54.3. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 1703

This sector is as shown below.

Saφ

aφ

A closed, densely defined linear operator A is called sectorial if for some sector asdescribed above, it follows that for all λ ∈ Saφ ,

(λ I−A)−1 ∈L (H,H) , (54.3.3)

and for some M ∥∥∥(λ I−A)−1∥∥∥≤ M|λ −a|

(54.3.4)

The following perturbation theorem is very useful for sectorial operators. I won’t use ithere, but in applications of this theory, it is useful. First note that for λ ∈ Saφ ,

A(λ I−A)−1 =−I +λ (λ I−A)−1 (54.3.5)

Also, if x ∈ D(A) ,(λ −A)−1 Ax =−x+λ (λ I−A)−1 x (54.3.6)

This follows from algebra and noting that λ I−A maps D(A) onto H because (λ I−A)−1 ∈L (H,H). Thus the above is true if and only if A =

(−I +λ (λ I−A)−1

)(λ I−A) which

is obviously true. 54.3.6 is similar. Thus from 54.3.5,∥∥∥A(λ I−A)−1∥∥∥≤ 1+ |λ |

∥∥∥(λ I−A)−1∥∥∥≤ 1+ |λ | M

|λ −a|≤C (54.3.7)

for some constant C whenever |λ | is large enough and in Saφ .

Proposition 54.3.2 Suppose A is a sectorial operator as defined above so it is a denselydefined closed operator on D(A)⊆ H which satisfies∥∥∥A(λ I−A)−1

∥∥∥≤C (54.3.8)

whenever |λ | ,λ ∈ Saφ , is sufficiently large and suppose B is a densely defined closed oper-ator such that D(B)⊇ D(A) and for all x ∈ D(A) ,

∥Bx∥ ≤ ε ∥Ax∥+K ∥x∥ (54.3.9)

where εC < 1. Then A+B is also sectorial.