54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1727
Now let 1−β = α and obtain∣∣∣∣(−A)α y∣∣∣∣≤ ε ||(−A)y||+Cε
−α/(1−α) ||y||
This proves 54.3.24.Finally choose ε to minimize the right side of the above expression. Thus let
ε =
(α ||y||C
||(−A)y||(1−α)
)1−α
Then the above expression becomes∣∣∣∣(−A)α y∣∣∣∣ ≤ ||(−A)y||
(α ||y||C
||(−A)y||(1−α)
)1−α
+C
((α ||y||C
||(−A)y||(1−α)
)1−α)−α/(1−α)
||y||
= ||(−A)y||α ||y||1−α
(αC
(1−α)
)1−α
+ ||(−A)y||α ||y||1−α
(αC
(1−α)
)−α
=
((αC
(1−α)
)1−α
+
(αC
(1−α)
)−α)||(−A)y||α ||y||1−α
≤ C′ ||(−A)y||α ||y||1−α
where C′ does not depend on α ∈ (0,1) . To see such a constant exists, note
limα→1
(αC
(1−α)
)1−α
= 1
and
limα→1
(αC
(1−α)
)−α
= 0
while
limα→0
(αC
(1−α)
)1−α
= 0, limα→0
(αC
(1−α)
)−α
= 1
Of course C′ depends on C but as shown above, this did not depend on α ∈ (0,1) . Thisproves 54.3.25.
The following corollary follows from the proof of the above theorem.
Corollary 54.3.20 Let α ∈ (0,1) . Then for all ε > 0, there exists a constant C (α,ε) suchthat ∣∣∣∣(−A)−α x
∣∣∣∣≤ ε ||x||+C (ε,α)∣∣∣∣∣∣(−A)−1 x
∣∣∣∣∣∣Also if A−1 is compact, then so is (−A)−α for all α ∈ (0,1).