit point 0 , that T is an admissible Fredholm perturbation, etc. We know about the summability of its eigenvalues and its trace if T belongs to the Hilbert-Schmidt class, the trace class or to one of the ideals extending the Schatten classes to the Banach space setting. To have a unified approach to many of these topics, we phrase our question as follows: Given an operator ideal A and a C_{0} semigroup (T_{t}) with generator A and resolvent R(λ, A) , how can we characterize “R(λ, A) ∈ A ” in terms of A and the semigroup T_{t} ?

11991 Mathematics Subject Classification: 47A60 , 47B10 , 47D06

Since the semigroup and the resolvent are connected by the Laplace transform

R(λ, A) = ∫_{0}^{∞} e^{−λ t} T_{t} dt (1)

the resolvent is a “smoothing” of the semigroup and one would generally expect that T_{t} ∈ A implies R(λ, A) ∈ A (see e.g. [V] for details) . But the inversion of the Laplace transform is very singular so that it is not surprising that the converse is false in general. E.g. a well-known result of Pazy shows that the compactness of R(λ, A) implies the compactness of the T_{t} only if the mapping t ↦ T_{t} is continuous in the operator norm for t > 0 . In \underline{Section 2} we show essentially that R(λ, A) belongs to an ideal A if and only if the integrated semigroup S_{t} := ∫_{0}^{t} T_{s} ds belongs to A. We also express “ T_{t} ∈ A ” in terms of growth conditions on the resolvent. In \underline{Section 3} we extend this result to the Phillips functional calculus: For a closed ideal A (e.g. if A is the ideal of compact or weakly compact operators) we have R(λ, A) ∈ A if and only if ĝ(−A) ∈ A for one or all functions g considered in this calculus (with a mild restriction) . We also generalize Pazy’s result on “ T_{t} ∈ K ”. As pointed out to us by the referee, one obtains a similar result for Lipschitz continuous integrated semigroups. In \underline{Section 4} we consider analytic semigroups. Here R(λ, A) ∈ A always implies T_{t} ∈ A, but if A is not closed the norm ∥ T_{t} ∥_{A} usually blows up for t → 0 (e.g. if A is the Hilbert-Schmidt or the trace class) . So the question arises how this growth of ∥ T_{t} ∥_{A} for t → 0 is related to the growth of the resolvent. Our main result here is that for all α > 0 and β ∈ (0, 1] one has

∥ T_{t} ∥_{A} ≤ C t^{β−α} on ℝ_{+}  ⟺  ∥ R(λ, A)^{α} ∥_{A} ≤ D λ^{−β} on ℝ_{+}

This result allows to obtain estimates on (T_{t}) directly from information on the imbeddings J_{α}(A) : D((−A)^{α}) ↪ X. Indeed, if J_{α}(A) ∈ A then ∥ T_{t} ∥_{A} ≤ C t^{−α} for small t. To illustrate the use of our general framework we discuss briefly some applications to ultracontractivity as well as to trace and Gaussian estimates for semigroups, all of which are related to (one-sided) operator ideals. Detailed expositions of these applications are given in [B2] and [B3] .

1 Preliminaries and notations

Let X be a complex Banach space with unit ball B_{X} and identity operator I_{X} . With the convention S_{0} := ℝ_{+} := \{ r ∈ ℝ; r > 0 \} we denote for all δ ∈ [ 0, \frac{π}{2} ) by Σ_{δ} and S_{δ} the open sectors

Σ_{δ} := \{λ ∈ ℂ; | arg(λ)| < \frac{π}{2} + δ \} , S_{δ} := \{z ∈ ℂ; | arg(z)| < δ \} .

Let (T_{t})_{t≥0} be a C_{0} semigroup on X with generator A. The family (S_{t})_{t≥0} given by

S_{t} x := ∫_{0}^{t} T_{s} x ds

is called the integrated semigroup and provides an integrated semigroup in the sense of [A]. We will also consider fractional powers of the resolvent (cf. 4.8(b) )

R(λ, A)^{α} = Γ(α)^{−1} ∫_{0}^{∞} e^{−λ t} t^{α−1} T_{t} dt , α > 0 . (2)

If A is the generator of a bounded holomorphic semigroup we denote by δ(A) its maximal angle of holomorphy and put for all 0 ≤ δ < δ(A)

R_{δ}(A) := sup_{λ∈ Σ_{δ}} ∥ λ R(λ, A) ∥ .

For a generator A the operator −A is non-negative in the sense of [MSM] , i.e. its fractional powers (−A)^{α} are well-defined for α > 0 . We denote by J_{α}(A) and ( if 0 ∈ ρ(A) ) J_{α}^{o}(A) the imbedding operators

D( (−A)^{α}) ↪ X  and  D^{o}( (−A)^{α}) ↪ X

where D^{o}( (−A)^{α}) is the usual domain equipped with the norm x ↦ ∥ (−A)^{α} x ∥.

We summarize briefly some relevant facts from the theory of operator ideals as developped in [P] . Suppose that for every pair of Banach spaces X and Y we are given a subspace A(X, Y) of the space L(X, Y) of all bounded linear operators that contains the finite dimensional operators and satisfies the following “ideal” property:

T ∈ L(X_{0}, X) , S ∈ A(X, Y) , R ∈ L(Y, Y_{0})  ⟹  RST ∈ A(X_{0}, Y_{0}) .

Then the union A of all A(X, Y) is called an operator ideal . Such an operator ideal is called a normed operator ideal if every component A(X, Y) carries a complete norm ∥ · ∥_{A} stronger than the operator norm with

∥ x^{∗} ⊗ y ∥_{A} = ∥ x^{∗} ∥ ∥ y ∥ for all x^{∗} ∈ X^{∗} , y ∈ Y

T ∈ L(X_{0}, X) , S ∈ A(X, Y) , R ∈ L(Y, Y_{0}) ⇒ ∥ RST ∥_{A} ≤ ∥ R ∥ ∥ S ∥_{A} ∥ T ∥ .

An operator ideal is called closed if ∥ · ∥_{A} = ∥ · ∥ and A(X, Y) is closed in L(X, Y) . E.g. the compact, weakly compact and strictly singular operators form closed operator ideals K, W and S, respectively. Other well-known examples are the operator ideals P_{r} and N_{r} of all absolutely rsumming operators and r-nuclear operators which coincide for 1 < r < ∞ and Hilbert spaces X, Y with the class S_{2} of Hilbert-Schmidt operators and therefore can be considered as extensions of S_{2} to the Banach space setting. The ideal S_{1} of operators with summable approximation numbers extends in this sense the trace class of Hilbert space operators, and there are further ideals extending the other Schatten classes S_{p} to the Banach space setting. For definitions and basic properties of these ideals see e.g. [P] . We will use the notation A(X) := A(X, X) and the following useful fact which is immediate from a convexity theorem for the Bochner integral (see e.g. [DU] , II.2.8 ).

Remark 1.1 Let I be an interval and F ∈ L_{1}(I, L(X)) satisfy F (I) ⊂ B_{A(X)} . If B_{A(X)} is closed in the operator norm then ∫_{I} F (t) dt ∈ |I| B_{A(X)} .

This observation can be applied not only to closed ideals but to most popular ideals.

Lemma 1.2 The unit ball B_{A(X)} is closed with respect to the strong operator topology of L(X) if

(a) A is maximal [P] (e.g. if A = P_{r}) or

(b) A = N_{r} for some 1 ≤ r < ∞ and X is a dual space such that X^{∗} has the Radon-Nikodym property (e.g. if X is reflexive) .

Proof. Let A^{max} denote the maximal hull of A ( [P], 8.7.1 ) . Then

B_{A(X)} ⊂ L − cl ( B_{A(X)} ) ⊂ s − cl ( B_{A(X)} ) ⊂ B_{A^{max}(X)} (3)

where s−cl denotes the closure in the strong operator topology. Here the last inclusion is shown similar to [P] , 10.3.4 . The maximality of A means A = A^{max}, hence in the case (a) the claim follows from (3) . The maximality of P_{r} is shown in [P] . In the case (b) the assumtions on X yield A(X) = A^{max}(X) ( [P], 19.2.1 ; [DF], 33.6.1 ) though A = N_{r} is not maximal and we can conclude as before. □

2 General semigroups

Let A be the generator of a C_{0} semigroup (T_{t}) on X. In this general situation we will show that the properties of the resolvent are reflected not necessarily by the semigroup (T_{t}) itself but rather in the properties of the integrated semigroup

S_{t} x := ∫_{0}^{t} T_{s} x ds

If the growth bound ω(T_{t}) is not negative we also need

S̃_{t} x := ∫_{0}^{t} e^{−bs} T_{s} x ds

for some b > ω(T_{t}) . (Of course, if ω(T_{t}) < 0 we might choose S̃_{t} = S_{t}. )

Theorem 2.1 For a C_{0} semigroup (T_{t}) with generator A and (S_{t}) , (S̃_{t}) as above the following statements are equivalent for every normed operator ideal A :

(a) R(λ, A) ∈ A(X) for one ( all ) λ ∈ ρ(A) .

(b) J_{1}(A) ∈ A(D(A), X) .

(c) S̃_{t} ∈ A(X) for one ( all ) t > 0. If B_{A(X)} is closed in L(X) we may add

(d) { t ∈ ℝ_{+} ; S_{t} ∈ A(X) } contains a set of positive Lebesgue measure.

Remark 2.2 (i) Condition (c) implies that (S̃_{t}) is ∥ · ∥_{A}-bounded while condition (d) ensures that (S_{t}) is ∥ · ∥_{A}-bounded on finite intervals and

lim sup_{t→∞} \frac{1}{t} log ∥ S_{t} ∥_{A} ≤ ω(T_{t}) ∨ 0

(ii) The proof will show that aside from the equivalence (a)⟺(b) all claims of Theorem 2.1 remain true if we assume A(X) only to be a one-sided ideal in the Banach algebra L(X) with norm estimate

∥ST ∥_{A} ≤ ∥S∥ ∥T ∥_{A}  or  ∥ST ∥_{A} ≤ ∥T ∥ ∥S∥_{A} . (4)

Contrary to what one might expect one cannot use the “translation” Ã = A − b to get (c) if S_{t} ∈ A(X) for some t ∈ ℝ_{+} . This is shown by the following example where { t ∈ ℝ_{+} ; S_{t} ∈ A(X) } is infinite.

Example 2.3 The bounded translation semigroup (T_{t}f)(s) := f(s − t) on L_{2}(T) is unitarily equivalent to the semigroup of diagonal operators

D_{t} x := (e^{−a_{n}t} x_{n}) on l_{2}(ℤ) (5)

for the special choice a_{n} := −i n and has the following properties :

(a) R(λ, A) ∈ S_{2} , but R(λ, A) ∉ S_{1} for all λ ∈ ρ(A) .

(b) S_{t} ∈ S_{1} for t = 2 π k, k ∈ ℕ.

Proof. For h_{t}(s) := ∑_{n} e^{−a_{n}t} e^{ins} a direct computation shows h_{t} ∗ f = T_{t} f for all f ∈ L_{2}(T) . Thus it is sufficient to establish the properties (a) and (b) for the semigroup (D_{t}) defined in (5) which admits the following representation :

(6)  R(λ, A) x = ( \frac{1}{λ + a_{n}} x_{n})  ,  S_{t} x = ( \frac{1 − e^{−a_{n}t}}{a_{n}} x_{n})

where expressions of the form \frac{1−e^{−at}}{a} are read as t if a = 0 . Now (a) is obvious because ( \frac{1}{λ−in}) ∉ l_{1} for all λ in the resolvent set ρ(A) . Furthermore in the case t = 2πk the operator S_{t} has rank 1 since

S_{t} x = (…, 0 , t x_{0} , 0 , …) . □

Example 2.3 shows also how far (T_{t}) can be from A if we only have R(λ, A) ∈ A. However, for a differentiable (in particular analytic) semigroup the T_{t} are bounded operators from X into D(A) and hence

Corollary 2.4 If (T_{t}) is a differentiable semigroup then R(λ, A) ∈ A(X) implies T_{t} ∈ A(X) for all t > 0 . The converse holds if B_{A(X)} is closed in L(X) .

In general, we have the following connection between the resolvent and the semigroup.

Proposition 2.5 Let B_{A(X)} be closed with respect to the strong operator topology of L(X) (cf. Lemma 1.2 ) . Then T_{t_{0}} ∈ A(X) for a fixed t_{0} > 0 if and only if there are constants C, λ_{0} ≥ 0 such that

∥ T_{t_{0}} R(λ, A) ∥_{A} ≤ C λ^{−1} for all λ ≥ λ_{0} .

Remark 2.6 (a) If T_{t_{0}} ∈ A(X) we trivially have for every ω > ω(T_{t}) that

∥ T_{t_{0}+t} ∥_{A} ≤ M e^{ω t} ∥ T_{t_{0}} ∥_{A} for all t ≥ 0 .

(b) If I_{X} ∉ A(X) then the strong convergence T_{t} → I_{X} implies ∥ T_{t} ∥_{A} → ∞ for t → 0 . In Section 4 we will relate this growth to the growth of ∥ R(λ, A) ∥_{A} for λ → ∞ in the case of analytic semigroups.

(c) For closed ideals A (for which B_{A(X)} is in general not closed with respect to the strong operator topology) we refer to Theorem 3.2 .

Proof of Proposition 2.5 . The “only if” part follows from the strong convergence of λ R(λ, A) T_{t_{0}} to T_{t_{0}} for λ → ∞. The “if” part is a consequence of 2.6(a) and

T_{t_{0}} R(λ, A) = ∫_{0}^{∞} e^{−λ t} T_{t+t_{0}} dt . □

For the proof of Theorem 2.1 we need

Lemma 2.7 Let f : [0, T] → L(X) be strongly continuous and bounded. For all ω ∈ ℝ consider the continuous function F_{ω} : [0, T] → L(X) defined by

F_{ω}(t) x := ∫_{0}^{t} e^{−ω s} f(s)x ds .

Assume that B_{A(X)} is closed in L(X) . If F_{ω_{0}} is A(X)-valued and ∥ · ∥_{A}-bounded for some ω_{0} ∈ ℝ then the same holds for all ω ∈ ℝ.

Proof. Fix ω ∈ ℝ. For g_{t} : [0, T] → ℝ , s ↦ (ω_{0}−ω)e^{−(ω_{0}−ω) (t−s)} a straightforward computation using Fubini’s theorem shows

F_{ω}(t) = e^{(ω_{0}−ω) t} (F_{ω_{0}}(t) − ∫_{0}^{t} F_{ω_{0}}(s) g_{t}(s) ds) . (7)

Hence the assertion follows from 1.1 and the assumtion on A. □

Proof of Theorem 2.1 . The resolvent identity shows the equivalence of the two versions of (a). For (a)⇔(b) we just have to observe that J_{1}(A) = (R(λ, A) : X → X) ∘ (A − λI : D(A) → X) and (R(λ, A) : X → X) = J_{1}(A) ∘ (R(λ, A) : X → D(A)) for all λ ∈ ρ(A) . (a)⇒(c) : Let M̃ := (1 + M) ∥ R(b, A) ∥_{A} and t ≥ 0 . From

( I − T̃_{t}) R(b, A) = ∫_{0}^{∞} T̃_{s} ds − ∫_{t}^{∞} T̃_{s} ds = S̃_{t}

we get S̃_{t} = (I − T̃_{t}) R(b, A) ∈ A(X) and ∥ S̃_{t} ∥_{A} ≤ ∥ I − T̃_{t} ∥ ∥ R(b, A) ∥_{A} ≤ M̃. (c)⇒(a) : We can assume that ∥ T̃_{t} ∥ < 1 and the representation

R(b, A) = ∫_{0}^{∞} T̃_{s} ds = ∑_{k=0}^{∞} T̃_{kt} ∫_{0}^{t} T̃_{s} ds = ( ∑_{k=0}^{∞} (T̃_{t})^{k}) S̃_{t}

reveals that R(b, A) = ( ∑_{k=0}^{∞}(T̃_{t})^{k}) S_{t} ∈ A(X) .

(c)⇒(d) By the proof of (a)⇒(c) we can assume in addition that (S̃_{t}) is ∥ · ∥_{A}- bounded on [0, T] . Now we can apply Lemma 2.7 to the function f(t) := T_{t}. (d)⇒(c) If we knew that (S_{t}) is ∥ · ∥_{A}-bounded on [0, T] for some T > 0 we could apply Lemma 2.7 again. As a first step towards this boundedness we show that there exists a non-empty open subset E of ℝ_{+} with the property that (S_{t}) is ∥·∥_{A}-bounded on E. Indeed, since {t ∈ ℝ_{+}; S_{t} ∈ A(X)} contains a set of positive Lebesgue measure it has a compact subset K still of positive Lebesgue measure. Now put K_{n} := { t ∈ K; S_{t} ∈ nB_{A(X)} }. If K_{n} ∋ t_{m} → t then S_{t_{m}} → S_{t} in L(X) , and since B_{A(X)} is closed in L(X) it follows that t ∈ K_{n}; i.e. K_{n} is closed. Since K = ⋃ K_{n} some K_{n} has positive Lebesgue measure and thus contains a non-empty open set E. Now we deduce the ∥ · ∥_{A}-boundedness of (S_{t}) on some intervall [0, T] as follows. By Corollary 20.17 in [HR] we have [0, T] ⊂ E − E for some T > 0 . An integrated semigroup satisfies the identity

S_{t} = T_{h} S_{t−h} + S_{h} , t ≥ h ≥ 0 .

If we choose for every h ∈ [0, T] some t, s ∈ E such that h = t − s then S_{h} = S_{t} − T_{h} S_{s} and

∥ S_{h} ∥_{A} ≤ ∥ S_{t} ∥_{A} + ∥ T_{h} ∥ ∥ S_{s} ∥_{A} ≤ 2 sup_{t≤T} ∥ T_{t} ∥ C . □

Proof of Remark 2.2 . (i) The A-boundedness of (S̃_{t}) is already shown in (a)⇒(c) above. The claim for (S_{t}) follows inductively from the identity

S_{nT} = T_{(n−1)T} S_{T} + S_{(n−1)T} .

Indeed, we get ∥ S_{nT} ∥_{A} ≤ n∥ T_{nT} ∥ ∥ S_{T} ∥_{A}. (ii) The above proof of Theorem 2.1 - aside from the equivalence (a)⇔(b) - only uses that A is a left ideal. Since the T_{t}, R(λ, A) and S_{t} commute the proof is also valid for right ideals. □

3 Application to the Phillips calculus

In this section we give a refinement of Theorem 2.1 in terms of the Phillips functional calculus for a C_{0} semigroup (T_{t}) with generator A. Let L_{1}^{+}(a) denote the space of all measurable ℂ-valued functions g on ℝ_{≥0} satisfying

∥ g ∥_{1,a} := ∫_{0}^{∞} | g(s) | e^{as} ds < ∞ .

If a > ω(T_{t}) the Phillips functional calculus ( [HP] , § 15 ) is defined by

ĝ (−A) x := ∫_{0}^{∞} T_{s} x g(s) ds , g ∈ L_{1}^{+}(a) .

Example 3.1 (a) Let Re(λ) > a, α > 0 . Then g_{λ,α}(s) := \frac{s^{α−1}}{Γ(α)} e^{−λs} ∈ L_{1}^{+}(a) and

ĝ_{λ,α}(−A) = R(λ, A)^{α} . (8)

(b) Let t > 0 . Then χ_{t} := χ_{[0,t]} ∈ L_{1}^{+}(a) and

χ̂_{t} (−A) = S_{t} . (9)

(c) For all t>0 and δ ∈ (0, 1) let f_{t,δ} denote the inverse Laplace transform of e^{−t z^{δ}} on ℂ_{+} . Then f_{t,δ} ∈ L_{1}^{+}(a) for a ≤ 0 , and the fractional power −(−A)^{δ} generates the holomorphic semigroup T_{t,δ} x := ∫_{0}^{∞} T_{s}x f_{t,δ}(s) ds ( [Y] , § IX.11 ) , i.e.

f̂_{t,δ}(−A) = T_{t,δ} . (10)

After a glance at (8) and (9) the equivalent statements of Theorem 2.1 are of the form ’ ĝ (−A) ∈ A(X) ’ for certain g ∈ L_{1}^{+}(a) . E.g. for the ideal A = K of compact operators this can be generalized.

Theorem 3.2 If A is closed then the following statements are equivalent :

(a) R(λ, A) ∈ A(X) for one ( all ) λ ∈ ρ(A) .

(b) S_{t} ∈ A(X) on [ 0, T ] for some T > 0 .

(c) ĝ (−A) ∈ A(X) for all g ∈ L_{1}^{+}(a) .

(d) ĝ_{o}(−A) ∈ A(X) for some g_{o} ∈ L_{1}^{+}(a) satisfying

(11)  ĝ_{o}(z) ≠ 0 for all Re(z) ≥ −a  and  0 ∈ supp(g_{o}) .

If in addition a ≤ 0 then another equivalent statement is :

(e) T_{t,δ} ∈ A(X) for one ( all ) t > 0 and one ( all ) 0 < δ < 1 .

Notice that χ̂_{t}(z) = \frac{1−e^{−tz}}{z} ≠ 0 for Re(z) > 0 , i.e. if a < 0 and b = 0 then the choice g_{o} := χ_{t} in (d) reproduces the condition (d) of 2.1 . Plugging g_{o} := g_{λ,α} in statement (d) leads to

Corollary 3.3 The resolvent is compact if and only if it is power compact .

This can also be seen from the formula [De]

R(λ, A) = (m − 1) ∫_{0}^{∞} μ^{m−2} R(λ + μ, A)^{m} dμ , λ > a .

The corollary has some interesting consequences.

Remark 3.4 (a) Let X be a space C(K) or L_{1}(Ω, μ) (or more generally a space with the so-called Dunford-Pettis property [DU] ). Then the product of two weakly compact operators is compact, hence Corollary 3.3 implies that a weakly compact resolvent is already compact. In other words, if A is a closed operator such that R(λ, A) is weakly compact but not compact, then A is not the generator of a C_{0} semigroup.

(b) Let X be a space L_{p}(Ω, μ) 1 < p < ∞. Then we obtain in a similar way from [M] and [W] that if R(λ, A) is strictly singular (or an admissable Fredholm perturbation) then R(λ, A) is already compact.

¿From (10) we derive

Corollary 3.5 If A is a closed ideal and R(λ, A) ∈ A(X) then the subordinated semigroups (T_{t,δ}) and the δ-times integrated semigroups

S_{t,δ} = Γ(δ)^{−1} ∫_{0}^{t}(t − s)^{δ−1} T_{s} ds

belong to A for all t > 0 and all δ ∈ (0, 1) .

In view of these consequences it is not surprising that the closedness of A is essential.

Example 3.6 For the semigroup in 2.3 we see from (6) that

R(λ, A)^{2} ∈ S_{1}  and  R(λ, A) ∉ S_{1} .

For the proof of Theorem 3.2 we need

Lemma 3.7 (a) { χ_{t} | t > 0 } is total in L_{1}^{+}(a) .

(b) Let g_{o} ∈ L_{1}^{+}(a) . Then g_{o} ∗ L_{1}^{+}(a) is dense in L_{1}^{+}(a) if and only if (11) holds.

Proof. (a) is the density of the step functions in L_{1}^{+}(a) .

(b) We can assume a = 0 by applying this special case to e^{a(·)} g_{o} ∈ L_{1}^{+} := L_{1}^{+}(0) otherwise. According to Nyman’s lemma ( [D] , 6.1 ) (11) holds if and only if the right translates {τ_{t} g_{o} | t > 0} of g_{o} are total in L_{1}^{+} . This is equivalent to the density of g_{o} ∗ L_{1}^{+} in L_{1}^{+} , see e.g. [K] , 2.3 . □

Proof of Theorem 3.2 . In view of (8), (9), (10) all statements are implied by (c) . (b)⇒(c) Due to 3.7 (a) every g ∈ L_{1}^{+}(a) is approximable in L_{1}^{+}(a) by linear combinations g_{n} of suitable χ_{t}. But (b) implies χ̂_{t} (−A) = S_{t} ∈ A(X) for all t ≥ 0 . Hence ĝ(−A) ∈ A(X) follows from the closedness of A and the continuity of the functional calculus :

A(X) ∋ ĝ_{n}(−A) → ĝ(−A) in L(X) .

(d)⇒(c) By means of 3.7 (b) we find some h ∈ L_{1}^{+}(a) such that ∥ g −g_{o} ∗h ∥_{1,a} ≤ ε. Recalling that ĝ_{o}(−A) ĥ(−A) ∈ A(X) yields the assertion because

∥ ĝ (−A) − ĝ_{o}(−A) ĥ(−A) ∥ = ∥ ĝ (−A) − g_{o} ∗ h^{}(−A) ∥
≤ M ∥ g − g_{o} ∗ h ∥_{1,a} ≤ M ε .

Now (a)⇒(c) and (e)⇒(c) follow from ĝ_{λ,1}(z) = (λ+z)^{−1} ≠ 0 and f̂_{t,δ}(z) = e^{−tz^{δ}} ≠ 0 for Re(z) ≥ −a. □

Next we present an extension of Pazy’s result on compact semigroups ( [Pa] , § I.2.3 ) .

Theorem 3.8 Let A be closed and (T_{t}) be norm-continuous for t > 0 . Then we may add in Theorem 3.2 the following equivalent statements :

(f) T_{t} ∈ A(X) for all t > 0 .

(g) μ̂(−A) ∈ A(X) for all μ ∈ M_{1}^{+}(a) .

Here we denote by M_{1}^{+}(a) the space of all measures μ on ℝ_{+} satisfying

∥ μ ∥_{1,a} := ∫_{0}^{∞} e^{as} d|μ|(s) < ∞ .

Remark 3.9 If X is a Hilbert space the norm-continuity of (T_{t}) is equivalent to

∥ R(a + ir, A) ∥ → 0 for |r| → ∞ and some a > ω(T_{t}) .

The same is true for positive semigroups on L_{p}(Ω, μ) , see [GW] for references. So in these cases statement (f) can be characterized by conditions on the resolvent only.

Proof of Theorem 3.8. (a)⇒(f) For a fixed t > 0 and all λ > 0 we have

λR(λ, A) T_{t} − T_{t} = λ ∫_{0}^{∞} e^{−λs}(T_{t+s} − T_{t}) ds

and thus for every δ > 0 and a > ω(T_{t})

∥ λR(λ, A) T_{t} − T_{t} ∥ ≤ sup_{s∈[0,δ]} ∥ T_{t+s} − T_{t} ∥ + C \frac{λ}{λ − ω} e^{a(t+δ)} e^{−δλ}

The first term tends to zero for δ → 0 while for every fixed δ the second term tends to zero for λ → ∞. Since λR(λ, A) T_{t} ∈ A(X) and A is closed it follows that T_{t} ∈ A(X). (f)⇒(g) Since ℝ_{+} ∋ t ↦ T_{t} ∈ L(X) is measurable the integral

μ̂(−A) = ∫_{0}^{∞} T_{t} dμ(t)

exists in L(X) for every μ ∈ M_{1}^{+}(a) with a > ω(T_{t}) . Since A is closed we deduce μ̂(−A) ∈ A(X) . (g)⇒(a) is clear. □

As pointed out to us by the referee, our Theorem 3.2 can be extended to bounded integrated semigroups (S_{t})_{t≥0} [A] satisfying the Lipschitz condition

lim sup_{h↘0} h^{−1}∥ S_{t+h} − S_{t} ∥ ≤ M for all t ≥ 0 .

By the integrated version of Widder’s Theorem [A, Cor. 1.2], the operators

R(λ) := λ ∫_{0}^{∞} e^{−λt}S_{t} dt , λ > 0 ,

provide a pseudo-resolvent (R(λ))_{λ>0} which is tempered at infinity in the sense of [CK]. Due to [CK, Thm. 2], there exists a continuous Banach algebra homomorphism H : L_{1}(ℝ_{+}) → L(X) such that

H(e^{−λ·}) = R(λ) and H(χ_{[0,t]}) = S_{t} for all λ > 0 , t ≥ 0 .

In this setting, the proof of Theorem 3.2 (for a = 0) shows the following version for integrated semigroups. By A we still denote a normed operator ideal.

Theorem 3.10 If A is closed then the following statements are equivalent :

(a) R(λ) ∈ A(X) for one ( all ) λ > 0 .

(b) S_{t} ∈ A(X) for all t ≥ 0 .

(c) H(g) ∈ A(X) for all g ∈ Ł_{1}(ℝ_{+}) .

(d) H(g_{o}) ∈ A(X) for some g_{o} ∈ Ł_{1}(ℝ_{+}) satisfying

ĝ_{o}(z) ≠ 0 for all Re(z) ≥ 0  and  0 ∈ supp(g_{o}) .

4 Analytic semigroups

In this section (T_{t}) is always a bounded holomorphic semigroup of angle δ(A) with generator A, and A is a one-sided operator-ideal in the sense of (4) . ¿From Corollary 2.4 we already know that R(λ, A) ∈ A if and only if T_{t} ∈ A but this qualitative information can be improved by using precise growth estimates on the A-norms of R(λ, A) and T_{t}, using among other things estimates by Prüss on the inverse Laplace transform [Pr] . If B_{A(X)} is closed with respect to the strong operator topology of L(X) (as it is shown in Lemma 1.2 for maximal ideals, e.g. the Hilbert-Schmidt and the trace class) and I_{X} ∉ A(X) then, of course, ∥ T_{t} ∥_{A} → ∞ blows up for t → 0 . The essential information of the next theorem is then how this growth is related to the growth of ∥ R(λ, A) ∥_{A} for |λ| → ∞. In our growth conditions we use the symbol ⪯ to indicate that the left hand side is dominated by a positive multiple of the right hand side.

Theorem 4.1 Let β ∈ [0, 1] , α ≥ β and δ ∈ [0, δ(A) ).

(a) If β > 0 and ∥ T_{t} ∥_{A} ⪯ t^{β−α} on ℝ_{+} then

∥ R(λ, A)^{α} ∥_{A} ⪯ | λ |^{−β} on ℂ_{+} .

(b) If ∥ R(λ, A)^{α} ∥_{A} ≤ D_{δ} | λ |^{−β} on Σ_{δ} then

∥ T_{z} ∥_{A} ≤ C_{β} Γ(α + 2) R_{δ}(A)^{2} D_{δ} | z |^{β−α} on S_{δ} .

Here we denote ℂ_{+} := Σ_{0} = {z ∈ ℂ; Re(z) > 0 }. Simple examples of diagonal operators of the type (5) show that the assertion (a) is not true for β = 0 [B1] . We postpone the proof of Theorem 4.1 to the end of the section and discuss first some consequences. Combining (a) and (b) of Theorem 4.1 leads to the following characterization.

Corollary 4.2 (a) If β ∈ (0, 1] then the following equivalences hold :

∥ T_{t} ∥_{A} ⪯ t^{β−α} on ℝ_{+}  ⇔  ∥ R(λ, A)^{α} ∥_{A} ⪯ λ^{−β} on ℝ_{+}
⇕ ⇕
∀δ : ∥ T_{z} ∥_{A} ⪯ | z |^{β−α} on S_{δ}  ⇔  ∀δ : ∥ R(λ, A)^{α} ∥_{A} ⪯ | λ |^{−β} on Σ_{δ}

All implications except for the two ’⇒’ hold also for β = 0 .

(b) Let ε>0 . If 0 ∈ ρ(A) then the following implications hold :

(−A)^{−α} ∈ A(X) ⇒ ∥ T_{z} ∥_{A} ⪯ | z |^{−α} on S_{δ} ⇒ (−A)^{−(α+ε)} ∈ A(X)

Example 4.3 Let (T_{t}) be a bounded holomorphic semigroup on L_{p}(G) for a measure space G. By considering the right-sided operator ideal A defined via A(X, Y) := L(X, L_{q}(G)) and using the notation ∥ · ∥_{p,q} := ∥ · ∥_{L(L_{p},L_{q})} this section establishes the following connections :

∥ T_{t} ∥_{p,q} ⪯ t^{−α} ⇌ (−A)^{−α} ∈ L(L_{p}, L_{q}) ⇌ ∥ R(λ, A)^{α+β} ∥_{p,q} ⪯ | λ |^{−β}

(The symbol ’⇌’ is used in order to indicate that, if necessary, the exponents α and β have to be interpreted ’up to an ε ’.) Characterizations of the above type are studied in the setting of ultracontractivity of semigroups, i.e. of ∥ T_{t} ∥_{1,∞} −estimates (cf. e.g. [C] ; [VSC] , § 2 ) .

In the following applications (mostly to differential operators) we use information on the imbedding J_{1}(A) : D(A) → X , provided by Sobolev imbeddings or the integral kernel of the resolvent, to obtain directly estimates for the semigroup.

Example 4.4 Let Ω be a bounded domain in ℝ^{n} and fix ε > 0 . Assume that A is a differential operator in L_{p}(Ω) with D((−A)^{α}) ⊂ W_{p}^{1+ε}(Ω) . If p = n the Sobolev imbedding theorem implies that R(λ, A)^{α}(L_{p}(Ω)) ⊂ C_{b}(Ω) ⊂ L_{p}(Ω) and R(λ, A) is a Hille-Tamarkin operator (which provide a one-sided operator ideal A in L(L_{p}(Ω)) [Pi] ) with ∥ R(λ, A)^{α} ∥_{A} ≤ C. Corollary 4.2 implies

∥ T_{t} ∥_{A} ⪯ t^{−α} .

In particular, the T_{t} are integral operators with p-summable eigenvalues. Indeed, since operators in A are absolutely p-summing with p = n we get from [Pi] , Thm. 3.7.3 that T_{t}^{n} has summable eigenvalues λ_{k}(T_{t}^{n}) . Hence T_{t} = (T_{t/n})^{n} yields

∑ |λ_{k}(T)| ≤ C_{1} ∥ T_{t/n} ∥_{A}^{n} ≤ C_{2} (n/t)^{αn} = C_{3} t^{−α n}

For a Schrödinger operator A one can choose α > \frac{1}{2} + ε and we almost get the trace estimate O(t^{−n/2}) obtained by direct estimation of the integral kernel of the T_{t}. But our approach applies to more general operators A also in L_{p}-spaces and it shows how the exponent in the Sobolev imbedding theorem determines the exponent in the trace estimate. In [B2] this method is refined to give precise estimates for elliptic operators on Hilbert spaces, even in the much more complicated case of unbounded domains.

Example 4.5 In [B3] we present an alternative approach to Gaussian heat kernel estimates . Instead of applying the standard method using Nash-type inequalities and Moser iteration we combine the corresponding Sobolev inequality and the ellipticity of A to

D((−Ã_{ρ})^{1/2}) ↪ W^{o} _{2}^{m}(Ω) ↪ L_{q}(Ω) (12)

where the Ã_{ρ} are suitable translations of the generators A_{ρ} of the perturbations (T_{t}^{ρ}) corresponding to Davies’ perturbation method [Da] . (12) implies J_{1/2}(Ã_{ρ}) ∈ A for the operator ideal A defined in Ex. 4.3 for p = 2 . Hence one obtains ∥ T̃_{t}^{ρ} ∥_{2,q} ⪯ t^{−1/2} uniformly in all perturbations from our perturbation result 4.6 and then ∥ T̃_{t}^{ρ} ∥_{1,∞} ⪯ t^{−N/2m} from an extrapolation result due to Coulhon [C] . Ultracontractive estimates of this type are equivalent to the desired Gaussian estimates (cf. [AE] , 3.3 ) .

In Corollary 4.2 we omitted explicit constants in the characterization

∥ T_{z} ∥_{A} ≤ C_{1} | z |^{β−α} ⟺ ∥ R(λ, A)^{α} ∥_{A} ≤ C_{2} | λ |^{−β} . (13)

Here we analyse the implication ’⇐’ and state explicitly which parameters of the semigroup enter in the constant C_{1} . This allows uniform ∥ · ∥_{A}-estimates for such perturbations (T̂_{t}) of the given semigroup (T_{t}) whose perturbative effect on these parameters is uniformly bounded. This is of crucial importance for both announced applications [B2] , [B3] to partial differential operators A. Hence we will express the right hand side of (13) in terms of the imbedding operator J_{α}(A) , i.e. in terms of the Sobolev imbedding corresponding to A. Like in Theorem 2.1 we need a \underline{right-sided} ideal property of ∥ · ∥_{A} here.

Proposition 4.6 If J_{α}(A) ∈ A and 0∈ρ(A) then

∥ T_{t} ∥_{A} ≤ C_{α} R_{0}(A)^{α+5} ∥ J_{α}^{o}(A) ∥_{A} t^{−α} on ℝ_{+} (14)

for a constant C_{α} of the type C_{α} = C′ C^{α} Γ(α + 2) .

Now we come to the proofs for this section. Recall that (T_{t}) is always a bounded holomorphic semigroup. In the proof of Theorem 4.1 we will use the ideal property to obtain A-norm estimates for resolvent powers R(λ, A)^{α} from the corresponding estimates for the operator norm. These resolvent estimates are collected in the following two lemmas.

Lemma 4.7 (a) For all λ ∈\overline{Σ_{δ}}, η > 0 we have ∥ η (η+λ−A)^{−1} ∥ ≤ (1+tan δ) R_{δ}(A) . In particular λ − A is non-negative for all such λ.

(b) D( (λ−A)^{α}) is independent of λ ∈ \overline{Σ_{δ}} with equivalent graphnorms.

(c) Let 0 < α < 1. Then we have for all λ_{1}, λ_{2} ∈ \overline{Σ_{δ}} and x ∈ D((−A)^{α}) :

∥ (λ_{1} − A)^{α}x − (λ_{2} − A)^{α}x ∥ ≤ C_{δ} R_{δ}(A)^{2} | λ_{1} − λ_{2} |^{α} ∥ x ∥ (15)

If λ_{1} = 0 we can replace C_{δ} by E_{α}:= \sqrt{π} Γ(1 − \frac{α}{2} )^{−1} Γ(\frac{1+α}{2} )^{−1} .

Proof. (a) is trivial since η + λ ∈ Σ_{δ} and η/| η + λ | ≤ 1 + tan δ =: C_{δ}.

(b) D((λ−A)^{α}) = D((λ+ε−A)^{α}) holds for all λ∈ \overline{Σ_{δ}} , ε>0 due to [MSM] , Theorem 2.1 . Applying this to rotations e^{iγ}A of A yields the first statement. Now let ∥ · ∥_{λ} denote the graphnorm of (λ−A)^{α}. D := D((−A)^{α}) = D((λ−A)^{α}) is a Banach space with respect to ∥ · ∥_{0} and ∥ · ∥_{λ} because (−A)^{α} and (λ − A)^{α} are closed operators. Hence D is also a Banach space with respect to ∥ · ∥_{0} + ∥ · ∥_{λ} which is stronger than and thus equivalent to ∥ · ∥_{0} and ∥ · ∥_{λ} .

(c) The standard representation for fractional powers of non-negative operators B

B^{α} x = \frac{sin απ}{π} ∫_{0}^{∞} η^{α−1} (η + B)^{−1} Bx dη , x ∈ D(B) (16)

([MSM] , Remarks 2.4 , 2.5 ) admits the following estimate for all x ∈ D(A) :

∥ x ∥^{−1} \frac{π}{sin απ} ∥ (λ_{1} − A)^{α}x − (λ_{2} − A)^{α}x ∥

≤ ∫_{0}^{∞} η^{α} ∥ (η + λ_{1} − A)^{−1} − (η + λ_{2} − A)^{−1} ∥ dη
≤ 2 C_{δ} R_{δ}(A) ∫_{0}^{| λ_{1}−λ_{2} |} η^{α−1} dη + C_{δ}^{2} R_{δ}(A)^{2} | λ_{1} − λ_{2} | ∫_{| λ_{1}−λ_{2} |}^{∞} η^{α−2} dη
≤ (2 C_{δ} α^{−1} + C_{δ}^{2} (1 − α)^{−1} ) R_{δ}(A)^{2} | λ_{1} − λ_{2} |^{α}

Thus (15) is shown for x ∈ D(A) . But D(A) is a core for (−A)^{α} and the general statement follows from (b). If λ_{1} = 0 and λ_{2} = λ ∈ ℂ_{+} the estimate reads as follows :

∥ x ∥^{−1} \frac{π}{sin απ} ∥ (−A)^{α}x − (λ − A)^{α}x ∥ ≤ ∫_{0}^{∞} η^{α} ∥ λ R(η + λ, A) R(η, A) ∥ dη

≤ R_{0}(A)^{2} ∫_{0}^{∞} η^{α−1} | λ |/\sqrt{η^{2} + |λ|^{2}} dη
= R_{0}(A)^{2} | λ |^{α} ∫_{0}^{∞} s^{α−1}/\sqrt{1 + s^{2}} ds

The assertion follows from the formula

∫_{0}^{∞} \frac{s^{α−1}}{\sqrt{1 + s^{2}}} ds = \frac{π}{sin απ} \frac{\sqrt{π}}{Γ(1 − \frac{α}{2}) Γ(\frac{1+α}{2})} (17)

and is transferred to λ_{2} ∈ \overline{Σ_{δ}} by means of the rotation argument from above. □

Lemma 4.8 (a) Let λ ∈ Σ_{δ} and μ ∈ \overline{Σ_{δ}} . Then R(λ, A) and (μ − A)R(λ, A) are non-negative, R(λ, A)^{α} ∈ L(X) has range D((−A)^{α}) , and we have

(μ − A)^{α} R(λ, A)^{α} = ((μ − A) R(λ, A))^{α} ∈ L(X) . (18)

(b) For all | γ | < δ(A) and all λ ∈ ℂ_{+} we have

R(e^{iγ} λ , A)^{α} = Γ(α)^{−1} e^{−iγα} ∫_{0}^{∞} e^{−λ t} t^{α−1} T_{e^{−iγ} t} dt . (19)

(c) For all λ ∈ Σ_{δ} we have

(20)  ∥ (λ R(λ, A))^{α} ∥ ≤ E_{α} R_{δ}(A) , 0 < α < 1
(21)  ∥ (−A)^{α}R(λ, A)^{α} ∥ ≤ C′ C^{α} R_{δ}(A)^{α+3}
(22)  ∥ (|λ| − A)^{α} R(λ, A)^{α} ∥ ≤ C_{δ} C^{α} R_{δ}(A)^{α+3}

Proof. (a) Since η(η+R(λ, A))^{−1} = I−η^{−1}(η^{−1}+λ−A)^{−1} and λ−A is non-negative this is also true for R(λ, A) . So R(λ, A)^{α} ∈ L(X) follows from [MSM] , Rem. 2.3 . Moreover (λ − A)^{α} = [R(λ, A)^{α}]^{−1} per definitionem ( [MSM] , 2.1 ) . For showing the non-negativity of (μ − A)R(λ, A) and the formula (18) one follows (the proof of) Thm 2.1 in [MSM] , and uses the above rotation argument.

(b) By means of the rotation argument we can assume γ = 0 . For α ∈ ℕ (19) follows inductively from (1) , for α ∈ (0, 1) it is obtained using the integral representation (16) . These two cases combine to the general statement by applying the additivity of the fractional powers [MSM] and Fubini’s theorem.

(c) Again we can assume δ = 0, i.e. λ ∈ ℂ_{+} . Then (20) follows from (17) and

∥ x ∥^{−1} \frac{π}{sin απ} ∥ R(λ, A)^{α}x ∥ = ∥ x ∥^{−1} ∥ ∫_{0}^{∞} η^{α−2} R(η^{−1} + λ, A)x dη ∥

≤ R_{0}(A) ∫_{0}^{∞} η^{α−2}/\sqrt{η^{−2} + |λ|^{2}} dη
= R_{0}(A) | λ |^{−α} ∫_{0}^{∞} s^{−α}/\sqrt{1 + s^{2}} ds

where we used (16) again. Now we verify (21) . For β := α − ⌊α⌋ ∈ [0, 1) we have

∥ (−A)^{α}R(λ, A)^{α} ∥ ≤ ( 1 + R_{δ}(A) )^{⌊α⌋} ∥ (−A)^{β}R(λ, A)^{β} ∥ .

Thus we can assume α ∈ (0, 1) . Using 4.7(c) for λ_{1} := 0 and λ_{2} := λ as well as (20) we deduce for all x ∈ X:

∥ (−A)^{α}R(λ, A)^{α} x − x ∥ ≤ E_{α} R_{δ}(A)^{2} | λ |^{α} ∥ R(λ, A)^{α} x ∥ ≤ E_{α}^{2} R_{δ}(A)^{3} ∥ x ∥

The proof of (22) is completely analogous. □

Remark 4.9 If 0∈ρ(A) one obtains analogously that 4.8(a),(b) hold also for λ = 0 . In this case we define the negative powers of −A by (−A)^{−α}:= R(0, A)^{α}.

In order to interpret the representations of R(λ, A)^{α} in (2) , (19) as an A(X)-valued Lebesgue integral (provided that T_{t} ∈ A(X) on ℝ_{+} with a suitable ∥ · ∥_{A}-behaviour) we have to check that the ∥ · ∥-smoothness of the semigroup implies at least its ∥ · ∥_{A}-measurability. Similarly, if the resolvent belongs to A, it will be necessary to transfer its ∥ · ∥-holomorphy into ∥ · ∥_{A}-holomorphy.

Lemma 4.10 (a) If T_{t} ∈ A(X) for all t > 0 then T_{(·)} : ℝ_{+} → A(X) is holomorphic.

(b) Let R(λ_{0}, A)^{α} ∈ A(X) for some λ_{0} ∈ Σ_{δ} . Then R(λ, A)^{α} ∈ A(X) for all λ ∈ Σ_{δ} and F_{α} : Σ_{δ} → A(X) , λ ↦ R(λ, A)^{α} is holomorphic with derivative

F′_{α}(λ) = −α R(λ, A)^{α+1} .

Proof. (a) follows by letting h → 0 in

∥ h^{−1} (T_{t+h} − T_{t}) − A T_{t} ∥_{A} ≤ ∥ T_{\frac{t}{2}} ∥_{A} ∥ h^{−1} (T_{\frac{t}{2} +h} − T_{\frac{t}{2}} ) − A T_{\frac{t}{2}} ∥ .

(b) The first part is trivial in view of the ’resolvent power identity’

R(λ, A)^{α} − R(μ, A)^{α} = ((μ − A)^{α} − (λ − A)^{α}) R(λ, A)^{α} R(μ, A)^{α} .

Now denote by ∥ · ∥_{λ} the graphnorm of (λ − A)^{α}. F_{α} is holomorphic into L(X) via differentiating under the integral in (19) . Let us assume for the moment that F_{α} is continuous into L := L(X, D((−A)^{α}) ) . Then F_{α} is even holomorphic into L since L ↪ L(X) (cf. [Wr] ). Due to R(λ, A)^{α+1} ∈ A(X) and ∥ · ∥_{λ} ⪯ ∥ · ∥_{0} we can estimate as follows :

∥ \frac{R(λ + h, A)^{α} − R(λ, A)^{α}}{h} + α R(λ, A)^{α+1} ∥_{A}

= ∥ (λ − A)^{α} ( \frac{R(λ + h, A)^{α} − R(λ, A)^{α}}{h} + α R(λ, A)^{α+1}) R(λ, A)^{α} ∥_{A}
⪯ ∥ \frac{R(λ + h, A)^{α} − R(λ, A)^{α}}{h} + α R(λ, A)^{α+1} ∥_{L} ∥ R(λ, A)^{α} ∥_{A}
→ 0 for h → 0

Therefore we will show inductively that for all n ∈ ℕ and all α ∈ (0, n) the map F_{α} : Σ_{δ} → L is continuous. For n = 1 this follows from 4.7(b),(c) :

∥ R(λ + h, A)^{α}x − R(λ, A)^{α}x ∥_{0} ⪯ ∥ R(λ + h, A)^{α}x − R(λ, A)^{α}x ∥_{λ}

⪯ ∥ ( (λ − A)^{α} − (λ + h − A)^{α} ) R(λ + h, A)^{α}x ∥
⪯ | h |^{α} ∥ R(λ + h, A)^{α}x ∥
≤ | h |^{α} C ∥ x ∥

for | h | small. For n > 1 the commutativity of the (λ − A)^{α} guarantees ∥ R(λ + h, A)^{α}x − R(λ, A)^{α}x ∥_{0}

⪯ ∥ ( (λ − A)^{α} − (λ + h − A)^{α} ) R(λ + h, A)^{α}x ∥
= ∥ ( (λ−A)^{α/n} − (λ+h−A)^{α/n} ) ∑_{j=0}^{n−1} (λ − A)^{α(n−1−j)/n} R(λ + h, A)^{α(n−j)/n} x ∥
⪯ | h |^{α/n} ∑_{j=0}^{n−1} ∥ (λ − A)^{α(n−1−j)/n} R(λ + h, A)^{α(n−j)/n} x ∥
≤ | h |^{α/n} C ∥ x ∥

for | h | small where we used the induction hypothesis in the last step. □ Now the preparations for the proofs of Section 4 are finished.

Proof of Theorem 4.1 . (a) By hypothesis we have ∥ T_{t} ∥_{A} ⪯ t^{β−α} on ℝ_{+} which extends to ∥T_{z}∥_{A}-estimates on sectors. Indeed, let δ′, δ′′ ∈ (δ, δ(A)) such that δ′ < δ′′ and r := sin(δ′′−δ′) . Since z ∈ S_{δ′} implies z − r|z| ∈ S_{δ′′} we have

∥ T_{z} ∥_{A} ≤ ∥ T_{r|z|} ∥_{A} ∥ T_{z−r|z|} ∥ ⪯ | z |^{β−α} on S_{δ′} . (23)

Now we continue with a slightly modified rotation argument. For | γ | ≤ δ′ the semigroup (T_{e^{iγ} t}) has the generator A_{γ} := e^{iγ} A and satisfies, due to Lemma 4.10(a), the representation (19) and the line (23) , the following estimate on ℂ_{+} :

∥ R(λ, A_{γ})^{α} ∥_{A} ≤ Γ(α)^{−1} D_{0} ∫_{0}^{∞} e^{−Re(λ) t} t^{β−1} dt = D Re(λ)^{−β}

with D independent of γ and λ. Thus we only have to show the estimate on S := Σ_{δ} \ S_{\frac{π}{2} −ε} , where ε := δ′ − δ. Now fix γ := −δ′. For all λ ∈ S with Im(λ) > 0 we have e^{iγ}λ ∈ S_{\frac{π}{2} −ε} and therefore

∥ R(λ, A)^{α} ∥_{A} = ∥ R(e^{iγ} λ, A_{γ})^{α} ∥_{A} ≤ D Re(e^{−iγ} λ)^{−β} ⪯ | λ |^{−β} .

On S ∩ {z | Im(z) < 0} we proceed analogously using γ := δ′ .

(b) Suppose first that δ = 0 . By 4.10(b) the function F : ℂ_{+} → A(X) , λ ↦ R(λ, A)^{α} is holomorphic and satisfies the following estimate on ℂ_{+} :

∥ F (λ) ∥_{A} , ∥ λ F′(λ) ∥_{A} ≤ max ( 1 , α R_{0}(A) ) D_{0} | λ |^{−β} . (24)

This can be seen from

∥ λ F′(λ) ∥_{A} ≤ α ∥ λ R(λ, A) ∥ ∥ R(λ, A)^{α} ∥_{A} ≤ α R_{0}(A) D_{0} | λ |^{−β} .

So if β > 0 we are in position to apply Theorems 1 and 2 of [Pr] and obtain a continuous function u : ℝ_{+} → A(X) having the properties

F (λ) = ∫_{0}^{∞} e^{−λ t} u(t) dt on ℂ_{+} (25)

∥ u(t) ∥_{A} ≤ C_{β} max ( 1 , α R_{0}(A) ) D_{0} t^{β−1} on ℝ_{+} .

In view of (19) the identity (25) holds for the function

u_{o}(t) = Γ(α)^{−1} t^{α−1} T_{t}

so that the uniqueness of the Laplace transform yields the assertion. Now we turn to the case β = 0 and set α̃ := α+1 , β̃ := 1 , F̃(λ) := R(λ, A)α̃ and D̃_{0} := R_{0}(A) D_{0} . Then (24) holds for these ˜ -quantities and our earlier conclusion shows

∥ T_{t} ∥_{A} ≤ C_{1} Γ(α̃) α̃ R_{0}(A) D̃_{0} t^{β̃−α̃} = C_{1} Γ(α + 2) R_{0}(A)^{2} D_{0} t^{β−α} .

For arbitrary δ ∈ [0, δ(A) ) we use the rotation argument again. Indeed, for | γ | < δ we get from the first part

∥ T_{e^{iγ}t} ∥_{A} ≤ C_{β} Γ(α + 2) R_{0}(e^{iγ}A)^{2} D_{δ} t^{β−α}
≤ C_{β} Γ(α + 2) R_{δ}(A)^{2} D_{δ} | e^{iγ}t |^{β−α} . □

Proof of Corollary 4.2 . (a) The ∥·∥_{A}-estimates for the semigroup on the halfline imply the analogous estimates on sectors as we showed in the proof of Theorem 4.1(a) . For the resolvent this can be seen as follows. If ∥ R(λ, A)^{α} ∥_{A} ≤ D | λ |^{−β} on ℝ_{+} then we obtain for all λ ∈ Σ_{δ} from (18) and (22) :

∥ R(λ, A)^{α} ∥_{A} = ∥ ((| λ | − A)^{α} R(λ, A)^{α} ) R(| λ |, A)^{α} ∥_{A}
≤ ∥ (| λ | − A)^{α} R(λ, A)^{α} ∥ ∥ R(| λ |, A)^{α} ∥_{A}
≤ C_{δ} C^{α} R_{δ}(A)^{α+3} D | λ |^{−β}

All other implications are trivial or proved in Theorem 4.1 .

(b) According to (21) we can estimate on Σ_{δ} as follows if (−A)^{−α} ∈ A(X) :

∥ R(λ, A)^{α} ∥_{A} ≤ ∥ (−A)^{α} R(λ, A)^{α} ∥ ∥ (−A)^{−α} ∥_{A} ⪯ 1

Hence 4.1 yields the first implication. The second follows directly from (19) for γ = 0 and λ = 0 (cf. 4.9 !) having in mind that (T_{t}) has negative type. □

Proof of Proposition. 4.6 . Due to (18) and (21) we have on ℂ_{+} :

(26)  ∥ R(λ, A)^{α} ∥_{A} ≤ ∥ (−A)^{α} R(λ, A)^{α} ∥ ∥ J_{α}^{o}(A) ∥_{A}
≤ C′ C^{α} R_{0}(A)^{α+3} ∥ J_{α}^{o}(A) ∥_{A} ,

and we can apply Theorem 4.1(b) for β = 0 . □

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Mathematisches Institut I, Universität Karlsruhe Englerstr. 2 , D-76128 Karlsruhe, Germany e-mail: soenke.blunck@math.uni-karlsruhe.de

lutz.weis@math.uni-karlsruhe.de