19.13. FRACTIONAL POWERS OF OPERATORS 573

Also, if α +β ≤ 1,

Aα Aβ = AA−(1−α)AA−(1−β ) = A2A−(1−α)A−(1−β )

Aα+β ≡ AA−(1−(α+β )) = A2A−1A−(1−(α+β ))

= A2A−β A−(1−β )A−(1−(α+β ))

= A2A−(1−α)A−(1−β ) = Aα Aβ

This shows the following.

Lemma 19.13.7 If α,β ∈ (0,1) , α +β ≤ 1, then Aα Aβ = Aα+β . Also Aα commutes withevery operator in L (X ,X) which commutes with A.

Proof: The last assertion follows right away from the fact noted above that A−(1−α)

commutes with all operators which commute with A and that so does A. Thus if C is sucha commuting operator,

CAα =CAA−(1−α) = ACA−(1−α) = AA−(1−α)C = AαC.

The next task is to remove the assumption that (Ax,x) ≥ ε |x|2 and replace it with(Ax,x)≥ 0.

Observation 19.13.8 First note that if Φ(t)= e−(εI+A)t , and if x0 is given, then if Φ(t)x0 =y(t) ,

y′ (t)+(εI +A)y(t) = 0, y(0) = x0

Then taking inner product of both sides with y(t) and integrating,

|y(t)|2

2− |x0|2

2≤ 0

and so |Φ(t)x0| = |y(t)| ≤ |x0| and so ∥Φ(t)∥ ≤ 1. This will be used in the followinglemma.

Lemma 19.13.9 Let (Ax,x)≥ 0. Then for α ∈ (0,1) ,

limε→0+

(εI +A)(εI +A)−α

exists in L (X ,X).

Proof: Let δ > ε and both ε and δ are small.((εI +A)(εI +A)−α − (δ I +A)(δ I +A)−α

)Γ(α) =∫

∞

0

((εI +A)e−(εI+A)t − (δ I +A)e−(δ I+A)t

)tα−1dt =