54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1725

and so on D((−A)α

),

S (s) = (−A)α S (s)(−A)−α , S (s)(−A)α = (−A)α S (s)

Now for x ∈ D((−A)α

),

||(S (t)− I)x|| =

∣∣∣∣∣∣∣∣−∫ t

0(−A)S (s)xds

∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣−∫ t

0(−A)1−α (−A)α S (s)xds

∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣−∫ t

0(−A)1−α S (s)(−A)α xds

∣∣∣∣∣∣∣∣≤

∫ t

0

∣∣∣∣∣∣(−A)1−α S (s)∣∣∣∣∣∣ds

∣∣∣∣(−A)α x∣∣∣∣

≤∫ t

0

C1−α

δ

1s1−α

e−δ sds∣∣∣∣(−A)α x

∣∣∣∣≤ C1−α

δ

1α

tα∣∣∣∣(−A)α x

∣∣∣∣and this shows 54.3.23.

Next consider 54.3.24. Let x ∈ H and β ∈ (0,1) . Then∣∣∣∣∣∣(−A)−β x∣∣∣∣∣∣= 1

Γ(β )

∣∣∣∣∣∣∣∣∫ ∞

0tβ−1S (t)xdt

∣∣∣∣∣∣∣∣=

1Γ(β )

∣∣∣∣∣∣∣∣∫ η

0tβ−1S (t)xdt +

∫∞

η

tβ−1S (t)xdt∣∣∣∣∣∣∣∣

≤ 1Γ(β )

∫η

0tβ−1 ||S (t)x||dt +

1Γ(β )

∣∣∣∣∣∣∣∣∫ ∞

η

tβ−1S (t)xdt∣∣∣∣∣∣∣∣

≤ CΓ(β )

ηβ

β||x||+ 1

Γ(β )

∣∣∣∣∣∣∣∣∫ ∞

η

tβ−1S (t)xdt∣∣∣∣∣∣∣∣

≤ CΓ(β )

ηβ

β||x||+

1Γ(β )

∣∣∣∣∣∣∣∣ηβ−1S (η)A−1x+(1−β )∫

∞

η

tβ−2S (t)A−1xdt∣∣∣∣∣∣∣∣

≤ 1Γ(β )

(Cηβ

β||x||+η

β−1 ∣∣∣∣A−1x∣∣∣∣+(1−β )

∣∣∣∣A−1x∣∣∣∣∫ ∞

η

tβ−2dt

)

=1

Γ(β )

(Cηβ

β||x||+2η

β−1 ∣∣∣∣A−1x∣∣∣∣) .