Journal of Convex Analysis Volume 7 (2000), No. 1, 115–128

Regular Maximal Monotone Operators and the Sum Theorem

Andrei Verona

Department of Mathematics, California State University, Los Angeles, CA 90032, USA. e-mail: averona@calstatela.edu

Maria Elena Verona

Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA. e-mail: verona@math.usc.edu

Received July 31, 1998 Revised manuscript received July 26, 1999

In this note, which is a continuation of [17], we study two classes of maximal monotone operators on general Banach spaces which we call C_{0} (resp. C_{1})-regular. All maximal monotone operators on a reflexive Banach space, all subdifferential operators, and all maximal monotone operators with domain the whole space are C_{1}-regular and all linear maximal monotone operators are C_{0}-regular. We prove that the sum of a C_{0} (or C_{1})-regular maximal monotone operator with a maximal monotone operator which is locally inf bounded and whose domain is closed and convex is again maximal monotone provided that they satisfy a certain “dom–dom” condition. From this result one can obtain most of the known sum theorem type results in general Banach spaces. We also prove a local boundedness type result for pairs of monotone operators.

1. Introduction

Let X be a Banach space and X^{∗} be its dual, endowed with the dual norm. We shall denote by B (resp. B^{∗}) the unit ball of X (resp. X^{∗}). Let also ⟨· , ·⟩ : X^{∗} × X → R denote the usual evaluation map, i.e. ⟨x^{∗}, x⟩ = x^{∗}(x).

Recall that a multivalued map T : X ⇉ X^{∗} is called a monotone operator if ⟨x^{∗} − y^{∗}, x − y⟩ ≥ 0 whenever x, y ∈ X, x^{∗} ∈ T (x), and y^{∗} ∈ T (y). The set of all x ∈ X such that T (x) ≠ ∅ is called the domain of T and is denoted D_{T}. If T is monotone and A is a subset of X, a pair (y, y^{∗}) ∈ A×X^{∗} is called monotonically related to T on A if ⟨x^{∗}−y^{∗}, x−y⟩ ≥ 0 whenever x ∈ A and x^{∗} ∈ T (x). The monotone operator T is called maximal on A if y^{∗} ∈ T (y) whenever the pair (y, y^{∗}) is monotonically related to T on A. (If A = X, we shall omit X from the above terminology.)

Let T : X ⇉ X^{∗} be a monotone operator, C be a closed convex subset of X, x ∈ C, and x^{∗} ∈ X^{∗}. Consider the following extended numbers (extended means that +∞ is a possibility)

L_{C}(x, x^{∗}, T) = 0 ∨ sup \{\frac{⟨z^{∗} − x^{∗}, x − z⟩}{∥x − z∥}; z ∈ C, z ≠ x, z^{∗} ∈ T (z)\}

ISSN 0944-6532 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

116 A. Verona, M. E. Verona / Regular maximal monotone operators and the sum theorem

M_{C}(x, x^{∗}, T) = 0 ∨ inf_{u^{∗}∈T (x)} sup \{\frac{⟨u^{∗} − x^{∗}, x − z⟩}{∥x − z∥}; z ∈ C, z ≠ x\}

(here a ∨ b = max{a, b}; we use the usual conventions that inf ∅ = +∞ and sup ∅ = −∞). When C = X we shall denote L_{C}(x, x^{∗}, T) by L(x, x^{∗}, T) and M_{C}(x, x^{∗}, T) by M(x, x^{∗}, T).

In the particular case when T is the subdifferential of a lower semicontinuous convex function and C = X, these numbers were introduced by Simons [11] who proved that they are equal to each other. In [17] we proved that these numbers are also equal when co(D_{T}) − C absorbs \overline{lin(D_{T} − C)} and either X is reflexive and T is maximal monotone or T is the subdifferential of a proper, convex, lower semicontinuous function on X. (Here “co” stands for “convex hull of” and “lin” stands for “linear span of”.) As a matter of fact, we proved their equality in a more general setting. Before stating this result we need to introduce some notation. For λ ≥ 0, let g_{λ,x} : X → R be defined by g_{λ,x}(u) = λ∥u−x∥. Let also I_{C} denote the indicator function of C (i.e. I_{C}(z) = 0 if z ∈ C and I_{C}(z) = +∞ otherwise). The following theorem was proved in [17] (see Theorem 1):

Theorem 1.1. Let C ⊆ X be nonempty, closed, convex, T : X ⇉ X^{∗} be monotone, and x ∈ X. Assume that T + ∂g_{λ,x} + ∂I_{C} is maximal monotone for any λ ≥ 0. Then L(x, x^{∗}, T + ∂I_{C}) = M(x, x^{∗}, T + ∂I_{C}). If x ∈ C, then L_{C}(x, x^{∗}, T) = M_{C}(x, x^{∗}, T).

For any monotone operator T : X ⇉ X^{∗} define

C_{0}(T) = {C ⊆ X; C is closed, convex, and D_{T} ∩ int(C) ≠ ∅}
C_{1}(T) = \{C ⊆ X; C is closed, convex, and ⋃_{λ>0} λ(co(D_{T}) − C) = \overline{lin(D_{T} − C)}\}

Clearly C_{0}(T) ⊆ C_{1}(T).

Definition 1.2.

(1) A maximal monotone operator T : X ⇉ X^{∗} is called X-regular if L(x, x^{∗}, T) = M(x, x^{∗}, T) for any x ∈ X and x^{∗} ∈ X^{∗}.

(2) Let i ∈ {0, 1}. A maximal monotone operator T : X ⇉ X^{∗} is called C_{i}-regular if L(x, x^{∗}, T + ∂I_{C}) = M(x, x^{∗}, T + ∂I_{C}) for any x ∈ X, x^{∗} ∈ X^{∗} and any C ∈ C_{i}(T).

Remark 1.3. (1) Of course the statement “T is C_{1}-regular” implies that “T is C_{0}-regular” which in turn implies that “T is X-regular”.

(2) If T is C_{1}-regular and C ∈ C_{1}(T) then D_{T} ∩ C ≠ ∅. Indeed, if D_{T} ∩ C = ∅, then L_{X}(x, x^{∗}, T + ∂I_{C}) = 0 ∨ sup ∅ = 0 ∨ {−∞} = 0 and thus M_{X}(x, x^{∗}, T + ∂I_{C}) = 0, implying that x ∈ D_{T} ∩ C for any (x, x^{∗}) ∈ X × X^{∗}, which is not true.

In [17], as a consequence of Theorem 1.1, we proved that maximal monotone operators on reflexive Banach spaces as well as subdifferential operators (on any Banach space) are C_{1}-regular.

Here are other results that we proved in [17] and will be used in this paper:

(i) If C ⊆ X is closed and convex and x ∈ C then L_{C}(x, x^{∗}, T) = L(x, x^{∗}, T + ∂I_{C}) and M_{C}(x, x^{∗}, T) = M(x, x^{∗}, T + ∂I_{C}).

(ii) M(x, x^{∗}, T) < +∞ if and only if x ∈ D_{T}. (iii) M(x, x^{∗}, T) = 0 if and only if x^{∗} ∈ \overline{T (x)}. (iv) If T is X-regular then \overline{D_{T}} is convex.

(v) If T is X-regular and x ∈ \overline{D_{T}} then T is locally bounded at x if and only if x ∈ int(D_{T}). (vi) A C_{0}-regular maximal monotone operator is maximal monotone locally (i.e. it is

maximal monotone on any open convex subset of X which intersects its domain).

One of the important open problems in convex analysis is to find conditions under which the sum T +S of two maximal monotone operators T and S is maximal monotone. When X is reflexive, Rockafellar [10] proved that a sufficient condition for the “sum theorem” to be true is that int(D_{T}) ∩ D_{S} be nonempty. This condition was relaxed by several authors (see [1], [3], [13]), the apparently least restrictive one being due to Simons (Theorem 26 in [13]) who proved that the sum theorem is true whenever S and T satisfy a certain χ constraint qualification (for example if co(D_{T}) − co(D_{S}) absorbs \overline{lin(D_{T} − D_{S})}. Later, in [14], Simons proved that these less restrictive conditions (his and the other ones) are equivalent to each other. The proofs of all these results rely heavily on the assumption that X is a reflexive Banach space. When X is a Banach space, not necessarily reflexive, the sum theorem is known to be true only in a few particular cases (see Chapter IX in [15]). Here are some of them:

(a) D_{T} = X = D_{S} (due to M. Heisler; see Section 3 in [5] or Theorem 40.4 in [15]).

(b) both T and S are linear and D_{S} = X (see Theorem 7.2 in [7] or Theorem 37.1 in [15]).

(c) T is the subdifferential of a proper lower semicontinuous convex function on X and S is linear with D_{S} = X (due to H. Bauschke; see Theorem 42.2 in [15]).

It is our aim in this paper to prove the sum theorem (in any Banach space X) in the case when T is C_{i}-regular (i ∈ {0, 1}), S is locally inf bounded (see the definition in the next section), D_{S} ∈ C_{i}(T), and D_{T} and D_{S} satisfy an additional condition. Since we shall also prove that a maximal monotone operator whose domain is X (or which is linear or which is the subdifferential of a proper lower semicontinuous convex function) is C_{0}-regular, and since a maximal monotone operator whose domain is X is locally inf bounded, our result is a generalization of (a), (b), and (c).

Finally we would like to mention that one can also introduce dual numbers

L_{V}^{∗} (x, x^{∗}, T) = 0 ∨ sup \{\frac{⟨z^{∗} − x^{∗}, x − z⟩}{∥x^{∗} − z^{∗}∥}; z^{∗} ∈ V, z^{∗} ≠ x^{∗}, z^{∗} ∈ T (z)\}

M_{V}^{∗} (x, x^{∗}, T) = 0 ∨ inf_{(z,x^{∗})∈G(T)} sup \{\frac{⟨u^{∗} − x^{∗}, x − z⟩}{∥x^{∗} − z^{∗}∥}; z^{∗} ∈ V, z^{∗} ≠ x^{∗}\}

for any (x, x^{∗}) ∈ X ×X^{∗}, V ⊆ X^{∗}. We shall study properties of these numbers in another paper.

2. A sum theorem

We begin with a construction and a lemma that will be useful in reducing statements about monotone operators on X to statements about monotone operators on a closed subspace.

Given T : X ⇉ X^{∗} and Y ⊆ X a closed subspace, define T |Y : Y ⇉ Y^{∗} by

(T |Y)(y) = {y^{∗} ∈ Y^{∗}; there exists x^{∗} ∈ T (y) such that y^{∗} = x^{∗}|Y }.

If C is a closed convex subset of Y, in addition to I_{C} : X → R ∪ {+∞} we shall also consider I_{C,Y} : Y → R ∪ {+∞}, the indicator function of C in Y, and its subdifferential ∂I_{C,Y} : Y ⇉ Y^{∗}. It is easy to verify that ∂I_{C,Y} = (∂I_{C})|Y.

Lemma 2.1. Let T : X ⇉ X^{∗} be monotone, Y ⊆ X be a closed subspace such that D_{T} ⊆ Y. Let also C be a closed convex subset of Y and x ∈ Y. Then

(a) T is maximal monotone ⟺ T = T +∂I_{Y} and T |Y : Y ⇉ Y^{∗} is maximal monotone.

(b) L_{X}(x, x^{∗}, T + ∂I_{C}) = L_{Y} (x, x^{∗}|Y, T |Y + ∂I_{C,Y}).

(c) M_{X}(x, x^{∗}, T + ∂I_{C}) = M_{Y} (x, x^{∗}|Y, T |Y + ∂I_{C,Y}).

Proof. (a) See Lemma 25 in [13]. To prove (b) notice that, since C ⊆ Y, ∂I_{C} = ∂I_{C} +∂I_{Y} and therefore

L_{X}(x, x^{∗}, T + ∂I_{C}) = L_{X}(x, x^{∗}, T + ∂I_{C} + ∂I_{Y})
= L_{Y} (x, x^{∗}, T + ∂I_{C}) = L_{Y} (x, x^{∗}|Y, (T + ∂I_{C})|Y)
= L_{Y} (x, x^{∗}|Y, T |Y + (∂I_{C})|Y) = L_{Y} (x, x^{∗}|Y, T |Y + ∂I_{C,Y})

the second equality following from Lemma 3 in [17] and the third one from the definitions. The proof of (c) is similar.

We recall now a construction and a related result due to Simons [12]. Given a monotone operator T : X ⇉ X^{∗} define ψ_{T} : X → R ∪ {+∞} by

ψ_{T} (u) = sup \{\frac{⟨z^{∗}, u − z⟩}{1 + ∥z∥}; z^{∗} ∈ T (z)\} .

Being the supremum of affine functions, ψ_{T} is convex and lower semicontinuous. If u ∈ D_{T} and u^{∗} ∈ T (u), then for any z ∈ D_{T} and any z^{∗} ∈ T (z) we have

\frac{⟨z^{∗}, u − z⟩}{1 + ∥z∥} = \frac{⟨z^{∗} − u^{∗}, u − z⟩}{1 + ∥z∥} + \frac{⟨u^{∗}, u − z⟩}{1 + ∥z∥} ≤ 0 + ∥u^{∗}∥ · \frac{∥u − z∥}{1 + ∥z∥} ≤ ∥u^{∗}∥(1 + ∥u∥)

which shows that ψ(u) < +∞. Thus

D_{T} ⊆ dom(ψ_{T}).  (2.1)

In [4] Coodey and Simons used a generalization of ψ_{T} to strengthen earlier results of Rockafellar [8] and Borwein and Fitzpatrick [2]. Among other results they proved that T is locally bounded at each surrounded point of co(D_{T}) (x ∈ X is a surrounded point of A if X \ {0} = ⋃_{λ>0} λ(A − x)). A variant of this result for a pair of monotone operators is presented next.

Proposition 2.2. Let T, S : X ⇉ X^{∗} be monotone operators such that co(D_{T}) − co(D_{S}) is absorbing. Then there exist r_{0} > 0 and c > 0 such that

∥t^{∗}∥, ∥s^{∗}∥ ≤ c(r_{0} +∥z∥)(r_{0} +∥t^{∗} +s^{∗}∥), whenever z ∈ D_{T} ∩D_{S}, t^{∗} ∈ T (z), and s^{∗} ∈ S(z).

Proof. Let ψ_{T} and ψ_{S} be defined as above. Since dom(ψ_{T}) and dom(ψ_{S}) are convex, it follows from (2.1) that co(D_{T}) ⊆ dom(ψ_{T}) and co(D_{S}) ⊆ dom(ψ_{S}). Thus co(D_{T}) − co(D_{S}) ⊆ dom(ψ_{T}) − dom(ψ_{S}) and our assumption implies that dom(ψ_{T}) − dom(ψ_{S}) is absorbing. From Corollary 4 in [13] it follows that there exist ε > 0 and r ≥ 1 such that

εB ⊆ {x ∈ X; ψ_{T} (x) ≤ r, ∥x∥ ≤ r} − {x ∈ X; ψ_{S}(x) ≤ r, ∥x∥ ≤ r}.

Let x ∈ εB, z ∈ D_{T} ∩ D_{S}, t^{∗} ∈ T (z), and s^{∗} ∈ S(z). Then x = a − b with ψ_{T} (a) ≤ r, ∥a∥ ≤ r, ψ_{S}(b) ≤ r, and ∥b∥ ≤ r. We have

⟨t^{∗}, x⟩ = ⟨t^{∗}, a − z⟩ + ⟨s^{∗}, b − z⟩ + ⟨t^{∗} + s^{∗}, z − b⟩
≤ ψ_{T} (a)(1 + ∥z∥) + ψ_{S}(b)(1 + ∥z∥) + ∥t^{∗} + s^{∗}∥(∥z∥ + r)
≤ (r + ∥z∥)(2r + ∥t^{∗} + s^{∗}∥)

from which it follows that

∥t^{∗}∥ ≤ \frac{(r + ∥z∥)(2r + ∥t^{∗} + s^{∗}∥)}{ε}.

A similar estimate can be obtained for ∥s^{∗}∥.

Corollary 2.3. Let T, S : X ⇉ X^{∗} be maximal monotone operators such that co(D_{T}) − co(D_{S}) is absorbing. Then T (z) + S(z) is a w^{∗}-closed subset of X^{∗} for any z ∈ D_{T} ∩ D_{S}.

Proof. Since T and S are maximal monotone, T (z) and S(z) are convex and therefore T (z) + S(z) is also convex. In view of the Krein-Šmulian theorem it is enough to prove that T (z) + S(z) is bw^{∗}-closed, that is every bounded w^{∗}-convergent net in T (z) + S(z) has its limit in T (z) + S(z).

Let {t_{i}^{∗}} ⊆ T (z) and {s_{i}^{∗}} ⊆ S(z), be nets such that the net {t_{i}^{∗} + s_{i}^{∗}} is bounded and w^{∗}- convergent to z^{∗}. By the previous proposition, the nets {t_{i}^{∗}} and {s_{i}^{∗}} are also bounded, so they are relatively w^{∗}-compact. By replacing them with subnets we may assume that w^{∗}-lim t_{i}^{∗} = t^{∗} and w^{∗}-lim s_{i}^{∗} = s^{∗}. Since T and S are maximal monotone, T (z) and S(z) are w^{∗}-closed and therefore t^{∗} ∈ T (z) and s^{∗} ∈ S(z). Then z^{∗} = t^{∗} +s^{∗} ∈ T (z)+S(z).

Corollary 2.4. Let T : X ⇉ X^{∗} be a maximal monotone operator, C ∈ C_{1}(T), and assume that L_{X}(x, x^{∗}, T + ∂I_{C}) = M_{X}(x, x^{∗}, T + ∂I_{C}) for any (x, x^{∗}) ∈ X × X^{∗}. Then T + ∂I_{C} is maximal monotone. In particular, if T : X ⇉ X^{∗} is C_{0} (resp. C_{1})-regular maximal monotone operator and C ∈ C_{0}(T) (resp. C ∈ C_{1}(T), then T + ∂I_{C} is maximal monotone.

Proof. Assume first that \overline{lin(D_{T} − C)} = X, i.e. co(D_{T}) − C is absorbing. Let (x, x^{∗}) ∈ X × X^{∗} be monotonically related to T + ∂I_{C}. Then L_{X}(x, x^{∗}, T + ∂I_{C}) = 0 and therefore M_{X}(x, x^{∗}, T + ∂I_{C}) = 0 too. This means that x^{∗} ∈ \overline{(T + ∂I_{C})(x)}. By Corollary 2.3, (T +∂I_{C})(x) is w^{∗}-closed and therefore norm-closed too. Thus x^{∗} ∈ (T +∂I_{C})(x), implying that T + ∂I_{C} is maximal monotone.

The general case can be reduced to the particular one considered above as follows. First we shall show that there is no loss of generality in assuming that 0 ∈ D_{T} ∩C. To this end, let c ∈ D_{T} ∩ C (this is possible because of Remark 1.3(2) in the introduction). Define T̃ : X ⇉ X^{∗} by T̃(x) = T (x + c) and let C̃ = C − c. It is easy to see that T̃ is maximal monotone, C̃ ∈ C_{1}(T̃), L_{X}(x, x^{∗}, T + ∂I_{C}) = L_{X}(x − c, x^{∗}, T̃ + ∂IC̃), M_{X}(x, x^{∗}, T + ∂I_{C}) = M_{X}(x − c, x^{∗}, T̃ + ∂IC̃), and that T + ∂I_{C} is maximal monotone if and only if T̃ + ∂IC̃ is maximal monotone. Since 0 ∈ DT̃ ∩ C̃, our assertion is proved.

Let Y = \overline{lin(D_{T} − C)}. Since 0 ∈ D_{T} ∩ C, it follows that D_{T} ⊆ Y and C ⊆ Y. Then co(D_{(T |Y)}) − C is absorbing (as a subset of Y) and, from Lemma 2.1, T |Y is maximal monotone and L_{X}(x, x^{∗}, T |Y + ∂I_{C,Y}) = M_{X}(x, x^{∗}, T |Y + ∂I_{C,Y}) for any (x, x^{∗}) ∈ Y × Y^{∗}. From the particular case proved at the beginning of the proof it follows that T |Y + ∂I_{C,Y} is maximal monotone. Finally, since (T + ∂I_{C})|Y = T |Y + ∂I_{C,Y}, from Lemma 2.1 we obtain that T + ∂I_{C} is maximal monotone.

Let C be a convex subset of X and let x ∈ C. Recall that the tangent cone to C at x, denoted C_{x}, and the normal cone to C at x, denoted N_{C}(x), are defined as follows

C_{x} = \overline{⋃_{t≥0} t(C − x)}

N_{C}(x) = {x^{∗} ∈ X^{∗}; ⟨x^{∗}, z − x⟩ ≤ 0, for any z ∈ C}.

It is easily verified that

N_{C}(x) = {x^{∗} ∈ X^{∗}; ⟨x^{∗}, v⟩ ≤ 0, for any v ∈ C_{x}}

and (by using a separation argument) that

C_{x} = {v ∈ X; ⟨x^{∗}, v⟩ ≤ 0, for any x^{∗} ∈ N_{C}(x)}.

As a matter of fact, N_{C}(x) = ∂I_{C}(x) whenever C is closed.

Lemma 2.5. Let T : X ⇉ X^{∗} be a maximal monotone operator, let C be a convex subset of X such that D_{T} ⊆ C, and let x ∈ D_{T}. Then T (x) + N_{C}(x) = T (x).

Proof. Let z ∈ D_{T}, z^{∗} ∈ T (z), x^{∗} ∈ T (x), and v^{∗} ∈ N_{C}(x). Then, from the monotonicity of T and the definition of N_{C}(x), we get

⟨x^{∗} + v^{∗} − z^{∗}, x − z⟩ = ⟨x^{∗} − z^{∗}, x − z⟩ + ⟨v^{∗}, x − z⟩ ≥ 0

and, since T is maximal monotone, it follows that x^{∗} + v^{∗} ∈ T (x). Thus T (x) + N_{C}(x) ⊆ T (x). Since 0 ∈ N_{C}(x), the other inclusion is obvious.

Before stating our main result, we need one more definition.

Definition 2.6. A multivalued map T : X ⇉ X^{∗} is called locally inf bounded if for every z ∈ D_{T} there exist ε > 0 and M > 0 such that for any u ∈ D_{T} with ∥u − z∥ ≤ ε there exists u^{∗} ∈ T (u) with ∥u^{∗}∥ ≤ M.

Example 2.7. (1) Any monotone operator T whose domain is open is locally inf bounded (because it is locally bounded, see for example Theorem 2.28 in [6]).

(2) If f : X → R ∪ {+∞} is a proper convex function which is locally Lipschitzian on its domain then D_{∂f} = dom(f) and ∂f is locally inf bounded (see for example [16], where locally inf bounded monotone operators where called locally efficient).

Theorem 2.8. Let T and S be maximal monotone operators on X and let i ∈ {0, 1}. Assume that T is C_{i}-regular, S is locally inf bounded, and D_{S} ∈ C_{i}(T). If i = 1 assume also that \overline{co(D_{T}) ∩ D_{S}} = \overline{D_{T}} ∩ D_{S} (for example, S satisfies all these conditions if D_{S} = X). Then T + S is maximal monotone.

Proof. Exactly as in the proof of Corollary 2.4, we can assume, without any loss of generality, that \overline{lin(D_{T} − D_{S})} = X. Thus the fact that D_{S} ∈ C_{i}(T) implies that

co(D_{T}) − D_{S} is absorbing.  (2.2)

Being maximal monotone, both S and T are w^{∗}-closed and convex valued. Thus, for any z ∈ D_{T} ∩ D_{S}, T (z) + S(z) is convex and, by (2.2) and Corollary 2.3, w^{∗}-closed. Assume that T + S is not maximal monotone. Then there exists a pair (x, y^{∗}) ∈ X × X^{∗} such that

⟨t^{∗} + s^{∗} − y^{∗}, z − x⟩ ≥ 0, whenever z ∈ D_{T} ∩ D_{S}, t^{∗} ∈ T (z), and s^{∗} ∈ S(z)  (2.3)

but y^{∗} ∉ (T + S)(x).

\underline{Claim I.} x ∈ D_{T} ∩ D_{S} (to be proved later).^{1}

Thus T (x) + S(x) ≠ ∅ and, as noticed above, T (x) + S(x) is w^{∗}-closed and convex. Since y^{∗} ∉ T (x) + S(x), by the separation theorem there exist u ∈ X, with ∥u∥ = 1, and a real number β such that

⟨t^{∗} + s^{∗}, u⟩ < β < ⟨y^{∗}, u⟩, for any t^{∗} ∈ T (x), s^{∗} ∈ S(x).  (2.4)

Since T is X-regular, \overline{D_{T}} is a closed convex set, (Theorem 7 in [17]), so D = \overline{D_{T}} ∩ D_{S} is also closed and convex. It is not difficult to see that

D = \overline{D_{T} ∩ int(D_{S})} if i = 0.  (2.5_{0})

The last hypothesis of the theorem implies that

D = \overline{co(D_{T}) ∩ D_{S}} if i = 1.  (2.5_{1})

Let D_{x} be the tangent cone to D at x.

\underline{Claim II.} u ∈ D_{x} (to be proved later).

We shall now choose some constants. First, since S is locally inf bounded, there exist r > 0 and α > 0 such that

S(z) ∩ rB^{∗} ≠ ∅ whenever z ∈ D_{S} and ∥z − x∥ ≤ α.  (2.6)

Choose also 0 < δ ≤ \frac{1}{2} such that rδ ≤ ε, where ε is defined next, independently of δ. Let

ε = \frac{1}{4}(⟨y^{∗}, u⟩ − β),

μ = sup_{x^{∗}∈T (x)}⟨x^{∗}, u⟩,

1In order to make the proof easier to follow, we shall claim several results and verify them later.

W = {x^{∗} ∈ X^{∗}; ⟨x^{∗}, u⟩ < β + ε − μ}.

(It follows from (2.4) that μ < +∞.) Then W is a w^{∗}-open subset of X^{∗} and (2.4) implies that S(x) ⊆ W. Finally, since S is maximal, we may also assume that α is small enough that

\underline{Claim III.} S(z) ∩ rB^{∗} ⊆ W whenever z ∈ D_{S} and ∥z − x∥ ≤ α (to be proved later).

We shall next use the fact that T is C_{i}-regular to derive a contradiction from (2.4). To this end consider the closed convex set

C = \{z ∈ X; ∥z−x∥ ≤ α, z = x+tv for some t > 0 and some v ∈ X with ∥v − u∥ < \frac{δ}{4}\}.

\underline{Claim IV.} ∥\frac{1}{∥z−x∥}(z − x) − u∥ ≤ δ for any z ∈ C (to be proved later).

Since x + \frac{α}{2}u ∈ int(C) and u ∈ D_{x}, from the definition of D_{x} it follows that

D ∩ int(C) ≠ ∅.  (2.7)

Let K = D_{S} ∩ C. If i = 0, then (2.7) and (2.5_{0}) imply that

D_{T} ∩ int(K) = D_{T} ∩ int(D_{S}) ∩ int(C) ≠ ∅

and therefore K ∈ C_{0}(T). If i = 1, then (2.7), and (2.5_{1}) imply that co(D_{T})∩D_{S}∩int(C) ≠ ∅. From (2.2) one can deduce now that co(D_{T})−K is absorbing and therefore K ∈ C_{1}(T).

Since T is C_{i}-regular, L(x, y^{∗}, T +∂I_{K}) = M(x, y^{∗}, T +∂I_{K}) < +∞ (the inequality because x ∈ D_{T} ∩ K). From the definition of M(x, y^{∗}, T + ∂I_{K}) there exists t_{0}^{∗} ∈ T (x) such that

\frac{⟨y^{∗} − t_{0}^{∗}, z − x⟩}{∥x − z∥} < L(x, y^{∗}, T + ∂I_{K}) + ε, for any z ∈ K.

Since u ∈ D_{x}, there exist w ∈ X and γ > 0 such that ⟨y^{∗} − t_{0}^{∗}, u⟩ ≤ ⟨y^{∗} − t_{0}^{∗}, w⟩ + ε and x + γw ∈ K. Then, from the above inequality, we obtain

⟨y^{∗} − t_{0}^{∗}, u⟩ ≤ ⟨y^{∗} − t_{0}^{∗}, w⟩ + ε = \frac{⟨y^{∗} − t_{0}^{∗}, (x + γw) − x⟩}{∥x − (x + γw)∥} < L(x, y^{∗}, T + ∂I_{K}) + 2ε.

From the definition of L(x, y^{∗}, T + ∂I_{K}), there exist z ∈ K ∩ D_{T} and t^{∗} ∈ T (z) such that

⟨y^{∗} − t_{0}^{∗}, u⟩ < \frac{⟨t^{∗} − y^{∗}, x − z⟩}{∥x − z∥} + 2ε.

This inequality can be rewritten as

⟨y^{∗} − t_{0}^{∗}, u⟩ < ⟨y^{∗} − t^{∗}, v⟩ + 2ε,  (2.8)

where v = (1/∥z − x∥)(z − x); since z ∈ C, by Claim IV,

∥v − u∥ < δ.  (2.9)

Choose now s^{∗} ∈ S(z) ∩ rB^{∗} (this is possible by (2.6)). From (2.2) we obtain

⟨t^{∗} + s^{∗} − y^{∗}, v⟩ ≥ 0

or

⟨y^{∗} − t^{∗}, v⟩ ≤ ⟨s^{∗}, v⟩.

Using (2.8) and this last inequality we get

⟨y^{∗}, u⟩ < ⟨t_{0}^{∗}, u⟩ + ⟨y^{∗} − t^{∗}, v⟩ + 2ε ≤ ⟨t_{0}^{∗}, u⟩ + ⟨s^{∗}, v⟩ + 2ε
= ⟨t_{0}^{∗}, u⟩ + ⟨s^{∗}, u⟩ + ⟨s^{∗}, v − u⟩ + 2ε

(by Claim III) ≤ ⟨t_{0}^{∗}, u⟩ + β + ε − μ + ∥s^{∗}∥ ∥v − u∥ + 2ε (by (2.9) and then by the choice of δ) ≤ ⟨t_{0}^{∗}, u⟩ − μ + β + 3ε + rδ ≤ 0 + β + 4ε = ⟨y^{∗}, u⟩

which is impossible. Thus T + S must be maximal monotone.

\underline{Proof of Claim I.} Let x_{0} ∈ D_{T} ∩ int(D_{S}) if i = 0 (resp. x_{0} ∈ D_{T} ∩ D_{S} if i = 1) and let [x, x_{0}] be the segment joining x and x_{0}. Since D_{S} is closed and convex there exists x_{1} ∈ [x, x_{0}] ∩ D_{S} such that [x, x_{0}] ∩ D_{S} = [x_{1}, x_{0}]. Since S is locally inf bounded and [x_{1}, x_{0}] is compact, there exists m > 0 and an open neighborhood A of [x_{1}, x_{0}] such that for any z ∈ A ∩ D_{S} there exists s^{∗} ∈ S(z) with ∥s^{∗}∥ ≤ m. For any ρ > 0 let K_{ρ} = {(1 − t)x + tu; 0 ≤ t ≤ 1, ∥u − x_{0}∥ ≤ ρ}. We claim that if ρ is small enough, then K_{ρ} ∩ D_{S} ⊆ A. If this was not true, then there would exist a sequence u_{n} → x_{0} in X and a sequence t_{n} → t in [0, 1] such that z_{n} = (1 − t_{n})x + tu_{n} ∈ D_{S} and z_{n} ∉ A. Then z = (1−t)x+tx_{0} = lim z_{n} ∉ A (since A is open), but z ∈ [x, x_{0}]∩D_{S} (since D_{S} is closed). Since [x, x_{0}] ∩ D_{S} = [x_{1}, x_{0}] ⊆ A it follows that z ∈ A, which is a contradiction. Thus there exists ρ > 0 such that K_{ρ} ∩ D_{S} ⊆ A. Let K = K_{ρ}.

Let z ∈ D_{T} ∩ K ∩ D_{S}, t^{∗} ∈ T (z), and u^{∗} ∈ ∂I_{K∩D_{S}}(z) = ∂(I_{K} + I_{D_{S}})(z). Since int(K) ∩ D_{S} ≠ ∅, from the sum formula for subdifferentials, there exist j^{∗} ∈ ∂I_{K}(z) and i^{∗} ∈ ∂I_{D_{S}}(z) such that u^{∗} = j^{∗}+i^{∗}. From the previous discussion, we can also choose s^{∗} ∈ S(z) with ∥s^{∗}∥ ≤ m. Then ⟨j^{∗}, x − z⟩ ≤ 0 and, by Lemma 2.5, s^{∗} + i^{∗} ∈ S(z). From (2.3) we obtain

\frac{⟨t^{∗} + u^{∗} − y^{∗}, x − z⟩}{∥x − z∥} = \frac{⟨t^{∗} + s^{∗} + i^{∗} − y^{∗}, x − z⟩}{∥x − z∥}+\frac{⟨j^{∗}, x − z⟩}{∥x − z∥}−\frac{⟨s^{∗}, x − z⟩}{∥x − z∥} ≤ 0+∥s^{∗}∥ ≤ m.

Thus L(x, y^{∗}, T +∂I_{K∩D_{S}}) < +∞. Since x_{0} ∈ int(K), then either x_{0} ∈ D_{T} ∩int(D_{S} ∩K) if i = 0 or, using (2.2), co(D_{T}) − D_{S} ∩ K absorbs X if i = 1. In both cases D_{S} ∩ K ∈ C_{i}(T). Since T is C_{i}-regular, it follows that M(x, y^{∗}, T + ∂I_{K∩D_{S}}) = L(x, y^{∗}, T + ∂I_{K∩D_{S}}) < +∞, hence x ∈ D_{T} ∩ D_{S} ∩ K (by (ii) in the Introduction). This proves Claim I.

\underline{Proof of Claim II.} Assume that u ∉ D_{x}. Then, by the separation theorem, there exists u^{∗} ∈ X^{∗} such that

⟨u^{∗}, z⟩ < ⟨u^{∗}, u⟩ for all z ∈ D_{x}.  (2.10)

Since 0 ∈ D_{x}, ⟨u^{∗}, u⟩ > 0; since tz ∈ D_{x} for any t > 0 and any z ∈ D_{x}, it follows from (2.10) that ⟨u^{∗}, z⟩ ≤ 0 for any z ∈ D_{x}, which means that u^{∗} ∈ ∂I_{D}(x) = ∂(I_{\overline{D}_{T}} + I_{D_{S}})(x). Since D_{T} −D_{S} is absorbing, the sum formula for subdifferentials implies that u^{∗} = v^{∗}+w^{∗}, v^{∗} ∈ ∂I_{\overline{D}_{T}} (x), w^{∗} ∈ ∂I_{D_{S}}(x). Choose t^{∗} ∈ T (x) and s^{∗} ∈ S(x). Then, by Lemma 2.5, t^{∗} + λv^{∗} ∈ T (x) and s^{∗} + λw^{∗} ∈ S(x) for any λ ≥ 0. It follows from (2.4) that ⟨u^{∗}, u⟩ = ⟨v^{∗}, u⟩ + ⟨w^{∗}, u⟩ ≤ 0 which contradicts our earlier finding that ⟨u^{∗}, u⟩ > 0. This contradiction proves Claim II.

\underline{Proof of Claim III.} If the claim was not true, (2.9) would imply that there exist a sequence {x_{n}} converging to x and a sequence {w_{n}^{∗}} with w_{n}^{∗} ∈ S(x_{n}) ∩ rB^{∗} but

⟨w_{n}^{∗}, u⟩ ≥ β + ε − μ.  (2.11)

If w^{∗} is a limit point of {w_{n}^{∗}} (there are such points because {w_{n}^{∗}} ⊂ rB^{∗}), then ⟨w^{∗} − s^{∗}, x − s⟩ ≥ 0 for any s ∈ D_{S} and any s^{∗} ∈ S(s) (because ⟨w_{n}^{∗} − s^{∗}, x_{n} − s⟩ ≥ 0). The maximal monotonicity of S implies that w^{∗} ∈ S(z). From (2.11) we obtain that ⟨w^{∗}, u⟩ ≥ β + ε − μ which contradicts the fact that S(x) ⊆ W.

\underline{Proof of Claim IV.} Given z = x + tv ∈ C, clearly \frac{1}{∥z−x∥}(z − x) = \frac{1}{∥v∥}v. We have

1 = ∥u∥ ≤ ∥v∥ + ∥u − v∥ ≤ ∥v∥ + \frac{δ}{4}

and thus

∥v∥ ≥ 1 − \frac{δ}{4} ≥ \frac{1}{2} ( since δ ≤ \frac{1}{2}).

Similarly

∥v∥ ≤ ∥v − u∥ + ∥u∥ ≤ \frac{δ}{4} + 1.

Combining the last inequalities we obtain

|1 − ∥v∥| ≤ \frac{δ}{4} and ∥v∥ ≥ \frac{1}{2} .

It follows that

∥\frac{1}{∥v∥}v − u∥ = \frac{∥v − ∥v∥u∥}{∥v∥} ≤ \frac{∥v − u∥ + ∥u − ∥v∥u∥}{∥v∥} ≤ \frac{\frac{δ}{4} + |1 − ∥v∥|}{∥v∥} ≤ \frac{\frac{δ}{4} + \frac{δ}{4}}{\frac{1}{2}} = δ

and thus ∥\frac{1}{∥z−x∥}(z − x) − u∥ ≤ δ for any z ∈ C. This proves Claim IV and completes the proof of the theorem.

Corollary 2.9. Let S : X ⇉ X^{∗} be a maximal monotone operator and f be a proper, convex, lower semicontinuous function on X. Assume that one of the following conditions is satisfied

(a) D_{S} = X.

(b) S is C_{0} (resp. C_{1})-regular, dom(f) ∈ C_{0}(S) (resp. dom(f)∈C_{1}(S) and \overline{co(D_{S})∩dom(f)} = \overline{D_{S}} ∩ dom(f)), and f is locally Lipschitz on dom(f).

Then ∂f + S is maximal monotone.

Proof. As mentioned in the introduction, ∂f is a C_{1}-regular maximal monotone operator. The maximal monotonicity of ∂f +S when (a) is satisfied follows from Theorem 2.8. If (b) is satisfied, then ∂f is locally inf bounded and the assertion follows again from Theorem 2.8.

Corollary 2.10. Let S : X ⇉ X^{∗} be a maximal monotone operator.

(i) S is C_{0}-regular if and only if S + ∂g_{λ,x} + ∂I_{C} is maximal monotone for any λ ≥ 0, any x ∈ X, and any C ∈ C_{0}(S).

(ii) If S is C_{0}-regular and C ∈ C_{0}(S) then S + ∂I_{C} is C_{0}-regular.

Proof. (i) The “if” part is an immediate consequence of Theorem 1.1 in the introduction, while the “only if” part follows from Corollary 2.9 because f = ∂(g_{λ,x} + I_{C}) satisfies condition (b) and ∂(g_{λ,x} + I_{C}) = ∂g_{λ,x} + ∂I_{C}.

(ii) It is enough to show that S + ∂g_{λ,x} + ∂I_{C} + ∂I_{K} is maximal monotone for any λ ≥ 0, any x ∈ X, and any K ∈ C_{0}(S + ∂I_{C}). Since D_{S+∂I_{C}} = D_{S} ∩ C and K ∈ C_{0}(S + ∂I_{C}), it follows that D_{S} ∩C ∩int(K) ≠ ∅ and therefore S +∂g_{λ,x} +∂I_{C} +∂I_{K} = S +∂g_{λ,x} +∂I_{C∩K}. If we can show that, D_{S} ∩ int(C ∩ K) ≠ ∅, then C ∩ K ∈ C_{0}(S) and (ii) follows from (i).

Choose u ∈ D_{S} ∩ int(C) (possible since C ∈ C_{0}(S)) and v ∈ D_{S+∂I_{C}} ∩ int(K) = D_{S} ∩ C ∩ int(K). If u = v, we are done. Otherwise, [u, v) ⊆ int(C), [u, v] ⊆ \overline{D_{S}} (because S is X-regular and therefore \overline{D_{S}} is convex), and v ∈ int(K). Thus there exists w on the open segment (u, v) such that w ∈ \overline{D_{S}} ∩ int(C) ∩ int(K). It is now obvious that D_{S} ∩ int(C ∩ K) ≠ ∅ and, as mentioned above, this proves (ii).

Corollary 2.11.

(i) If S : X ⇉ X^{∗} is maximal monotone, locally inf bounded, and D_{S} is closed and convex then S is C_{1}-regular. In particular, a maximal monotone operator S with D_{S} = X is C_{1}-regular.

(ii) If T, S : X ⇉ X^{∗} are maximal monotone, locally inf bounded, D_{T} , D_{S} are closed and convex, and ⋃_{λ>0} λ(D_{T} − D_{S}) = \overline{lin(D_{T} − D_{S})}, then T + S is maximal monotone.

Proof. (i) Let C ∈ C_{1}(S) and λ ≥ 0. T = ∂g_{λ,x} + ∂I_{C} = ∂(g_{λ,x} + I_{C}) is C_{1}-regular and D_{T} = C. Since co(D_{T}) = C = co(C) and co(D_{S}) = D_{S}, we have

⋃_{λ>0}(co(D_{T}) − co(D_{S})) = ⋃_{λ>0}(C − D_{S}) = \overline{lin(C − D_{S})} = \overline{lin(D_{T} − D_{S})}

hence D_{S} ∈ C_{1}(T). Also, since D_{T} = C and D_{S} are closed,

\overline{D_{T} ∩ D_{S}} = \overline{C ∩ D_{S}} = C ∩ D_{S} = \overline{D_{T}} ∩ D_{S}.

By Theorem 2.8, T + S = S + ∂g_{λ,x} + ∂I_{C} is maximal monotone. Thus, by Theorem 1.1 in the introduction, S is C_{1}-regular.

(ii) By (i), T is C_{1}-regular. Since the conditions of Theorem 2.8 are satisfied, T + S is maximal monotone.

3. Linear maximal monotone operators

A single valued monotone operator T is called linear if D_{T} is a linear subspace of X and T : D_{T} → X^{∗} is linear. We begin by recalling a result proved in [7] and [15] (we shall reformulate it so that we can use our notation). Assume that T : D_{T} → X^{∗} is linear and monotone. Then

T is maximal monotone ⟺ D_{T} is dense in X and D_{T} = {x ∈ X; L_{X}(x, 0, T) < +∞}. (3.1)

Lemma 3.1. Let T : D_{T} → X^{∗} be linear and maximal monotone. Let C be a closed convex subset of X such that int(C) ∩ D_{T} ≠ ∅, let x ∈ X and x^{∗} ∈ X^{∗}. If L_{X}(x, x^{∗}, T + ∂I_{C}) < +∞ then x ∈ D_{T}.

Proof. Assume first that 0 ∈ int(C)∩D_{T}. In view of (3.1) and the maximal monotonicity of T, it is enough to show that L_{X}(x, 0, T) < +∞. To this end let g : X → R denote the gauge function associated to C, i.e. g(z) = inf{t ≥ 0; z ∈ tC}. Since int(C) ≠ ∅, g is a continuous semi-norm on X and therefore it is Lipschitzian. Let α denote the Lipschitz constant of g. Set M = (L_{X}(x, x^{∗}, T + ∂I_{C}) + ∥x^{∗}∥)(α∥x∥ + 1) and let z ∈ D_{T}.

Case I: z ∈ C. We have

\frac{⟨T (z), x − z⟩}{∥x − z∥} = \frac{⟨T (z) − x^{∗}, x − z⟩}{∥x − z∥} + \frac{⟨x^{∗}, x − z⟩}{∥x − z∥}
≤ \frac{⟨T (z) − x^{∗}, x − z⟩}{∥x − z∥} + ∥x^{∗}∥
≤ L_{X}(x, x^{∗}, T + ∂I_{C}) + ∥x^{∗}∥ ≤ M (since 0 ∈ ∂I_{C}(z)).

Case II: z ∉ C. Let t = g(z) and u = \frac{1}{t}z. Since C is closed, t > 1 and u ∈ C. We have

\frac{⟨T (z), x − z⟩}{∥x − z∥} = \frac{⟨T (tu), x − tu⟩}{∥x − tu∥} = \frac{t⟨T (u), x − u⟩ + t⟨T (u), (1 − t)u⟩}{∥x − tu∥}

(use the facts that ⟨T (u), u⟩ ≥ 0 and t > 1)

≤ \frac{t⟨T (u), x − u⟩}{∥x − tu∥} = \frac{t⟨T (u), x − u⟩}{t∥x − u∥} · \frac{∥tx − tu∥}{∥x − tu∥}
= \frac{⟨T (u), x − u⟩}{∥x − u∥} · \frac{∥tx − tu∥}{∥x − tu∥}

(use a computation similar to that in Case I)

≤ (L_{X}(x, x^{∗}, T + ∂I_{C}) + ∥x^{∗}∥)\frac{∥tx − tu∥}{∥x − tu∥}
≤ (L_{X}(x, x^{∗}, T + ∂I_{C}) + ∥x^{∗}∥)\frac{∥tx − x∥ + ∥x − tu∥}{∥x − tu∥}
= (L_{X}(x, x^{∗}, T + ∂I_{C}) + ∥x^{∗}∥) (\frac{(t − 1)∥x∥}{∥x − tu∥} + 1)
≤ (L_{X}(x, x^{∗}, T + ∂I_{C}) + ∥x^{∗}∥) (\frac{(g(z) − g(x))∥x∥}{∥x − z∥} + 1)
≤ (L_{X}(x, x^{∗}, T + ∂I_{C}) + ∥x^{∗}∥)(α∥x∥ + 1) ≤ M.

Thus L_{X}(x, 0, T) < +∞ and, as mentioned above, this proves the lemma in the case when 0 ∈ D_{T} ∩ int(C).

To prove the general case, let c ∈ int(C) ∩ D_{T}. A fairly trivial verification shows that

L_{X}(x − c, x^{∗} − T (c), T + ∂I_{C−c}) = L_{X}(x, x^{∗}, T + ∂I_{C}) < +∞.

Since 0 ∈ int(C − c), it follows that x − c ∈ D_{T}. Since c ∈ D_{T} and since D_{T} is a linear subspace of X, it follows that x ∈ D_{T}.

Proposition 3.2. Let T : D_{T} → X^{∗} be linear and maximal monotone. Then T is C_{0}- regular.

Proof. Let C be a closed convex subset of X such that int(C) ∩ D_{T} ≠ ∅ and let x ∈ C. Since always 0 ≤ L_{X}(x, x^{∗}, T +∂I_{C}) ≤ M_{X}(x, x^{∗}, T +∂I_{C}) (Lemma 1 in [17]), it is enough to assume that L_{X}(x, x^{∗}, T + ∂I_{C}) is finite and that M_{X}(x, x^{∗}, T + ∂I_{C}) > 0 and show that M_{X}(x, x^{∗}, T + ∂I_{C}) ≤ L_{X}(x, x^{∗}, T + ∂I_{C}).

First notice that, by Lemma 3.1, x ∈ D_{T}. Let ε > 0 and M < M_{X}(x, x^{∗}, T + ∂I_{C}). In view of (3.1), D_{T} is dense in X. Therefore we can find z ∈ C ∩ D_{T} such that

M ≤ \frac{⟨T (x) − x^{∗}, x − z⟩}{∥x − z∥}

(notice that, since T is single valued, the “inf” from the definition of M_{X}(x, x^{∗}, T + ∂I_{C}) disappears).

Let 0 < λ < 1 and let z_{λ} = (1 − λ)x + λz ∈ C ∩ D_{T}. Clearly x − z_{λ} = λ(x − z). Choose λ so small that

\frac{λ⟨T (x − z), x − z⟩}{∥x − z∥} ≤ ε.

We have

M ≤ \frac{⟨T (x) − x^{∗}, x − z⟩}{∥x − z∥} = \frac{⟨T (x) − x^{∗}, x − z_{λ}⟩}{∥x − z_{λ}∥}

≤ \frac{⟨T (x) − T (z_{λ}), x − z_{λ}⟩}{∥x − z_{λ}∥} + \frac{⟨T (z_{λ}) − x^{∗}, x − z_{λ}⟩}{∥x − z_{λ}∥}

≤ \frac{⟨T (x − z_{λ}), x − z⟩}{∥x − z∥} + L_{X}(x, x^{∗}, T + ∂I_{C})

≤ \frac{λ⟨T (x − z), x − z⟩}{∥x − z∥} + L_{X}(x, x^{∗}, T + ∂I_{C}) ≤ ε + L_{X}(x, x^{∗}, T + ∂I_{C}) .

Since this is true for any ε > 0 and any M < M_{X}(x, x^{∗}, T + ∂I_{C}), it follows that M_{X}(x, x^{∗}, T + ∂I_{C}) ≤ L_{X}(x, x^{∗}, T + ∂I_{C}) and the theorem is completely proved.

Combining Proposition 3.2 with Theorem 2.8 we obtain the next corollary which was proved by different methods in both [7] and [15].

Corollary 3.3. If T and S are linear maximal monotone operators on X and D_{S} = X then T + S is (linear) maximal monotone.

128 A. Verona, M. E. Verona / Regular maximal monotone operators and the sum theorem

Since a C_{0}-regular maximal monotone operator satisfies the conditions of Theorem 10 in [17], we obtain another proof of the following result due to Phelps and Simons [7] and Simons [15]. Corollary 3.4. A linear maximal monotone operator is maximal monotone locally (i.e. it is maximal monotone on any open convex subset of X which intersects its domain).

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