42.1. BASIC THEORY 1369

Lemma 42.1.3 Aλ x ∈ AJλ x and |Aλ x| ≤ |y| for all y ∈ Ax whenever x ∈ D(A) . Also Aλ ismonotone.

Proof: Consider the first claim. From the definition,

Aλ x≡ 1λ

x− 1λ

Jλ x

Is1λ

x− 1λ

Jλ x ∈ AJλ x?

Isx− Jλ x ∈ λAJλ x?

Isx ∈ Jλ x+λAJλ x?

Isx ∈ (I +λA)Jλ x?

Certainly so. This is how Jλ is defined.Now consider the second claim. Let y ∈ Ax for some x ∈ D(A) . Then by monotonicity

and what was just shown

0≤ (Aλ x− y,Jλ x− x) =−λ (Aλ x− y,Aλ x)

and so|Aλ x|2 ≤ (y,Aλ x)≤ |y| |Aλ x|

Finally, to show Aλ is monotone,

(Aλ x−Aλ y,x− y) =(1λ

x− 1λ

Jλ x−(

1λ

y− 1λ

Jλ y),x− y

)

=1λ|x− y|2− 1

λ(Jλ x− Jλ y,x− y)

≥ 1λ|x− y|2− 1

λ|x− y| |Jλ x− Jλ y|

≥ 1λ|x− y|2− 1

λ|x− y|2 = 0

and this proves the lemma.

Proposition 42.1.4 Suppose D(A) is dense in H. Then for all x ∈ H,

|Jλ x− x| → 0