Kenneth Kuttler

54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1733

=sin(πβ )

∫∞

0λ−β (λ I−B)−1 (A−B)(λ I−A)−1 dλ . (54.3.31)

Therefore, letting x ∈ D(A) and letting C denote a generic constant which can be changedfrom line to line and using 54.3.29 and 54.3.30,∣∣∣∣∣∣x− (−B)β (−A)−β x

∣∣∣∣∣∣≤ C

∫∞

λβ

∣∣∣∣∣∣(−B)β (λ I−B)−1 (A−B)(λ I−A)−1 x∣∣∣∣∣∣dλ

The reason (−B)β goes inside the integral is that it is a closed operator. Then the above

≤ C∫

∞

λβ (λ +δ )1−β

∣∣∣∣∣∣(A−B)(−A)−α (−A)α (λ I−A)−1 x∣∣∣∣∣∣dλ

≤ C∫

∞

λβ (λ +δ )1−β

∣∣∣∣∣∣(−A)α (λ I−A)−1 x∣∣∣∣∣∣dλ

≤ C∫

∞

λβ (λ +δ )1−β

(λ +δ )1−αdλ ||x||=C ||x|| .

It follows (−B)β (−A)−β is bounded on D(A).Next reverse A and B in 54.3.31. This yields

(−A)−β − (−B)−β =sin(πβ )

∫∞

0λ−β (λ I−A)−1 (B−A)(λ I−B)−1 dλ .

Letting x ∈ D(A) , ∣∣∣∣∣∣x− (−A)β (−B)−β x∣∣∣∣∣∣

≤ C∫

∞

0λ−β

∣∣∣∣∣∣(−A)β (λ I−A)−1 (B−A)(λ I−B)−1 x∣∣∣∣∣∣dλ

≤ C∫

∞

λβ (λ +δ )1−β

∣∣∣∣∣∣(B−A)(−B)−α (−B)α (λ I−B)−1 x∣∣∣∣∣∣dλ (54.3.32)

≤ C∫

∞

λβ (λ +δ )1−β (λ +δ )1−α

dλ ||x||=C ||x|| (54.3.33)

This shows (−A)β (−B)−β is bounded on D(A) = D(B) . Note the assertion these arebounded refers to the norm on H.

It remains to verify D((−A)β

)= D

((−B)β

). Since D(A) is dense in H there exists

a unique L(A,B) ∈L (H,H) such that L(A,B) = (−A)β (−B)−β on D(A). Let L(B,A) bedefined similarly as a continuous linear map which equals (−B)β (−A)−β on D(A) . Then

(−A)−β L(A,B) = (−B)−β

(−B)−β L(B,A) = (−A)−β

54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1733= NP) 1-8 (ara) (A—B) (ATA) Nd. (4.3.31)1 0Therefore, letting x € D(A) and letting C denote a generic constant which can be changedfrom line to line and using 54.3.29 and 54.3.30,I cotcart< cf Ve P (ar—B) '(A—B) (AA) ‘aaaThe reason (—B)P goes inside the integral is that it is a closed operator. Then the aboveB)(—A)~% (—A)® (AI—A)~ 'x|[aaCl aracara llCl aye % (AT—A)~ 'x||da0 1 1c| dA ||x|| =C| |x|].0 ab (A+6)!-P (Aa+6)!-% | | || ||It follows (—B)P (—A)-P is bounded on D(A).Next reverse A and B in 54.3.31. This yields(—A)-6 —(-B) FP = a (AI—A)~'(B—A) (AI—B)~! da.nu 0Letting x € D(A),fecareo|< cf one B (ar—A)~! (B—A) (AI—B)~ 'x||aaCl ayppralle-aarec B)* (AI—B)~ | x|| dA (543.32)°° 1Ci AP (A +6)! P(A +8)'%dh ||x|| =C||-| (54.3.33)This shows (—A)? (—B)~* is bounded on D(A) = D(B). Note the assertion these arebounded refers to the norm on H.It remains to verify D ((-4)*) =D ((-8)*) . Since D(A) is dense in H there existsa unique L(A,B) € @ (H,#H) such that L(A, B) = (—ayP (—B)? on D(A). Let L(B,A) bedefined similarly as a continuous linear map which equals (—B)P (—a)F on D(A). Then(A) PL(A,B) = (—B)(-B) PL(B,A) = (-A)”