17.4. SECTORIAL OPERATORS AND ANALYTIC SEMIGROUPS 431

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) , (17.11)

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

(17.12)

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 (17.13)

Also, if x ∈ D(A) ,

(λ −A)−1 Ax =−x+λ (λ I−A)−1 x (17.14)

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. 17.14 is similar. Thus from 17.13,∥∥∥A(λ I−A)−1∥∥∥≤ 1+ |λ |

∥∥∥(λ I−A)−1∥∥∥≤ 1+ |λ | M

|λ −a|≤C (17.15)

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

Proposition 17.4.4 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 (17.16)

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∥ (17.17)

for some K, where εC < 1. Then A+B is also sectorial.