592 CHAPTER 19. HILBERT SPACES

Definition 19.14.16 Let A be a densely defined closed operator on H a real Banach space.Then A∗ is defined as follows.

D(A∗)≡{

y∗ ∈ H ′ : |y∗ (Ax)| ≤C ||x|| for all x ∈ D(A)}

Then since D(A) is dense, there exists a unique element of H ′ denoted by A∗y such that

A∗ (y∗)(x) = y∗ (Ax)

for all x ∈ D(A) .

Lemma 19.14.17 Let A be closed and densely defined on D(A)⊆H, a Banach space. ThenA∗ is also closed and densely defined. Also (A∗)∗ = A. In addition to this, if (λ I−A)−1 ∈L (H,H) , then (λ I−A∗)−1 ∈L (H ′,H ′) and((

(λ I−A)−1)n)∗

=((λ I−A∗)−1

)n

Proof: Denote by [x,y] an ordered pair in H×H. Define τ : H×H→ H×H by

τ [x,y]≡ [−y,x]

A similar notation will apply to H ′×H ′. Then the definition of adjoint implies that forG (B) equal to the graph of B,

G (A∗) = (τG (A))⊥ (19.14.99)

For S⊆ H×H, define S⊥ by{[a∗,b∗] ∈ H ′×H ′ : a∗ (x)+b∗ (y) = 0 for all [x,y] ∈ S

}If S⊆ H ′×H ′ a similar definition holds.

{[x,y] ∈ H×H : a∗ (x)+b∗ (y) = 0 for all [a∗,b∗] ∈ S}

Here is why 19.14.99 is so. For [x∗,A∗x∗] ∈ G (A∗) it follows that for all y ∈ D(A)

x∗ (Ay) = A∗x∗ (y)

and so for all [y,Ay] ∈ G (A) ,

−x∗ (Ay)+A∗x∗ (y) = 0

which is what it means to say [x∗,A∗x∗] ∈ (τG (A))⊥ . This shows

G (A∗)⊆ (τG (A))⊥

To obtain the other inclusion, let [a∗,b∗] ∈ (τG (A))⊥ . This means that for all [x,Ax] ∈G (A) ,

−a∗ (Ax)+b∗ (x) = 0