Linear Monotone Subspaces

of Locally Convex Spaces

M.D. Voisei∗ and C. Zalinescu†

AbstractThe main focus of this paper is to study multi-valued linear monotone operators in

the context of locally convex spaces via the use of their Fitzpatrick and Penot functions.Notions such as maximal monotonicity, uniqueness, negative-infimum, and (dual-) repre-sentability are studied and criteria are provided.

Key Words: Fitzpatrick function, Penot function, monotone multifunction, skew linearsubspace, monotone linear subspace, representable multifunction, locally convex spaceMathematics Subject Classification 2000: Primary 47A06, 47H05; Secondary 26B25,52A41.

1 Preliminaries

Motivated by the facts that, besides the subdifferentials, single-valued linear maximal mono-tone operators enjoy a set of stronger monotonicity properties and belong to most of thespecial classes of maximal monotone operators introduced in the non-reflexive Banach spacesettings (see [1, 2]), the linear monotone subspaces of a non-reflexive Banach space made theobject of extensive studies in [18, 3].

It is hoped that a linear maximal monotone would be a part of a counter-example tothe celebrated Rockafellar Conjecture (see e.g. [5]), that is why a thorough study of linearmonotone operators could help accomplish that goal.

The vast majority of the results concerning monotone operators are stated in the contextof Banach spaces with a few exceptions (see e.g. [6, 8]). In the context of non-reflexive Banachspaces it is natural to use topologies compatible with the simple duality (X ×X∗, X∗ ×X)instead of the strong topologies. This leads to the context of locally convex spaces underwhich, fortunately, many of the results stated in Banach spaces for monotone operators stillhold.

Our goal in this paper is twofold: - to extend and add new properties to the knownresults concerning maximal monotone multi-valued operators in the context of locally convexspaces and - to provide general criteria for various classes of linear monotone operators whileperforming a comparison of the main types of linear maximal monotone operators throughexamples and counter-examples.

It is interesting to mention that in a locally convex space context, currently, there are twocharacterization of maximal monotonicity of a general operator:

∗Towson University, Department of Mathematics, Towson, Maryland, USA, email: mvoisei@towson.edu.†University “Al.I.Cuza” Iasi, Faculty of Mathematics, 700506-Iasi, Romania, and Institute of Mathematics

Octav Mayer, Iasi, Romania, email: zalinesc@uaic.ro.

1

• (Fitzpatrick [8, Theorem 3.8]) Let X be a locally convex space and T : X ⇒ X∗ bemonotone. Then T is maximal monotone iff ϕT (x, x∗) = sup{x∗(y)+y∗(x−y) | (y, y∗) ∈gph T} > x∗(x), for every (x, x∗) ∈ X ×X∗ \ gph T .

• (Voisei & Zalinescu [31, Theorem 1]) Let X be a locally convex space and T : X ⇒ X∗.Then T is maximal monotone iff T is representable and T is NI in X ×X∗.

Here, we provide another characterization for the maximality of a monotone operator interms of uniqueness, namely

• Let X be a locally convex space and T : X ⇒ X∗ be monotone. Then T is maximalmonotone iff T is unique and dual-representable (see Theorem 6 below).

The plan of the paper is as follows. In the second section the main notions and notationtogether with the most representative results concerning these notions are presented. Section3 studies general monotone operator properties such as uniqueness, NI type, graph-convexity,(dual-)representability, and maximal monotonicity. Section 4 deals with skew linear monotoneoperators. In Section 5 the main types of linear monotone subsets are studied and criteriaare provided.

2 Main notions and notations

For a locally convex space (E, τ) and A ⊂ E, we denote by “clτ A” the τ−closure of A,“conv A” the convex hull of A, convτA := clτ (conv A) the closed convex hull of A, “aff A”the affine hull of A, “linA” the linear hull of A, linτ

A := clτ (linA) the closed linear hull ofA, “Ai” the algebraic interior of A, “iA” the relative algebraic interior of A with respect toaff A, while icA := iA if aff A is τ−closed and icA := ∅ otherwise, is the relative algebraicinterior of A with respect to affτ

A := clτ (aff A). In the sequel when the topology is implicitlyunderstood we avoid the use of the τ−notation. A subset A is a cone if R+A = A while A isa double-cone if RA = A.

For f, g : E → R := R ∪ {−∞, +∞} we set [f ≤ g] := {x ∈ E | f(x) ≤ g(x)}; the sets[f = g], [f < g] and [f > g] are defined similarly.

Throughout this paper, if not otherwise explicitly mentioned, (X, τ) is a non trivial (thatis, X 6= {0}) separated locally convex space, X∗ is its topological dual endowed with the weak-star topology ω∗, the topological dual of (X∗, ω∗) is identified with X, and the weak topologyon X is denoted by ω. The duality product of X×X∗ is denoted by 〈x, x∗〉 := x∗(x) =: c(x, x∗)for x ∈ X, x∗ ∈ X∗.

To an operator (or multifunction) T : X ⇒ X∗ we associate its graph: gph T = {(x, x∗) ∈X × X∗ | x∗ ∈ T (x)}, inverse: T−1 : X∗ ⇒ X, gph T−1 = {(x∗, x) | (x, x∗) ∈ gphT},domain: dom T := {x ∈ X | T (x) 6= ∅} = PrX(T ), and range: ImT := {x∗ ∈ X∗ | x∗ ∈T (x) for some x ∈ X} = PrX∗(T ). Here PrX and PrX∗ are the projections of X ×X∗ ontoX and X∗, respectively. When no confusion can occur, T will be identified with gphT .

On X, we consider the following classes of functions and operators:

Λ(X) is the class of proper convex functions f : X → R. Recall that f is proper if dom f :={x ∈ X | f(x) < ∞} is nonempty and f does not take the value −∞;

2

Γτ (X) is the class of functions f ∈ Λ(X) that are τ–lower semicontinuous (τ–lsc for short);when the topology is implicitly understood we use the notation Γ(X);

M(X) is the class of monotone operators T : X ⇒ X∗. Recall that T : X ⇒ X∗ is monotoneif 〈x1 − x2, x

∗1 − x∗2〉 ≥ 0 for all x∗1 ∈ Tx1, x∗2 ∈ Tx2;

M(X) is the class of maximal monotone operators T : X ⇒ X∗. The maximality is under-stood in the sense of graph inclusion as subsets of X ×X∗.

Notions associated to a proper function f : X → R:

epi f := {(x, t) ∈ X × R | f(x) ≤ t} is the epigraph of f ;

clτ f : X → R, the τ–lsc hull of f , is the greatest τ–lsc function majorized by f ; henceepi(clτ f) = clτ (epi f);

conv f : X → R, the convex hull of f , is the greatest convex function majorized by f ; hence(conv f)(x) := inf{t ∈ R | (x, t) ∈ conv(epi f)} for x ∈ X;

convτf : X → R, the τ−lsc convex hull of f , is the greatest τ–lsc convex function majorizedby f ; hence epi(convτf) := convτ (epi f);

f∗ : X∗ → R is the convex conjugate of f : X → R with respect to the dual system (X, X∗),f∗(x∗) := sup{〈x, x∗〉 − f(x) | x ∈ X} for x∗ ∈ X∗;

∂f(x) is the subdifferential of the proper function f : X → R at x ∈ X; ∂f(x) := {x∗ ∈X∗ | 〈x′ − x, x∗〉 + f(x) ≤ f(x′), ∀x′ ∈ X} for x ∈ X (it is clear that ∂f(x) := ∅ forx 6∈ dom f). Recall that NC = ∂ιC is the normal cone of C, where ιC is the indicatorfunction of C ⊂ X defined by ιC(x) := 0 for x ∈ C and ιC(x) := ∞ for x ∈ X \ C.

Let Z := X ×X∗. It is known that (Z, τ × ω∗)∗ = Z via the coupling

z · z′ := ⟨x, x′∗

⟩+

⟨x′, x∗

⟩, for z = (x, x∗), z′ = (x′, x′∗) ∈ Z.

For a proper function f : Z → R all the above notions are defined similarly. In addition, withrespect to the natural dual system (Z,Z) induced by the previous coupling, the conjugate off is given by

f¤ : Z → R, f¤(z) = sup{z · z′ − f(z′) | z′ ∈ Z},and by the biconjugate formula, f¤¤ = convτ×ω∗f whenever f¤ (or convτ×ω∗f) is proper.

To a multifunction T : X ⇒ X∗ we associate the following functions: cT : Z → R,cT := c + ιT , ψT : Z → R, ψT := convτ×ω∗cT is the Penot function of T , ϕT : Z → R,ϕT := c¤

T = ψ¤T is the Fitzpatrick function of T .

From the definition of ϕT one has (as observed in [28, Proposition 2])

T ⊂ (domT ×X∗) ∪ (X × Im T ) ⊂ [ϕT ≥ c]. (1)

Moreover, as observed in several places (see e.g. [14, 15, 26, 28, 29, 31]),

T ∈M(X) ⇐⇒ conv cT ≥ c ⇐⇒ ψT ≥ c ⇐⇒ T ⊂ [ϕT = c] ⇐⇒ T ⊂ [ϕT ≤ c], (2)T ∈M(X) ⇒ T ⊂ [ψT = c] ⊂ [ϕT = c], (3)

3

andT ∈ M(X) ⇐⇒ T = [ϕT ≤ c] ⇐⇒ [

ϕT ≥ c and T = [ϕT = c]]. (4)

To the equivalences in (4) we add the following

T ∈ M(X) ⇐⇒ [T ∈M(X), [ϕT ≤ c] ⊂ T

]. (5)

While, from the last equivalence in (4), the direct implication is obvious, for the converseimplication we have, by (1), that [ϕT ≤ c] ⊂ T ⊂ [ϕT ≥ c]; whence ϕT ≥ c and [ϕT = c] ⊂ T .Since T is monotone, by (3), we get T = [ϕT = c] and so, from (4), T ∈ M(X).

Note that condition T ∈ M(X) is not superfluous in (5). Take for example the non-monotone T = {(x, y) ∈ R2 | xy ≥ 0} for which [ϕT ≤ c] = {(0, 0)} ⊂ T .

The preceding relations suggest the introduction of the following classes of functions:

F := F(Z) := {f ∈ Λ(Z) | f ≥ c},R := R(Z) := Γτ×ω∗(Z) ∩ F(Z),

D := D(Z) := {f ∈ R(Z) | f¤ ≥ c}.It is known that [f = c] ∈M(X) for every f ∈ F(Z) (see e.g. [15]).We consider the following classes of multifunctions T : X ⇒ X∗:

• T is representable in Z if T = [f = c] for some f ∈ R; in this case f is called arepresentative of T . We denote by RT the class of representatives of T .

• T is dual-representable if T = [f = c] for some f ∈ D; in this case f is called ad–representative of T . We denote by DT the class of d-representatives of T .

• T is of negative infimum type in Z (NI for short) if ϕT ≥ c in Z. Note that in the caseX is a Banach space, this notion is a weaker form relative to X∗ ×X∗∗ of the originalversion introduced by Simons [22] in the sense that T is NI in the sense of Simons meansthat T is maximal monotone in X×X∗ and T−1 is NI in X∗×X∗∗ in the present sense.However, if T−1 is NI in X∗×X∗∗ in the present sense, then T is NI in X×X∗. In thisform this notion was first considered in [29, Remark 3.5].

• T is unique in Z if T is monotone and admits a unique maximal monotone extension inZ. In the context of Banach spaces, in [14] the preceding notion was considered underthe name of pre-maximal monotone operator. Previously, the uniqueness notion wasused in [10, 21, 1]; T is unique in their sense iff T−1 (as a subset of X∗×X∗∗) is uniquein X∗ ×X∗∗ in the present sense.

Fitzpatrick proved in [8, Theorem 2.4] that

f ∈ F =⇒ [f = c] ⊂ [f¤ = c], (6)

from which, it follows that

[f = c] = [f¤ = c] ∀f ∈ D. (7)

As observed in [29, Remark 3.6] (see also [31]), if f ∈ RT , that is, T is representable with fa representative of T , then we have

ϕT ≤ f ≤ ψT , ϕT ≤ f¤ ≤ ψT . (8)

4

Hence, if T ∈ M(X) and f ∈ RT then f ∈ DT . Moreover,(f ∈ R, T ⊂ [f = c]

)=⇒ (

f ≤ ψT , [ψT = c] ⊂ [f = c]). (9)

Indeed, T ⊂ [f = c] implies f ≤ cT , whence f ≤ ψT (because f ∈ R); therefore [ψT = c] ⊂[f = c].

In particular, (9) shows that [ψT = c] is the smallest representable extension of T ∈M(X)in the sense of graph inclusion and so [ψT = c] =

⋂{M | T ⊂ M, M representable}.

Remark 1 For a Banach space X with topological dual X∗, both endowed with their strongtopologies denoted by “s”, there have been considered three different notions of representabil-ity of a monotone operator T : X ⇒ X∗:

• the one above (first introduced in [26]) that will be used throughout this article,

• Martinez-Legaz–Svaiter (MLS) representability. In [14] one says that T is representableif there exists an s × s−lsc convex function h : X × X∗ → R such that h ≥ c andT = [h = c]. Note that these two types of representability coincide if X is reflexive butthey are different in the non-reflexive case. Indeed, consider E a non-reflexive Banachspace, X := E∗ and S := {0} × E ⊂ X × X∗ (here E is identified with its image bythe canonical injection of E into E∗∗ = X∗). Clearly S is a skew strongly closed linearspace in X × X∗, and S = [ιS = c] which makes S MLS-representable. However S isnot representable in our sense because its Penot function in X × X∗, namely, ψS =cls×ω∗ ιS = ι{0}×X∗ 6= cls×s ιS = ι{0}×E and so [ψS = c] = {0} ×X∗ = cls×ω∗ S ! S.

• Borwein representability involves no topology (see [5, Section 2.2]) or the strong topologyon X×X∗ (see [4, page 3918]) and the equality T = [h = c] in the MLS-representabilityis replaced by the inclusion T ⊂ [h = c] for an s×s−lsc convex function h : X×X∗ → Rwith h ≥ c. While it is clear that Borwein representability of a monotone operator isdifferent from the other two, in the case of a maximal monotone operator Borwein’srepresentability coincides with the MLS-representability.

3 Types of monotone operators

The next characterizations of representability and maximality were stated in the contextof Banach spaces but their arguments work in a locally convex settings, too (see also [31,Theorem 1]).

Theorem 1 ([26, Theorems 2.2, 2.3]) Let T : X ⇒ X∗.(i) T is representable iff T ∈ M(X) and T = [ψT = c], that is, ψT is a representative of

T (or ψT ∈ RT ),(ii) T is maximal monotone iff T is representable and T is of negative infimum type, that

is, ϕT is a (d–)representative of T (or ϕT ∈ RT , or ϕT ∈ DT ).

It is important to notice that, in the context of non-reflexive Banach spaces, the pre-vious characterization of maximality fails if our representability is replaced by the MLS-representability. Indeed, as in Remark 1, consider E a non-reflexive Banach space, X := E∗

and S := {0} ×E ( {0} ×X∗ ⊂ Z. As seen in Remark 1, S is MLS-representable since ιS is

5

strongly lsc convex, ιS ≥ c, and S = [ιS = c]; moreover, S is NI in Z because ϕS = ι{0}×X∗ ,and clearly S is not maximal monotone (in Z).

As in [14] (for X a Banach space and with a different notation), for a subset A of Z weset

A+ := [ϕA ≤ c] = {z ∈ Z | c(z − w) ≥ 0, ∀w ∈ A},the set of all z ∈ Z that are monotonically related to (m.r.t. for short) A and A++ := (A+)+.Note that ∅+ = Z, Z+ = ∅, A ⊂ B ⊂ Z implies B+ ⊂ A+ and (∪i∈IAi)

+ = ∩i∈IA+i for any

family (Ai)i∈I of subsets of Z. Moreover, for A ⊂ Z one has:

A ∈M(X) ⇔ A ⊂ A+, A ∈ M(X) ⇔ A = A+. (10)

For T ∈M(X) we have

T+ =⋃{M | M ∈ M(X), T ⊂ M}, (11)

and soT++ =

⋂{M | M ∈ M(X), T ⊂ M}. (12)

In particular, for T ∈ M(X) we have that T++ ∈ M(X) and T ⊂ T++ ⊂ T+. Moreover,from (11) it is easily noticed that T ∈M(X) is unique iff T+ is maximal monotone.

Note that the above mentioned properties of T+ can also be found in [14].

Lemma 2 (i) For every A ⊂ Z we have ψA ≥ ϕA+ in Z.(ii) If T ∈M(X) then T+ is NI, that is ϕT+ ≥ c in Z.

Proof. (i) Notice that A ⊂ A++ = [ϕA+ ≤ c] for every A ⊂ Z. Hence ϕA+ ≤ cA and soϕA+ ≤ ψA in Z since ϕA+ is convex and ω × ω∗−lsc.

(ii) Let M ⊂ Z be a maximal monotone extension of T . Then M ⊂ T+, and so ϕT+ ≥ϕM ≥ c in Z.

For uniqueness one has the following characterizations (see also Proposition 33 below fordouble-cones); these characterizations can also be found in [14, Proposition 36]), [21, Theorem19], and [1, Fact 2.6].

Proposition 3 Let T ∈M(X). TFAE:(i) T is unique,(ii) T+ is monotone,(iii) T+ is maximal monotone,(iv) T+ = T++,(v) T++ is maximal monotone.In this case the unique maximal monotone extension of T is T+ = T++.

Proof. Note that (i) ⇒ (iii) is true because of (11), (iii) ⇒ (ii) is obvious, while (iii) ⇔ (iv)and (iv) ⇒ (v) follow from the second part of (10). The implication (ii) ⇒ (iv) holds sincewhenever T+ is monotone we have, by the first part in (10), that T+ ⊂ T++; the reversedinclusion being true because T is monotone. To complete the proof it suffices to show (v) ⇒(i). To this end we see from (12) that if T++ is maximal monotone then T++ is the uniquemaximal monotone extension of T .

6

As it would be expected from Theorem 1, the following characterization theorem showsthat the NI condition for T ∈ M(X) is actually equivalent to the maximality of [ψT = c]which is the minimal representable operator that contains T .

Proposition 4 Let T ∈M(X). Then(i) ϕT = ϕ[ψT =c] and ψT = ψ[ψT =c]; in particular T+ = [ψT = c]+.(ii) T is NI iff [ψT = c] is NI iff [ψT = c] is maximal monotone iff T has a unique

representable extension.(iii) If T is NI then T is unique and

T+ = [ψT = c] = [ϕT = c] = [ϕT ≤ c]

is the unique representable extension and the unique maximal monotone extension of T .

Proof. Set R := [ψT = c] the smallest representable extension of T ∈M(X).(i) From (3), we see that T ⊂ R, and so ϕT ≤ ϕR. Conversely, by the Fenchel inequality,

for all z, w ∈ ZϕT (z) ≥ z · w − ψT (w).

Pass to supremum over w ∈ R to find ϕT (z) ≥ ϕR(z) for every z ∈ Z. The other relation isobtained by conjugation.

(ii) According to (i) and from Theorem 1 (ii) applied for R we get that R is maximalmonotone iff R is NI iff T is NI.

Assume now that R is maximal monotone and h ∈ R is such that T ⊂ [h = c]. By (9) wehave R ⊂ [h = c]. Because R is maximal monotone and [h = c] is monotone we obtain thatR = [h = c], and so R is the unique representable extension of T . Conversely, assume thatT has a unique representable extension. Since any maximal monotone extension of T is alsorepresentable it follows that R is maximal monotone and the unique representable extensionof T .

(iii) Assume that T is NI. Hence T+ = [ϕT ≤ c] = [ϕT = c]. By (ii) R is the uniquerepresentable extension of T ; in particular R is the unique maximal monotone extension ofT , and so, according to Proposition 3, R = T+.

Recently, in [3, Theorem 4.2] it is proved, in the context of a Banach space, that amaximal monotone operator with a convex graph is in fact affine. This result holds in locallyconvex spaces as a consequence of the broader fact that the affine hull of a monotone convexmultifunction remains monotone.

Proposition 5 Let T ⊂ X ×X∗. TFAE:(i) cT is convex,(ii) T is monotone and convex.In this case aff T is monotone and T ⊂ aff T ⊂ T+. In particular, if T is maximal

monotone and convex then T is affine.

Proof. (i) ⇒ (ii) Since cT is convex so is dom cT = T , while cT ≥ c and T = [cT = c] showthat T is monotone.

7

(ii) ⇒ (i) For all λ ∈ [0, 1] and z, z′ ∈ T = dom cT we have λz + (1− λ)z′ ∈ T since T isconvex and

cT (λz + (1− λ)z′)− λcT (z)− (1− λ)cT (z′) = −λ(1− λ)c(z − z′) ≤ 0,

because T is monotone. Hence cT is convex.Assume that T is nonempty, convex and monotone. Because (T − z)+ = T+− z for every

z ∈ Z, we may (and do) assume that 0 ∈ T . In this case aff T = R+T−R+T since R+T−R+Tis linear.

Let z, z′ ∈ T , α, α′ ∈ R+. For α + α′ 6= 0 we have αα+α′ z, α′

α+α′ z′ ∈ T because T is convex

and 0 ∈ T . Since T is monotone we get

c(αz − α′z′) = (α + α′)2c( α

α + α′z − α′

α + α′z′

) ≥ 0,

and this implies that aff T is monotone.To prove that aff T ⊂ T+ first note that [1,∞)T+ ⊂ T+. Indeed for z ∈ T+, t ≥ 1,

and an arbitrary u ∈ T we have t−1u ∈ T since T is convex and 0 ∈ T . Consequently, themonotonicity of T provides us with

c(tz − u) = t2c(z − t−1u) ≥ 0,

that is, tz ∈ T+ for all z ∈ T+ and t ≥ 1.Recall that (see (10)) T monotone is equivalent to T ⊂ T+, from which [1,∞)T ⊂

[1,∞)T+ ⊂ T+. Since by the convexity of T , [0, 1]T = T , we obtain R+T ⊂ T+.It is time to show that aff T = R+T − R+T ⊂ T+. Indeed, let α, α′ ≥ 0 and z, z′ ∈ T .

Again, for arbitrary u ∈ T we have

c(αz − α′z′ − u) = (1 + α′)2c( α′

1 + α′z′ +

11 + α′

u− α

1 + α′z) ≥ 0,

because α′1+α′ z

′ + 11+α′u ∈ T and α

1+α′ z ∈ R+T ⊂ T+; hence αz − α′z′ ∈ T+.If, in addition, T is maximal monotone then either from aff T monotone or from T =

T+ = aff T we get that T is affine.

Remark 2 For T monotone and convex the sets aff T and T+ can be very different. Forexample, take T := {(0, 0)} ⊂ R×R. Then aff T = {(0, 0)} while T+ covers quadrants I andIII.

Also, we cannot expect T+ to be convex or a cone. Take T := {(x, 0) | 0 ≤ x ≤ 1} ⊂ R×R.Then T+ = {(x, y) | x ≤ 0, y ≤ 0} ∪ T ∪ {(x, y) | x ≥ 1, y ≥ 0} is neither a cone nor convexsince for example (1, 1), (0, 0) ∈ T+ and (1/2, 1/2) 6∈ T+.

However it is easily checked that T+ is a (double-) cone whenever T is a (double-) cone.

It is clear that every maximal monotone operator is dual-representable with its Fitzpatrickor Penot functions as (d-)representatives. The converse of the previous fact is currently anopen problem. In reflexive spaces dual-representable operators are maximal monotone (seee.g. [7, Theorem 3.1]). Under the uniqueness property dual-representable operators becomemaximal even in locally convex spaces.

Theorem 6 The operator T : X ⇒ X∗ is maximal monotone iff T is dual-representable andunique.

8

Proof. While the direct implication is clear (see e.g. Theorem 1 (ii)), for the converse let Tbe unique and dual-representable with a d-representative h. From (8) we have that ϕT ≤ h.For the maximality of T it suffices to prove that T is NI or T+ = [ϕT ≤ c] ⊂ [ϕT = c].

Assume by contradiction that there exists z0 ∈ T+ \ [ϕT = c] = [ϕT < c]. Then

domh ⊂ {z0}+ = {z ∈ Z | c(z − z0) ≥ 0}. (13)

Indeed, if there is a z ∈ domh (⊂ domϕT ) such that c(z − z0) < 0 then z 6∈ T+ since T+

is monotone and z0 ∈ T+. This implies that z ∈ [ϕT > c] ∩ domϕT . From the continuityof ϕT − c on the segment [z, z0] := {tz + (1 − t)z0 | 0 ≤ t ≤ 1} there exists t ∈ (0, 1) suchthat w := tz + (1 − t)z0 ∈ [ϕT = c] ⊂ T+. It follows that c(w − z0) = t2c(z − z0) < 0. Thiscontradicts the monotonicity of T+. Therefore (13) holds.

The inclusion in (13) yields that

h(z) ≥ c(z)− c(z − z0) = z · z0 − c(z0) ∀z ∈ Z, (14)

or c(z0) ≥ z · z0 − h(z), for every z ∈ Z. After passing to supremum over z ∈ Z we find thatz0 ∈ [h¤ = c] = T ⊂ [ϕT = c]. This is in contradiction with the choice of z0. Hence T is NIand consequently maximal monotone.

Corollary 7 The monotone operator T : X ⇒ X∗ admits a unique dual-representable exten-sion iff T admits a unique maximal monotone extension.

Proof. The direct implication is plain since every maximal monotone operator is dual-representable. For the converse implication let h ∈ D be such that T ⊂ [h = c] =: D. SinceT is unique, so is D, hence, according to Theorem 6, D is maximal monotone. Therefore Dis the unique maximal monotone and dual-representable extension of T .

Remark 3 Note that a unique and non-maximal monotone operator is not dual-representablewhile a representable and non-maximal monotone operator is not of NI type. Therefore aunique representable non-maximal monotone operator is neither NI nor dual-representable.

In view of Theorem 6 a path to a proof of the Rockafellar Conjecture is provided by thefollowing result.

Corollary 8 Let X,Y be Banach spaces, A : X → Y a continuous linear operator, M ∈M(X) and N ∈ M(Y ) with 0 ∈ ic(conv(domN −A(domM))). Then M +A>NA is maximalmonotone iff M + A>NA is unique. Here A> : Y ∗ → X∗ stands for the adjoint of A.

Proof. As seen in [31, Remark 1] the mentioned qualification constraint ensures the dualrepresentability of M + A>NA, a d-representative for it being given by

h¤(x, x∗) = min{f¤(x, x∗ −A>y∗) + g¤(Ax, y∗) | y∗ ∈ Y ∗}, ∀(x, x∗) ∈ X ×X∗,

where f ∈ RM , g ∈ RN . The stated equivalence follows directly from Theorem 6.

Proposition 9 Let T : X ⇒ X∗ be a monotone operator. Then T is unique and not ofNI type iff domϕT is monotone and [ϕT = c] ( domϕT . In this case [ψT = c] is uniquerepresentable and neither NI nor dual representable, and domϕT is affine and the uniquemaximal monotone extension of both T and [ψT = c].

9

Proof. For the direct implication, since T is not of NI type the set [ϕT < c] is non-empty;hence [ϕT = c] 6= domϕT . Let z0 ∈ [ϕT < c], z ∈ domϕT and zt = (1−t)z0 +tz for 0 ≤ t ≤ 1.The function [0, 1] 3 t → (ϕT − c)(zt) is continuous and limt↓0(ϕT − c)(zt) = (ϕT − c)(z0) < 0.This shows that for every z ∈ domϕT there is δ ∈ (0, 1] such that (1 − t)z0 + tz ∈ [ϕT ≤ c]for every t ∈ [0, δ].

Therefore, for z1, z2 ∈ domϕT there is λ ∈ (0, 1) such that (1−λ)z0+λz1, (1−λ)z0+λz2 ∈[ϕT ≤ c]. This yields that c(z1 − z2) ≥ 0, since T is unique or equivalently T+ = [ϕT ≤ c]is (maximal) monotone. We proved that domϕT is monotone. But T+ ⊂ domϕT and T+

is maximal monotone. Hence T+ = domϕT is maximal monotone with a convex graph thusaffine (see e.g. [3, Theorem 4.2] or Proposition 5).

Conversely, whenever domϕT is monotone we get, as above, that T+ = domϕT is theunique maximal monotone extension of T and ϕT (z) ≤ c(z) for every z ∈ domϕT . Togetherwith [ϕT = c] ( domϕT this shows that [ϕT < c] 6= ∅, i.e., T is not NI.

The remaining conclusions follow from Remark 3.

Corollary 10 Let T ⊂ Z be monotone. Then domϕT is monotone iff domϕT is maximalmonotone. In this case T is unique and domϕT is the unique maximal monotone extensionof T .

Remark 4 Since every unique non-NI operator has its unique maximal monotone extensionan affine operator it is natural to search for such an object in the class of linear monotoneoperators (see Example 19 below).

A monotone operator T is not of NI type iff T is not unique or T is unique but not NI.This allows us to characterize the NI operator class.

Proposition 11 Let T ⊂ Z be monotone. Then T is NI iff either [ϕT ≤ c] is monotone anddomϕT is non-monotone, or [ϕT = c] = domϕT .

Proof. If T is NI then T is unique, i.e., T+ = [ϕT ≤ c] is monotone. If domϕT is non-monotone then we are done while if domϕT is monotone then as previously seen [ϕT ≤ c] =[ϕT = c] = domϕT .

For the converse implication, notice that in both cases T is unique. The conclusion followsimmediately using Proposition 9.

Remark 5 For a monotone operator T ⊂ Z, the condition [ϕT = c] = domϕT is equivalent toϕT = cM , for some affine maximal monotone set M ⊂ Z. In this case M = [ϕT = c] = domϕT

is the unique maximal monotone extension of T .

In terms of its Fitzpatrick and Penot functions a monotone operator T is maximal mono-tone iff T = [ψT = c] and either [ϕT ≤ c] is monotone and domϕT is non-monotone or[ϕT = c] = domϕT .

Sections 4, 5 below deal with strengthening these results in the (skew-) linear cases.

10

4 Skew double-cones and skew linear subspaces

As in the previous section, in this section X denotes a separated locally convex space if nototherwise explicitly specified. A subset A of Z = X×X∗ is called non-negative if c(z) ≥ 0, forevery z ∈ A and skew if c(z) = 0, for every z ∈ A. Our main goal in this section is to studyskew linear subspaces and skew double-cones of Z via their Fitzpatrick and Penot functions.

An important example of monotone operator (multifunction) is provided by non-negativelinear subspaces L ⊂ Z = X ×X∗, that is, L is a linear space with c(z) ≥ 0 for every z ∈ L(one obtains immediately that L is monotone) and its particular case formed by skew linearsubspaces of Z.

For A ⊂ Z we set

−A := {(x, x∗) ∈ Z | (x,−x∗) ∈ A},A⊥ := {z′ ∈ Z | 〈z, z′〉 = 0, ∀z ∈ A},

andA0 := {z ∈ A⊥ | c(z) = 0} = A⊥ ∩ [c = 0]. (15)

It is obvious that A⊥ = (linω×ω∗A)⊥, A⊥⊥ = linω×ω∗

A, A⊥ is a ω×ω∗−closed linear subspaceof Z, A0 is a double-cone, and linA = aff A = conv A whenever A is a double-cone. Recallthat ω denotes the weak topology on X and ω∗ denotes the weak-star topology on X∗. Also,note that A ⊂ A⊥ implies that A is skew.

Proposition 12 Let D ⊂ Z be a skew double-cone. Then(i) ϕD = ιD⊥ and ψD = ι

linω×ω∗

D;

(ii) z is m.r.t. D iff z ∈ D⊥ and c(z) ≥ 0, or equivalently

D+ = [ϕD ≤ c] = {z ∈ D⊥ | c(z) ≥ 0}; (16)

(iii) [ψD = c] = {z ∈ linω×ω∗D | c(z) = 0} = (D⊥)0;

Proof. (i) Since D is a skew double-cone we have

ϕD(z) = sup{z · z′ − c(z′) | z′ ∈ D} = sup{z · z′ | z′ ∈ D} = ιD⊥ .

Assertions (ii) and (iii) follow from the expressions of ϕD and ψD in (i) and (15).

In the next result we provide a characterization of skew monotone double-cones.

Proposition 13 Let D ⊂ Z be a double-cone. TFAE: (a) D is skew and monotone, (b) c(z+z′) = 0 for all z, z′ ∈ D, (c) D ⊂ D⊥, (d) linD is a skew linear space, (e) linω×ω∗

D is a skewlinear space.

In particular a linear subspace S ⊂ Z is skew iff S ⊂ S⊥; moreover clω×ω∗ S is skew, forevery skew linear subspace S ⊂ Z.

Proof. Set L := linω×ω∗D; of course, L is a linear subspace. The implications (e) ⇒ (d) ⇒

(b) ⇒ (a) are obvious.(a) ⇒ (b) Because D is skew and monotone, by Proposition 12 we have ιL = ψD ≥ c;

hence c(z) ≤ 0 for every z ∈ L. Take z, z′ ∈ D ⊂ L; it follows that z + z′ ∈ L, and soc(z +z′) = c(z)+z ·z′+ c(z′) = z ·z′ ≤ 0. Since −z ∈ D we get z ·z′ = 0, and so c(z +z′) = 0.

11

(b) ⇒ (c) In the proof of (a) ⇒ (b) we obtained that z · z′ = 0 for all z, z′ ∈ D, and soD ⊂ D⊥.

(c) ⇒ (b) We have that z · z′ = 0 for all z, z′ ∈ D. Taking z′ = z we get c(z) = 0 forz ∈ D. Then c(z + z′) = c(z) + z · z′ + c(z′) = z · z′ = 0 for all z, z′ ∈ D.

(b) ⇒ (e). Since D is a double-cone, from (b) ⇒ (a) we have that D and −D are skew andmonotone. As in the proof of (a) ⇒ (b) we obtain that c(z) ≤ 0 for all z ∈ L and c(z′) ≤ 0for all z′ ∈ −L; therefore, c(z) = 0 for every z ∈ L, that is, L is skew. The proof is complete.

Corollary 14 Let D ⊂ Z be a skew monotone double-cone. Then L := linω×ω∗D is a skew

ω × ω∗–closed linear space, ψD = ψL = ιL, ϕD = ϕL = ιL⊥ and [ψD = c] = domψD = L. Inparticular, D is representable iff D = L, that is, D is linear and ω × ω∗–closed.

Proof. Because D⊥ = L⊥, using Proposition 12 (i) we obtain that ψD = ψL = ιL andϕD = ϕL = ιL⊥ . Moreover, by Proposition 13 we have that L is a skew linear space. Itfollows that L ⊂ [ψL = c] = [ψD = c] ⊂ domψD = domψL = L. Hence L = [ψL = c], and soL is representable. If D is representable then D = [ψD = c] = L. The conclusion follows.

An immediate consequence of the previous result is that the affine hull of a skew monotonedouble-cone remains skew (and implicitly monotone). In section 5 we will see that there existsa monotone double-cone whose affine hull is not monotone. Moreover, Proposition 13 andCorollary 14 show that when dealing with skew monotone double-cones, we may assumewithout loss of generality that they are linear spaces, even ω × ω∗–closed linear spaces. Forthis reason in the sequel, in this section, we work only with skew linear spaces.

Proposition 15 Let S ⊂ Z be a skew linear space. TFAE: (a) S is NI, (b) clω×ω∗ S is NI,(c) clω×ω∗ S is maximal monotone, (d) S has a unique representable extension, (e) −S⊥ ismonotone.

In this case

cl ω×ω∗S = [ψS = c] = [ϕS ≤ c] = [ϕS = c] = {z ∈ S⊥ | c(z) = 0} = S0, (17)

S, −S, − clω×ω∗ S are unique in Z, clω×ω∗ S is the unique maximal monotone extension andthe unique representable extension of S in Z, and −S⊥ is the unique maximal monotoneextension, as well as the unique dual-representable extension of −S and of − clω×ω∗ S in Z.

Proof. By Corollary 14 we have that L := cl ω×ω∗S = [ψS = c]. The equivalence of conditions(a), (b) and (d) follows from Proposition 4 (ii), while the equivalence of (b) and (e) followsimmediately from the relation ϕS = ϕL = ιL⊥ = ιS⊥ mentioned in Corollary 14. In this case,from Proposition 4 (iii) and Corollary 14, we find (17).

Assume now that S is NI. By Proposition 4 (iii) S is unique with L its unique maximalmonotone extension and unique representable extension.

Since S is NI we have that −S⊥ = −L⊥ is monotone. Because S is skew, so is −S, and so(−S)+ = [ϕ−S ≤ c] = [ϕ−L ≤ c] = [ι−L⊥ ≤ c] = −L⊥ is monotone. According to Proposition3 and Corollary 7, −L⊥ is the unique maximal monotone extension and the unique dualrepresentable extension of −S and −L in Z.

Direct consequences of the previous result follow.

12

Corollary 16 Let S ⊂ Z be a skew linear space such that S⊥ is skew. Then S is NI,cl ω×ω∗S = S⊥ and S⊥ is the unique maximal monotone extension of S in Z.

Proposition 17 Let S ⊂ Z be a skew linear space and S0 := [ϕS = c]. TFAE: (i) S isunique, (ii) S0 is monotone, (iii) S⊥ is monotone or −S⊥ is monotone, (iv) S0 is linear,(v) S0 is convex, (vi) cl ω×ω∗S is unique.

In this case S0 = cl ω×ω∗S.

Proof. By Corollary 14 we have that L := cl ω×ω∗S is skew and ϕS = ϕL; hence (i) ⇔ (vi).Moreover, the implication (iv) ⇒ (v) is obvious.

(i) ⇒ (ii) According to Proposition 3 it is clear that S unique implies that S0 (⊂ [ϕS ≤ c])is monotone.

(ii) ⇒ (iii) For this assume that S⊥ and −S⊥ are not monotone. Then there exist z, z′ ∈S⊥ such that c(z′) < 0 < c(z). For t ∈ R and η(t) := c(z + tz′) = c(z) + t〈z, z′〉+ t2c(z′) wehave that η(0) > 0 and lim|t|→∞ η(t) = −∞. Therefore there exist distinct t1, t2 ∈ R such thatη(t1) = η(t2) = 0, and so z1 := z+t1z

′, z2 := z+t2z′ ∈ S0. Since c(z1−z2) = (t1−t2)2c(z′) < 0

we see that S0 is not monotone.The implication (iii) ⇒ (i) is straightforward from the second part of Proposition 15

applied for S or −S.(iii) ⇒ (iv) If −S⊥ is monotone, using by Proposition 15 we obtain that S is NI and

S0 = L (see (17)). Similarly, if S⊥ is monotone then −S0 = (−S)0 = −L. Hence in bothcases S0 = L, and so S0 is linear.

(v) ⇒ (ii) If S0 is convex then ιS0 is convex, ιS0 ≥ c, and S0 = [ιS0 = c]; hence S0 ismonotone. The proof is complete.

Proposition 18 Let S ⊂ Z be a skew linear space which is not unique. Then S0 = [ϕS = c]is an NI double-cone which is neither monotone nor convex, S⊥ and −S⊥ are not monotone,cl ω×ω∗S is not unique, (S0)⊥ is skew, convω×ω∗S0 = S⊥, and

S++ = (cl ω×ω∗S)++ = (S0)⊥ = cl ω×ω∗S ( S0 ( S+. (18)

Proof. Because ϕS = ιS⊥ and S0 = [ϕS = c], it is clear that S0 is a skew double-cone.Because S ⊂ Z is a skew linear space which is not unique, by the preceding proposition, weknow that L := cl ω×ω∗S is not unique, S0 is not monotone or convex, and S⊥, −S⊥ are notmonotone. Because S ⊂ [ψS = c] ⊂ [ϕS = c] = S0 ⊂ S⊥ we get L = S⊥⊥ ⊂ (S0)⊥ ⊂ S⊥.Since S0 is a skew double-cone we have ϕS0 = ι(S0)⊥ .

Assume, by contradiction, that S0 is not NI. Then, there is z ∈ [ϕS0 < c], that is,z ∈ (S0)⊥ ⊂ S⊥ with c(z) > 0. Clearly z is m.r.t. S0. Because S⊥ is not monotone there isz′ ∈ S⊥ such that c(z′) < 0. Let η(t) := c(z + tz′) for t ∈ R. Note that η(0) = c(z) > 0 andlim|t|→∞ η(t) = −∞. Therefore there is t0 6= 0 such that η(t0) = 0, i.e., z0 := z + t0z

′ ∈ S0.Since z is m.r.t. S0, we get the contradiction c(z0 − z) = t20c(z

′) ≥ 0. Therefore S0 is NI andthis translates into ϕS0 ≥ c or −(S0)⊥ is monotone.

From the equivalence of (i) and (iii) in Proposition 17 we have that −S is not unique, andso, by the above argument, −((−S)0)⊥ = (S0)⊥ is monotone. Hence (S0)⊥ is skew.

From Corollary 14 and (3) we have

S ⊂ L = [ψS = c] ⊂ [ϕS = c] = S0 ⊂ S+.

13

This leads to (see Lemma 2 (i))

ιL = ψS ≥ ϕS+ ≥ ϕS0 = ι(S0)⊥ , (19)

whence L ⊂ (S0)⊥ ⊂ S⊥.Assume that z ∈ (S0)⊥ \L. By a separation theorem, there is w ∈ S⊥ such that z ·w 6= 0.

Since z ∈ (S0)⊥ (⊂ S⊥) we know that w 6∈ S0, whence c(w) 6= 0. Notice that, since (S0)⊥ isskew, c(z) = 0 and so

c(z + λ0w) = λ0z · w + λ20c(w) = 0, for λ0 := − 1

c(w)z · w 6= 0,

that is z +λ0w ∈ S0. From z ∈ (S0)⊥ we get the contradiction 0 = z · (z +λ0w) = λ0z ·w 6= 0.Therefore L = (S0)⊥, and by (19), this yields ψS = ϕS+ = ιL. Since L is skew and S+ is NIby Lemma 2 (ii), we deduce S++ = [ϕS+ ≤ c] = [ϕS+ = c] = L. By Proposition 4 (i) we haveS+ = [ψS = c]+ = L+, and so S++ = L++.

The strict inclusion L ( S0 comes from the fact that L is a skew subspace while S0

is not monotone, while the strict inclusion S0 ( S+ comes from the fact that −S⊥ is notmonotone. The other conclusions follow from Proposition 17 while from (S0)⊥ = cl ω×ω∗S wefind convω×ω∗S0 = (S0)⊥⊥ = S⊥.

We comprise some of the information on a skew linear space (or skew monotone double-cone) S ⊂ Z in the following chart:

1. S is representable iff S is linear and ω × ω∗−closed2. S is NI iff −S⊥ is monotone3. S is unique iff S⊥ is monotone or −S⊥ is monotone4. S is unique and not NI iff S⊥ is monotone and non-skew5. S is maximal monotone iff S is linear and ω × ω∗−closed, and −S⊥ is monotone

These results on skew linear subspaces prefigure the results we will obtain in Section 5 forlinear multifunctions.

In the preceding results the topology ω × ω∗ on Z can be replaced by any locally convextopology σ compatible with the natural duality (Z,Z), that is, for which (Z, σ)∗ = Z, forexample the topology τ × ω∗, where τ is the initial topology of X.

Example 19 In [10, page 89] Gossez considered the linear operator T : `1 → `∞ defined byT ((xn)n≥1) := (yn)n≥1 with yn :=

∑k≥1 xk +xn−2

∑nk=1 xk. The operator T is skew, that is,

〈x, Tx〉 = 0 for every x ∈ `1, or equivalently, 〈x, Ty〉 = −〈y, Tx〉 for all x, y ∈ `1 (in the dualityof `1 × `∞). In fact for x ∈ `1 we have that Tx ∈ c, the subspace of convergent sequencesof `∞. Indeed, limTx = −∑

k≥1 xk = −〈x, e〉, where en := 1 for n ≥ 1. Consequently,T1x := Tx + 〈x, e〉 e ∈ c0 for every x ∈ `1. From Proposition 23 below it follows that both{(T1x, x) | x ∈ `1} ⊂ c0 × `1 and {(x, T1x) | x ∈ `1} ⊂ `1 × `∞ are maximal monotone in thecorresponding spaces.

Consider

S := {(Tx, x) | x ∈ `1, 〈x, e〉 = 0} ⊂ c0 × `1, (20)R := −S = {(−Tx, x) | x ∈ `1, 〈x, e〉 = 0} . (21)

14

Let us first determine S⊥ (in c0×`1). Consider (u, v) ∈ S⊥. Then 〈u, x〉c0×`1+〈Tx, v〉c0×`1

= 0for every x ∈ `1 with 〈x, e〉 = 0. Since 〈u, x〉c0×`1

= 〈x, u〉 and 〈Tx, v〉c0×`1= 〈v, Tx〉 =

−〈x, Tv〉, we obtain that

[x ∈ `1, 〈x, e〉 = 0] ⇒ 〈x, u− Tv〉 = 0,

and so necessarily u − Tv = γe, that is, u = Tv + γe, for some γ ∈ R. Since u ∈ c0, itfollows that γ = 〈v, e〉. Hence S⊥ ⊂ {(Tv + 〈v, e〉 e, v) | v ∈ `1}. Conversely, if v, x ∈ `1 and〈x, e〉 = 0 then Tx ∈ c0, Tv + 〈v, e〉 e ∈ c0 and

〈Tx, v〉c0×`1+ 〈Tv + 〈v, e〉 e, x〉c0×`1

= 〈v, Tx〉+ 〈x, Tv〉+ 〈v, e〉 〈x, e〉 = 0,

which proves thatS⊥ = {(Tv + 〈v, e〉 e, v) | v ∈ `1} .

Moreover, S = S⊥⊥. Indeed, because S ⊂ S⊥⊥, let us prove the converse inclusion.Consider (y, x) ∈ S⊥⊥ (⊂ c0 × `1). Then

0 = 〈y, v〉c0×`1+ 〈Tv + 〈v, e〉 e, x〉c0×`1

= 〈v, y〉+ 〈x, Tv + 〈v, e〉 e〉 = 〈y − Tx + 〈x, e〉 e, v〉

for every v ∈ `1, and so y − Tx + 〈x, e〉 e = 0. Taking the limit we get 2 〈x, e〉 = 0, and soy = Tx. Therefore, (y, x) ∈ S, and so S = clω×ω∗ S = S⊥⊥. Since R = −S, we obtain thatR = clω×ω∗ R = R⊥⊥, too.

We know that ϕS = ιS⊥ and ϕR = ιR⊥ = ι−S⊥ . Because 〈Tv + 〈v, e〉 e, v〉 = 〈v, e〉2 and〈v, e〉2 > 0 for certain v, we get that S⊥ is monotone and non-skew. Therefore, according toPropositions 15 and 17, S (= clω×ω∗ S) is representable, unique and not NI with S+ (= S⊥ =domϕS) its unique maximal monotone extension; moreover, S is not dual-representable andS = [ϕS = c] = [ιS⊥ = c].

On the other hand, because −R⊥ = S⊥ is monotone, by Proposition 15 we have that R(= clω×ω∗ R) is maximal monotone.

Example 20 Let us consider the sets S, R defined in (20) and (21) as subsets of `∞× `1, `∞being endowed with the weak∗-topology ω∗ and `1 with the weak topology ω. Then S and Rare skew linear subspaces. The calculus above shows that S⊥ = {(Tv + γe, v) | v ∈ `1, γ ∈ R}.Then one obtains immediately that S⊥⊥ = S. Hence S = clω×ω∗ S = S⊥⊥ and R =clω×ω∗ R = R⊥⊥. It follows that S and R are representable (because S = clω×ω∗ S andR = clω×ω∗ R), not unique (because S⊥ and −S⊥ are not monotone), and consequently, notNI.

Remark 6 Because R is maximal monotone in c0× `1 but it is not NI in `∞× `1, it furnishesan example of maximal monotone operator which is not NI in the sense of Simons.

5 Monotone double-cones and monotone linear subspaces

In the beginning of this section we investigate monotone operators that admit affine maximalmonotone extensions.

Lemma 21 Every affine monotone subset of X × X∗ admits an affine maximal monotoneextension.

15

Proof. Let L0 be affine monotone in Z. Without loss of generality we assume that 0 ∈ L0.Consider

L = {L ⊂ Z | L0 ⊂ L, L is a linear monotone subspace}ordered by inclusion. Every chain {Li}i in L admits ∪iLi as an upper bound in L. By theZorn Lemma there exists a maximal element in L denoted by L.

Let z ∈ L+ and L′ = L + Rz. Clearly, L′ is linear and L′ ⊃ L0. Moreover, for everyu ∈ L and every α ∈ R we have that c(u + αz) = c(u) ≥ 0 if α = 0 and c(u + αz) =α2c(z − α−1(−u)) ≥ 0 if α 6= 0 since z ∈ L+ and α−1(−u) ∈ L. Therefore L′ ∈ L, and soL′ = L, whence z ∈ L. This proves that L is maximal monotone.

Theorem 22 Every convex monotone subset of X×X∗ admits an affine maximal monotoneextension.

Proof. Let T be convex monotone in X × X∗. According to Proposition 5, aff T is (affineand) monotone thus, by the previous lemma, it admits an affine maximal monotone extensionthat is also an extension for T .

The possibility of finding closed forms for the Fitzpatrick and Penot functions allowedus in the previous section to provide characterizations of the main classes of skew operators.Similar ideas are used in this section for the multi-valued linear monotone case; the goal beingto offer criteria for a linear monotone operator to belong to a certain class in terms of itsPenot and Fitzpatrick functions or through the monotonicity of sets directly associated tothe operator via decomposition and orthogonality.

An important example of linear monotone subspace is that of non-negative single-valuedlinear operator, that is, A : domA = X → X∗ with 〈x,Ax〉 ≥ 0 for every x ∈ X; if 〈x,Ax〉 = 0for every x ∈ X then A is skew. Of course, the linear operator A : X → X∗ is skew iff A and−A are non-negative. The next two results refer to such situations.

Proposition 23 (i) If A : X → X∗ is a non-negative linear operator then A is maximalmonotone.

(ii) If A′ : X∗ → X is a non-negative linear operator, that is 〈A′x∗, x∗〉 ≥ 0 for everyx∗ ∈ X∗, then A′ is maximal monotone.

(iii) Assume that X is a Banach space and let A : X → X∗ and A′ : X∗ → X be non-negative operators. Then A and A′ are continuous (for X and X∗ endowed with the normtopologies).

Proof. (i) Assume that A : X → X∗ is a non-negative linear operator and let (u, u∗) ∈ X×X∗

be m.r.t. A. Then 〈u− x, u∗ −Ax〉 ≥ 0 for x ∈ X. Taking x = u − ty with t > 0, y ∈ Xwe get 〈y, u∗ −Au + tAy〉 ≥ 0. Letting t → 0, then replacing y by −y we obtain that〈y, u∗ −Au〉 = 0 for every y ∈ X. Hence u∗ = Au, which shows that A is maximal monotone.

(ii) Apply (i) for A replaced by A′ (or use a similar argument).(iii) Assume now that X is a Banach space. Since A is maximal monotone, then gph A is

strongly closed and so A is continuous. Similarly for A′.

Note that the graph of a non-negative linear operator A : X → X∗ is a non-negative linearsubspace of Z and the graph of a skew linear operator A : X → X∗ is a skew subspace of Z.

16

Proposition 24 Let S ⊂ Z be a skew linear subspace. Then PrX(S) = X iff S is thegraph of a skew linear operator A : X → X∗. Hence if PrX(S) = X then S = S⊥ (inparticular S is ω × ω∗-closed) and S is maximal monotone. Similarly, PrX∗(S) = X∗ iffS−1 := {(x∗, x) | (x, x∗) ∈ S} is the graph of a skew linear operator A′ : X∗ → X. Hence ifPrX∗(S) = X∗ then S = S⊥ and S is maximal monotone.

Proof. Assume that PrX(S) = X. Let u∗ ∈ X∗ be such that (0, u∗) ∈ S. For every x ∈ Xthere exists x∗ ∈ X∗ with (x, x∗) ∈ S. Because S is a linear subspace, (x, x∗+u∗) ∈ S. Hence0 = 〈x, x∗ + u∗〉 = 〈x, x∗〉 + 〈x, u∗〉 = 〈x, u∗〉. Since x ∈ X is arbitrary, we get u∗ = 0. Itfollows that, for every x ∈ X, S(x) = {Ax} is a singleton. The operator A obtained in thisway is a skew linear operator. From Proposition 23 we obtain that A is maximal monotone,and so S is maximal monotone. Similarly since PrX(−S) = X, we know that −S is maximalmonotone, therefore −S is NI. From Proposition 15 we have that −(−S)⊥ = S⊥ is (theunique) maximal monotone extension of S, and so S = S⊥.

Applying the previous result for S−1 in the case PrX∗(S) = X∗ or repeating the aboveargument we get the last assertion.

To a subset A ⊂ Z we associate its

• skew part SA := Skew(A) := {z ∈ A | c(z) = 0} = A ∩ [c = 0],

• positive part PA := Pos(A) := {z ∈ A | c(z) > 0} = A ∩ [c > 0],

• negative part NA := Neg(A) := {z ∈ A | c(z) < 0} = A ∩ [c < 0],

• unitary part UA := Unit(A) := {z ∈ A | c(z) = 1} = A ∩ [c = 1],

• crown CA := Crown(A) = convω×ω∗UA;

clearly A = SA ∪ PA ∪NA.We also use the notation U = [c = 1], P = [c > 0], S = [c = 0], and introduce the map

ζ : P→ Z, ζ(z) = z/√

c(z) (z ∈ P). (22)

Note that c(ζ(z)) = 1 for every z ∈ P, whence ζ(P) = U, ζ(z) = z for every z ∈ U, andζ(tz) = ζ(z) for all t ∈ R∗ := R \ {0} and z ∈ P.

Whenever D is a double-cone, RSD = SD, R∗PD = PD = (0,∞)UD = R∗UD, R∗ND =ND. Our analysis is based on the study of the Fitzpatrick and Penot functions for double-cones. Note that it is readily seen that whenever D is a double-cone with a nonempty negativepart ND = D ∩ [c < 0] its Fitzpatrick function is identically equal to +∞ while its Penotfunction is improper; more precisely ψD(z) = −∞ for z ∈ convω×ω∗D and ψD(z) = +∞ forz ∈ Z \ convω×ω∗D. Indeed, taking z0 ∈ ND and t ∈ R we have that

ϕD(z) = sup{z · z′ − c(z′) | z′ ∈ D} ≥ sup{tz · z0 + t2(−c(z0)) | t ∈ R} = ∞,

for all z ∈ Z. Hence ϕL = ∞ and ϕ¤L = −∞.

That is why, from this point of view, it is natural to study double-cones D that are non-negative (i.e., with an empty negative part D ∩ [c < 0] or equivalently c(z) ≥ 0, for everyz ∈ D); in particular our main interests lie in studying monotone double-cones or linear

17

operators. It is clear that two non-negative double-cones coincide if they have the same skewand unitary parts.

Recall that the support function to A ⊂ Z is given by σA(z) = supu∈A z · u for z ∈ Z,while barA := domσA denotes the barrier cone of A, and barA = bar(convω×ω∗A). Byconvention sup ∅ := −∞ and inf ∅ := +∞; hence σ∅ = −∞. Moreover, the Minkowskifunctional associated to A ⊂ Z is given by pA(z) = inf{t > 0 | z ∈ tA}. It is well-knownthat, when A is a convex set containing 0, pA is a sublinear function whose domain is R+A;moreover, if A is symmetric then pA is even, and so pA is a semi-norm when restricted to itsdomain which is a linear space. When A is a symmetric closed convex set then tA = [pA ≤ t]for every t > 0 and [pA = 0] = A∞; in particular pA is lsc. Recall that the asymptotic coneof the nonempty closed convex set C ⊂ Z is C∞ = ∩t>0t(C − c) for some (every) c ∈ C andthe polar of C is C◦ := [σC ≤ 1]. Of course, C∞ is a closed convex cone; C∞ is a linear spacewhen C is symmetric. Recall also that D0 := D⊥ ∩ [c = 0] = Skew(D⊥). We also use theconventions 0 · (−∞) := 0 and 0 · (+∞) := +∞; therefore 0f = ιdom f .

Proposition 25 Let D ⊂ Z be a non-negative double-cone with skew part S, non-emptypositive part P , unitary part U , and crown C. Then

(i) ϕD = 14σ2

C + ιS⊥, or in extended form:

ϕD(z) = supw∈P

|z · w|24c(w)

if z ∈ S⊥, ϕD(z) = ∞ otherwise.

(ii) domϕD = S⊥ ∩ barC, and so domϕD is a linear subspace of Z.(iii) ϕD(tz) = t2ϕD(z), for all z ∈ Z and t ∈ R.(iv) ψD = clω×ω∗ [p2

C¤ι∗S⊥ ], where (f¤g)(z) := infw∈Z [f(w) + g(z − w)] for z ∈ Z.

(v) ψD(tz) = t2ψD(z), for all z ∈ Z and t ∈ R, domψD is a linear space, and lin D ⊂domψD ⊂ linω×ω∗

D.(vi) [ϕD = 0] = D⊥, [σC = 0] = C⊥ = U⊥ = P⊥, D⊥ = S⊥ ∩ C⊥.(vii) D+ = D0 ∪

(S⊥ ∩ ζ−1(2C◦)

)with Skew(D+) = D0 := Skew(D⊥), Unit(D+) =

(2C◦) ∩ S⊥ ∩ U and Pos(D+) = S⊥ ∩ ζ−1(2C◦).

Proof. (i) Let z ∈ Z. Then

ϕD(z) = sup{z · z′ − c(z′) | z′ ∈ S ∪ RU

}

= max(sup

{z · z′ | z′ ∈ S

}, sup

{λz · z′ − λ2 | z′ ∈ U, λ ∈ R})

= max(ιS⊥(z), sup

{14(z · z′)2 | z′ ∈ U

})= max

(ιS⊥(z), 1

4σ2U (z)

)

= ιS⊥(z) + 14σ2

U (z) = ιS⊥(z) + 14σ2

C(z).

The first part in (ii) is obvious, while the linearity of domϕD and (iii) follow from the factthat σU is an extended seminorm (since U = −U).

(iv) Let us determine ψD. Because ϕD = 14σ2

C + ιS⊥ , and σ2C , ιS⊥ are proper (ω×ω∗−) lsc

convex functions with 0 ∈ domσ2C ∩ dom ιS⊥ , we have that ψD = ϕ¤

D = clω×ω∗ [(14σ2

C)∗¤ι∗S⊥ ].

Clearly, ι∗S⊥ = ι

linω×ω∗

S).

The crown C is a symmetric set because so is U . Note that σ∗U = σ∗C = ιC (see [32,Theorem 2.4.14]). Let us determine (1

4σ2C)∗. To do this we apply [32, Theorem 2.8.10 (iii)]

for X replaced by X × X∗, Y = R, Q = R+, f := 0, g(t) := 0 for t < 0, g(t) := 14 t2 for

t ≥ 0 and H := σC ; clearly g is Q–increasing, convex and continuous, hence continuous at

18

any x0 ∈ dom f ∩H−1(dom g) = domσC . Moreover, g∗(s) = ∞ for s < 0 and g∗(s) = s2 fors ≥ 0. Applying [32, Theorem 2.8.10 (iii)] we obtain that

(14σ2

C)∗(z) = min {(sσC)∗(z) + g∗(s) | s ≥ 0}= min

{(0σC)∗(z), inf

{s(σC)∗(s−1z) + s2 | s > 0

}}

= min{(ιdom σC

)∗(z), inf{sιC(s−1z) + s2 | s > 0

}}.

Butinf

{sιC(s−1z) + s2 | s > 0

}= inf

{s2 | s > 0, (z) ∈ sC

}= (pC(z))2 ,

where pC is the Minkowski functional associated to C; pC is a seminorm on the linear spacedom pC = R+C. Because for f ∈ Γ(E) one has f∞ = σdom f∗ , we have that clω∗(dom f∗) =∂f∞(0) (see e.g. [32, Exercise 2.23]). Taking f = ιC , we obtain that clω×ω∗(domσC) =∂ιC∞(0) = (C∞)0. Hence (ιdom σC

)∗ = ι(dom σC)0 = ιC∞ . Since 0 ∈ C we have that C =C + C∞ ⊃ C∞, and so (pC(z))2 = 0 = (ιdom σC

)∗(z) for (z) ∈ C∞. Therefore,

(14σ2

C)∗(z) = (pC(z))2 ∀z ∈ Z.

It follows that ψD = (clω×ω∗ µ)2, where µ := pC¤ιL with L := linω×ω∗S.

(v) Because pC and ιL are symmetric sublinear functionals, so is µ; hence clω×ω∗ µ is asymmetric sublinear functional, too. It follows that domψD = domµ is a linear space. Thementioned inclusions follow from the fact that ψD = convω×ω∗cD ≤ cD, and so

D ⊂ [ψD = c] ⊂ domψD = dom(clω×ω∗cD) ⊂ convω×ω∗(dom cD) = convω×ω∗D.

(vi) Since D is non-negative and 0 ∈ D, clearly D⊥ ⊂ [ϕD = 0]. Let z ∈ [ϕD = 0]; thenz · z′ ≤ c(z′) for every z′ ∈ D, whence tz · z′ = z · (tz′) ≤ c(tz′) = t2c(z′) for all t ∈ R andz′ ∈ D. Dividing by t > 0, then by t < 0, and letting t → 0 we get z · z′ = 0 for z′ ∈ D,and so z ∈ D⊥. Hence D⊥ = [ϕD = 0]. A simpler argument (using the symmetry of C)gives [σC = 0] = C⊥. It is obvious that P⊥ = U⊥ = C⊥. Thus, from D = S ∪ P we getD⊥ = S⊥ ∩ P⊥ = S⊥ ∩ C⊥.

(vii) We know that D+ = [ϕD ≤ c] = S⊥∩ [σ2C ≤ 4c] is a non-negative double-cone. Hence

Skew([σ2C ≤ 4c]) = [σC = 0] ∩ [c = 0] = C⊥ ∩ [c = 0]; therefore Skew(D+) = S⊥ ∩ P⊥ ∩ [c =

0] = Skew(D⊥) =: D0 while Pos([σ2C ≤ 4c]) = ζ−1(2C◦). The proof is complete.

Let us call a double-cone D positive if D is non-negative and Skew(D) = {0}. In this caseϕD = 1

4σ2C , ψD = p2

C , where C = Crown(D), and Unit(D+) = (2C◦)∩U, D+ = D0∪ζ−1(2C◦).Therefore a positive double-cone D is monotone iff D ⊂ D+ iff C ⊂ 2C◦.

Proposition 26 Let D ⊂ Z be a non-negative double-cone with skew part S, non-emptypositive part P , unitary part U , and crown C. TFAE:

(a) D is monotone,(b) 〈z, z′〉2 ≤ 4c(z) · c(z′) for all z, z′ ∈ D,(c) S ⊂ D0 and U ⊂ S⊥ ∩ (2C◦),(d) C ⊂ S⊥ ∩ (2C◦) and S ⊂ S⊥.In this case S is a skew monotone double-cone, D ⊂ S⊥, and convω×ω∗S ⊂ D⊥.

19

Proof. (a) ⇔ (b) Assume that D is monotone and fix z, z′ ∈ D. Then tz ∈ D for t ∈ R, andso

c(tz − z′) = t2c(z)− t⟨z, z′

⟩+ c(z′) ≥ 0 ∀t ∈ R,

which is equivalent to c(z), c(z′) ≥ 0 and 〈z, z′〉2 ≤ 4c(z) · c(z′).(a) ⇔ (c) Both D and D+ are non-negative double-cones with Skew(D+) = D0 and

Pos(D+) = ζ−1(2C◦) ∩ S⊥. Clearly, D is monotone iff D ⊂ D+ iff S ⊂ Skew(D+) andU ⊂ Pos(D+). Note that U ⊂ ζ−1(2C◦) ∩ S⊥ iff U ⊂ S⊥ ∩ 2C◦.

(c) ⇔ (d) The direct implication is obvious. For the converse we have to prove thatS ⊂ D0, or equivalently S ⊂ D⊥ (= S⊥ ∩ P⊥). First observe that P⊥ = U⊥ = C⊥. SinceC ⊂ S⊥, we have that S ⊂ S⊥⊥ ⊂ C⊥ = P⊥. Since by our hypothesis we have S ⊂ S⊥, weget S ⊂ D⊥.

The inclusion D ⊂ S⊥ (⇔ S ⊂ D⊥) follows from (b), while from S ⊂ D⊥ we getconvω×ω∗S ⊂ D⊥.

Remark 7 It is interesting to observe that if D ⊂ Z is a monotone double-cone, aff L could benon-monotone. For this claim consider X := R2 identified with its dual, z1 := ((0, 1), (1, 1)),z2 := ((1, 1), (1, 0)), z3 := ((1, 0), (1, 0)) and di := Rzi. We have that c(z1) = c(z2) =c(z3) = 1 ≥ 0; moreover 〈z1, z2〉 = 〈z2, z3〉 = 2, 〈z1, z3〉 = 1, and so 〈z1, z2〉2 ≤ 4c(z1) · c(z2),〈z2, z3〉2 ≤ 4c(z2) · c(z3), 〈z1, z3〉2 ≤ 4c(z1) · c(z3). It follows that D := ∪3

i=1Rzi is a monotonedouble-cone. However, z := z1 − 2z2 + z3 = ((−1,−1), (0, 1)) ∈ aff D and c(z) = −1 < 0.Hence aff L is not monotone.

It is clear that a linear subspace L is monotone iff L is non-negative iff L = SL ∪ PL.

Theorem 27 Let L ⊂ Z be linear monotone subspace with skew part S, nonempty positivepart P , unitary part U , and crown C. Then

(i) S is a skew linear subspace of Z.(ii) S + U = U , S + P = P and S ⊂ conv U ⊂ C; in particular S ⊂ C∞, and so

barU = barC ⊂ S⊥.(iii) conv U = {z ∈ L | c(z) ≤ 1} = [cL ≤ 1].(iv) ϕL = 1

4σ2C , domϕL = barC ⊂ S⊥ is a linear subspace of Z and ϕL(tz) = t2ϕL(z),

for all z ∈ Z and t ∈ R.(v) ψL = p2

C , C = [ψL ≤ 1], C∞ = [ψL = 0], ψL(tz) = t2ψL(z) for all z ∈ Z and t ∈ R,domψL = RC is a linear space, and L ⊂ domψL ⊂ clω×ω∗ L.

(vi) −C∞ is a ω × ω∗-closed monotone linear subspace.(vii) If, moreover, L is ω × ω∗-closed then C∞ = S; in particular C∞ is skew and S is

ω × ω∗-closed.

Proof. (i) From Proposition 26 we have that L ⊂ S⊥. Consider z, z′ ∈ S; then z + z′ ∈ Land c(z + z′) = c(z) + z · z′ + c(z′) = 0, and so z + z′ ∈ S. Since S is a double-cone, it followsthat S is linear.

(ii) Because 0 ∈ S, clearly U ⊂ S + U . Let z0 ∈ S and z ∈ U . Since z ∈ L ⊂ S⊥, we havethat c(z0 + z) = c(z0) + z0 · z + c(z) = c(z) = 1; hence z0 + z ∈ U . It follows that S + U ⊂ U ,whence S + U = U . Similarly, S + P = P .

Fix z1 ∈ U ; hence −z1 ∈ U . Then for z0 ∈ S we have that z0 + z1, z0 − z1 ∈ U , and soz0 = 1

2(z0 + z1)+ 12(z0− z1) ∈ conv U . Hence S ⊂ conv U ⊂ C. Since C is a symmetric closed

convex set we have that C∞ is a linear space and S ⊂ C∞.

20

As seen in the proof of Proposition 25(iv), we have that cl(barC) = (C∞)0. Because C∞is a linear space we have that (C∞)0 = (C∞)⊥, and so barU = barC ⊂ (C∞)⊥ ⊂ S⊥.

(iii) Let z ∈ L be such that t := c(z) ≤ 1. If t = 0 then z ∈ S ⊂ conv U . If t > 0 thenz′ := t−1/2z ∈ U ⊂ conv U , and so z = (1− t1/2)0+ t1/2z′ ∈ conv U . Hence [cL ≤ 1] ⊂ conv U .Since cL is convex and U ⊂ [cL ≤ 1] we also have conv U ⊂ [cL ≤ 1].

(iv) & (v) From (ii) we see that domσC ⊂ S⊥; hence Proposition 25 gives ϕL = 14σ2

C +ιS⊥ = 1

4σ2C , and so ψL = p2

C , [ψL ≤ 1] = [pC ≤ 1] = C, [ψL = 0] = [pC = 0] = C∞. Theremaining properties follow from Proposition 25, too.

(vi) The set C∞ is linear since C is symmetric. Also, C∞ = [ψL = 0] is dissipative, thatis, C∞ ⊂ [c ≤ 0] since ψL ≥ c; hence −C∞ is also monotone.

(vii) Assume that L is ω×ω∗-closed. Then C∞ ⊂ C ⊂ L, and so C∞ is monotone. Using(vi) we obtain that C∞ is skew, which implies C∞ ⊂ S. From (ii) we get C∞ = S.

We complete Theorem 27 with the following result.

Proposition 28 Let L ⊂ Z be a linear monotone subspace with skew part S, nonemptypositive part P , unitary part U , and crown C. Then

(i) [ψL = c] is linear and monotone, C is not a linear space and C⊥ = L⊥; moreover,

[ψL = c] = {z | c(z) ≥ 0, z ∈√

c(z) · C},

where 0 · C := C∞, and

[ϕL = c] = Skew(L⊥) ∪ ζ−1([σC = 2]) = Skew(C⊥) ∪ ζ−1([σC = 2]).

(ii) Skew([ψL = c]) = Skew(C∞) and Skew([ψL = c]) is a ω×ω∗−closed (skew) and linearsubspace, Pos([ψL = c]) = ζ−1(C), Unit([ψL = c]) = Unit(C), and Crown([ψL = c]) = C.

(iii) [ψL = c] ⊂ domψL = RC ⊂ clω×ω∗ L ⊂ S⊥[ψL=c], ϕ[ψL=c] = 14σ2

C , ψ[ψL=c] = p2C .

Proof. (i) To prove that [ψL = c] is linear it suffices to show that z + z′ ∈ [ψL = c], for everyz, z′ ∈ [ψL = c]. To this end fix z′ ∈ L. Then the function ηz′ : Z → R ∪ {∞} defined by

ηz′(z) = ψL(z + z′)− z · z′ − c(z′) (z ∈ Z),

is proper convex and lsc. For z ∈ L we have z + z′ ∈ L, and so ψL(z + z′) = c(z + z′) =c(z) + z · z′ + c(z′). It follows that ηz′ ≤ cL. This yields

ψL(z + z′) ≤ ψL(z) + z · z′ + c(z′) ∀z ∈ Z, ∀z′ ∈ L.

Hence if z ∈ [ψL = c] then z + z′ ∈ [ψL ≤ c] = [ψL = c]. We proved z + z′ ∈ [ψL = c] for allz ∈ [ψL = c] and z′ ∈ L. Now consider the function ηz′ from above with z′ ∈ [ψL = c] andrepeat the argument to conclude that [ψL = c] is linear.

The formula for [ψL = c] follows from ψL = p2C and Theorem 27 (v). Since ϕL = 1

4σ2C ≥ 0

(and using Proposition 25 (vi)), we have that

[ϕL = c] = ([σC = 0] ∩ [c = 0]) ∪ ([σ2

C = 4c] ∩ [c > 0])

= Skew(C⊥) ∪ ζ−1([σC = 2]).

By Theorem 27 (ii) we have S ⊂ C, whence C⊥ ⊂ S⊥; using Proposition 25 (vi) we getL⊥ = C⊥ ∩ S⊥ = C⊥. Assume that C is linear. Then C = C∞ = [ψL = 0]. It follows that

21

c(z) ≤ ψL(z) = 0 for every z ∈ C, and so we get the contradiction 1 = c(z) ≤ 0 for everyz ∈ U (⊂ C). (ii) From Theorem 27 (v) we get

Skew([ψL = c]) = [ψL = c] ∩ [c = 0] = [ψL = 0] ∩ [c = 0] = C∞ ∩ [c = 0] = Skew(C∞).

One obtains similarly the other equalities. Because −C∞ is a ω× ω∗–closed monotone linearsubspace, using Theorem 27 (vii) we obtain that Skew(C∞) = −Skew(−C∞) is ω×ω∗–closedand linear.

(iii) All the facts are consequences of Theorem 27, Proposition 26, (i), and Proposition 4(i).

In the sequel we discuss the representability of a linear subspace. Note that the small-est representable operator that contains a monotone double-cone D, namely [ψD = c], is a(monotone) double-cone. In this case it makes sense to talk about the skew, positive, andunitary parts as well as the crown of [ψD = c].

Proposition 29 If L ⊂ Z is linear and representable then SL is a ω×ω∗-closed skew linearsubspace.

Proof. We know that SL is a skew linear subspace from Theorem 27. Hence, by Proposition13, clω×ω∗ SL is skew and ιclω×ω∗ SL

= ψSL≥ ψL ≥ c because L is monotone. This yields

clω×ω∗ SL ⊂ [ψL = c] ∩ [c = 0] = L ∩ [c = 0] = SL.

Let us define the ω × ω∗−natural convergence (ν for short) of nets (zi)i∈I in Z as

zi →ν z ⇐⇒ [c(zi) ≤ c(z) ∀i ∈ I, and zi →ω×ω∗ z in Z],

and the ω × ω∗−natural closure of a subset A ⊂ Z, denoted by clν A, as the set {z ∈ Z |∃(zi)i ⊂ A : zi →ν z}. A set A ⊂ Z is called ν−closed if A = clν A. Similarly, a functionf : Z → R is ν−lsc if f(z) ≤ limi f(zi) whenever zi →ν z (zi, z ∈ Z).

Observe that the ω×ω∗−natural convergence introduced above is different of the conver-gence induced by the natural topology defined by Penot in [16].

Note that for every T ∈ M(X), [ψT = c] is ν–closed; in particular every representableoperator is ν–closed. Indeed, let (zi)i∈I ⊂ [ψT = c] such that zi →ν z ∈ Z. Then

ψT (z) ≤ lim infi∈I

ψT (zi) ≤ lim supi∈I

ψT (zi) = lim supi∈I

cT (zi) ≤ c(z),

which proves that z ∈ [ψT = c]. Moreover ψT (z) = limi ψT (zi) and similarly f(z) = limi f(zi)for every f ∈ RT and (zi)i∈I ⊂ [f = c] with zi →ν z. The following theorem shows that amonotone linear subspace L of Z is representable iff L is ν−closed.

Theorem 30 Let L ⊂ Z be a linear monotone subspace with nonempty positive part. TFAE:(i) L is representable,(ii) Unit([ψL = c]) ⊂ L,(iii) cL is ν–lsc,(iv) L is ν–closed.

22

Proof. The implication (i) ⇒ (ii) is straightforward.(ii) ⇒ (i) Let z ∈ [ψL = c]. If c(z) > 0 then 1√

c(z)z ∈ U[ψL=c] ⊂ L and so z ∈ L. By

Proposition 28 (i) we have that [ψL = c] is linear; since L ⊂ [ψL = c], clearly [ψL = c]has nonempty positive part. If c(z) = 0, by Theorem 27 (ii), then z ∈ Skew([ψL = c]) ⊂conv(Unit([ψL = c])) ⊂ L. Hence L = [ψL = c].

(i) ⇒ (iii) Assume that L is representable. To prove (iii) it is sufficient to consider the net(zi)i∈I ⊂ L with zi →ν z, that is, zi → z for the topology ω × ω∗ and c(zi) ≤ c(z) for everyi ∈ I. Note that

c(z) ≤ ψL(z) ≤ lim infi∈I

ψL(zi) = lim infi∈I

c(zi) ≤ c(z),

which yields that z ∈ [ψL = c] = L and so cL is ν–lsc.The implication (iii) ⇒ (iv) is plain.(iv) ⇒ (ii) Let z ∈ Unit([ψL = c]), that is, ψL(z) = c(z) = 1. Using Proposition 28 (ii)

and Theorem 27 (iii) we get z ∈ C = clω×ω∗ [cL ≤ 1], and so there exists (zi)i∈I ⊂ [cL ≤ 1](⊂ L) with zi → z for the topology ω × ω∗ in Z. Since c(z) = 1 and L is ν–closed we getz ∈ L.

Remark 8 Note that whenever S is a skew linear subspace of Z, clν S = clω×ω∗ S sinceclν S ⊂ clω×ω∗ S and clω×ω∗ S is skew. In this case S is representable iff S is ω × ω∗−closed,that is, we recover part of Corollary 14.

Corollary 31 If L ⊂ Z is a linear monotone and ω × ω∗–closed subspace then L is repre-sentable.

Corollary 32 If L ⊂ Z is a linear representable subspace and (zi)i∈I ⊂ L is such thatzi →ν z ∈ Z, then c(z) = limi c(zi), and f(z) = limi f(zi) for every f ∈ RL.

In the next result we characterize the uniqueness of monotone double-cones.

Proposition 33 Let D ⊂ Z be a monotone double-cone. Then D is unique iff [ϕD = c] ismonotone.

Proof. According to Proposition 3 it is clear that D unique implies that D1 := [ϕD = c] ismonotone. For the converse it suffices to prove that D non-unique implies that D1 is non-monotone. Since D is not unique we can find z1, z2 ∈ [ϕD ≤ c] such that c(z1 − z2) < 0. Letd := {tz1 + (1− t)z2 | t ∈ R} be the line through z1, z2; because z1, z2 ∈ domϕD and domϕD

is a linear space (see Proposition 25), we have that d ⊂ domϕD. Because c(z1 − z2) < 0and ϕD is convex, the function R 3 t 7→ η(t) := (ϕD − c)(tz1 + (1 − t)z2) is finite-valued,continuous and coercive, i.e., lim|t|→+∞ η(t) = +∞. Since η(0) ≤ 0 and η(1) ≤ 0, there existt0 ≤ 0 and t1 ≥ 1 such that η(t0) = η(t1) = 0. Then z′1 := t0z1 + (1 − t0)z2 ∈ D1 andz′2 := t1z1 + (1− t1)z2 ∈ D1. Because c(z′1 − z′2) = (t1 − t0)2c(z2 − z1) < 0, we have that D1

is not monotone.

Corollary 34 Let L ⊂ Z be linear monotone with non-empty crown C. TFAE: (a) L isunique, (b) [ϕL = c] is monotone; (c) ζ−1(2C◦) ∪ Skew(L⊥) is monotone; (d) ζ−1([σC =2]) ∪ Skew(L⊥) is monotone.

From Propositions 11 and 33 we get immediately the next result.

23

Corollary 35 Let D ⊂ Z be a monotone double-cone. Then D is NI iff either [ϕD = c] ismonotone and domϕD is not monotone or domϕD = [ϕD = c].

This result together with Proposition 28 (i) and Remark 5 yield the next result.

Corollary 36 Let L ⊂ Z be linear monotone with non-empty crown C and set M := [ϕL =c] = ζ−1([σC = 2]) ∪ Skew(L⊥). Then L is NI iff either M is monotone and barC is notmonotone or ϕL = cM .

Remark 9 When S is a skew linear subspace of Z Corollary 35 spells S is NI iff eitherS0 := S⊥ ∩ [c = 0] is monotone and S⊥ is not monotone or S0 = S⊥ (that is S⊥ is skew).According to Corollary 16 and Proposition 17, this comes to S is NI iff −S⊥ is monotone,and we recover part of Proposition 15.

The following chart comprises the information on a linear non-skew monotone subspaceL with crown C.

1. L is representable iff L is ν−closed2. L is unique and not NI iff barC is monotone and ζ−1([σC = 2]) ∪ Skew(L⊥) ( barC3. L is unique iff ζ−1([σC = 2]) ∪ Skew(L⊥) is monotone4. L is NI iff either ζ−1([σC = 2]) ∪ Skew(L⊥) is monotone and barC is not monotone

or ϕL = c[ϕL=c]

5. L is maximal monotone iff L is ν−closed and NI.

References

[1] H.H. Bauschke, J.M. Borwein, Maximal monotonicity of dense type, local maximal mono-tonicity, and monotonicity of the conjugate are all the same for continuous linear oper-ators, Pacific J. Math. 189 (1999), 1–20.

[2] H.H. Bauschke, S. Simons, Stronger maximal monotonicity properties of linear operators,Bull. Austral. Math. Soc. 60 (1999), 163–174.

[3] H.H. Bauschke, X. Wang, L. Yao, Monotone linear relations: Maximality and Fitzpatrickfunctions, arXiv:0805.4256v1 (2008).

[4] J.M. Borwein, Maximality of sums of two maximal monotone operators in general Banachspace, Proc. Am. Math. Soc. 135 (2007), 3917–3924.

[5] J.M. Borwein, Maximal monotonicity via convex analysis, J. Convex Anal. 13 (2006),561–586.

[6] F.E. Browder, The fixed point theory of multi-valued mappings in topological vectorspaces, Math. Ann. 177 (1968), 283–301.

[7] R. Burachik, B.F. Svaiter, Maximal monotonicity, conjugation and the duality product,Proc. Am. Math. Soc. 131 (2003), 2379–2383.

[8] S. Fitzpatrick, Representing monotone operators by convex functions, Workshop / Mini-conference on Functional Analysis and Optimization (Canberra, 1988), in: Proc. CentreMath. Anal. Austral. Nat. Univ. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65.

24

[9] S. Fitzpatrick, R.R. Phelps, Some properties of maximal monotone operators on nonre-flexive Banach spaces, Set-Valued Anal. 3 (1995), 51–69.

[10] J.-P. Gossez, On the range of a coercive maximal monotone operator in a nonreflexiveBanach space, Proc. Amer. Math. Soc. 35 (1972), 67–70.

[11] J.-P. Gossez, On the extensions to the bidual of a maximal monotone operator, Proc.Amer. Math. Soc. 62 (1977), 88–92.

[12] R. Holmes, Geometric Functional Analysis and its Applications, Graduate Texts in Math-ematics, No. 24. Springer-Verlag, New York, 1975.

[13] M. Marques Alves, B.F. Svaiter, Brønsted–Rockafellar property and maximality of mono-tone operators representable by convex functions in non-reflexive Banach spaces, J. Con-vex Anal. 15 (2008), No. 3, arXiv:0802.1895v1 (2008).

[14] J. E. Martınez-Legaz, B.F. Svaiter, Monotone operators representable by l.s.c. convexfunctions, Set-Valued Anal. 13 (2005), 21–46.

[15] J.-P. Penot, The relevance of convex analysis for the study of monotonicity, NonlinearAnal. 58 (2004), 855–871.

[16] J.-P. Penot, Natural closures, natural compositions and natural sums of monotone oper-ators, J. Math. Pures Appl. 89 (2008), 523–537.

[17] J.-P. Penot, C. Zalinescu, Some problems about the representation of monotone operatorsby convex functions, ANZIAM J. 47 (2005), 1–20.

[18] R.R. Phelps, S. Simons, Unbounded linear monotone operators on nonreflexive Banachspaces, J. Convex Anal. 5 (1998), 303–328.

[19] R.T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math.J. 16 (1969), 397–407.

[20] R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, 1970.

[21] S. Simons, The range of a monotone operator, J. Math. Anal. Appl. 199 (1996), 176–201.

[22] S. Simons, Minimax and Monotonicity, Lecture Notes in Mathematics, 1693, Springer-Verlag, Berlin, 1998.

[23] S. Simons, Dualized and scaled Fitzpatrick functions, Proc. Amer. Math. Soc. 134 (2006),2983–2987.

[24] S. Simons, From Hahn–Banach to Monotonicity, Lecture Notes in Mathematics, 1693(2nd Ed.), Springer-Verlag, Berlin, 2008.

[25] S. Simons, C. Zalinescu, A new proof for Rockafellar’s characterization of maximal mono-tone operators, Proc. Amer. Math. Soc. 132 (2004), 2969–2972.

[26] M.D. Voisei, A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach spaces, Math. Sci. Res. J. 10 (2) (2006), 36–41.

25

[27] M.D. Voisei, The sum theorem for linear maximal monotone operators, Math. Sci. Res.J. 10 (4) (2006), 83–85.

[28] M.D. Voisei, Calculus rules for maximal monotone operators in general Banach space, J.Convex Anal. 15 (2008), 73–85.

[29] M.D. Voisei, The sum and chain rules for maximal monotone operators, Set-Valued Anal.16 (2008), 461–476.

[30] M.D. Voisei, C. Zalinescu, Strongly-representable monotone operators, J. Convex Anal.(to appear), arXiv:0802.3640v1 (2008).

[31] M.D. Voisei, C. Zalinescu, Maximal monotonicity criteria for the composition and thesum under minimal interiority conditions, preprint 2008.

[32] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore(2002).

26