1714 CHAPTER 54. FUNCTIONAL ANALYSIS APPLICATIONS

Corollary 54.3.11 If for some a ∈ R, the numerical values of −aI + A are in the set{λ : |λ | ≥ π−φ} where 0 < φ < π/2, and a ∈ r (A) then A is sectorial.

Proof: By assumption, 0∈ r (−aI +A) and also from Proposition 54.3.10, for µ ∈ S0,φ ′

where π/2 > φ′ > φ ,

((−aI +A)−µI)−1 ∈L (H,H) ,∥∥∥((−aI +A)−µI)−1

∥∥∥≤ M|µ|

Therefore, for µ ∈ S0,φ ′ ,µ +a ∈ r (A) . Therefore, if λ ∈ Sa,φ ′ ,λ −a ∈ S0,φ ′∥∥∥(A−λ I)−1∥∥∥= ∥∥∥(A−aI− (λ −a) I)−1

∥∥∥≤ M|λ −a|

54.3.2 An Interesting ExampleIn this section related to this example, for V a Banach space, V ′ will denote the space ofcontinuous conjugate linear functions defined on V . Usually the symbol has meant thespace of continuous linear functions but here they will be conjugate linear. That is f ∈ V ′

meansf (ax+by) = a f (x)+b f (y)

and f is continuous.Let Ω be a bounded open set in Rn and define

V0 ≡{

u ∈C∞(Ω)

: u = 0 on Γ}

where Γ is some measurable subset of the boundary of Ω and C∞(Ω)

denotes the restric-tions of functions in C∞

c (Rn) to Ω. By Corollary 15.5.11 V0 is dense in L2 (Ω) . Now definethe following for u,v ∈V0.

A0u(v)≡−a∫

Ω

uvdx−∫

Ω

a(x)∇u ·∇vdx

where a > 0 and a(x)≥ 0 is a C1(Ω)

function. Also define the following inner product onV0.

(u,v)1 ≡∫

Ω

(auv+a(x)∇u ·∇v

)dx

Let ||·||1 denote the corresponding norm.Of course V0 is not a Banach space because it fails to be complete. u∈V will mean that

u ∈ L2 (Ω) and there exists a sequence {un} ⊆V0 such that

limm,n→∞

||un−um||1 = 0

andlimn→∞|un−u|L2(Ω) = 0.