Chapter 42

Maximal Monotone Operators, HilbertSpace42.1 Basic Theory

Here is provided a short introduction to some of the most important properties of maximalmonotone operators in Hilbert space. The following definition describes them. It is morespecialized than the earlier material on maximal monotone operators from a Banach spaceto its dual and therefore, better results can be obtained. More on this can be read in [24]and [116].

Definition 42.1.1 Let H be a real Hilbert space and let A : D(A)→P (H) have the fol-lowing properties.

1. For each y ∈ H there exists x ∈ D(A) such that y ∈ x+Ax.

2. A is monotone. That is, if z ∈ Ax and w ∈ Ay then

(z−w,x− y)≥ 0

Such an operator is called a maximal monotone operator.

It turns out that whenever A is maximal monotone, so is λA for all λ > 0.

Lemma 42.1.2 Suppose A is maximal monotone. Then so is λA. Also Jλ ≡ (I +λA)−1

makes sense for each λ > 0 and is Lipschitz continuous.

Proof: To begin with consider (I +A)−1. Suppose

x1,x2 ∈ (I +A)−1 (y)

Then y ∈ (I +A)xi and so y− xi ∈ Axi. By monotonicity

(y− x1− (y− x2) ,x1− x2)≥ 0

and so0≥ |x1− x2|2

which shows J1 ≡ (I +A)−1 makes sense. In fact this is Lipschitz with Lipschitz constant1. Here is why. x ∈ (I +A)J1x and y ∈ (I +A)J1y. Then

x− J1x ∈ AJ1x, y− J1y ∈ AJ1y

and so by monotonicity

0≤ (x− J1x− (y− J1y) ,J1x− J1y)

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