Location, identification, and representability of monotone operators in locally convex spaces
aa r X i v : . [ m a t h . F A ] D ec Location, identification, and representability of monotoneoperators in locally convex spaces
M.D. Voisei
Abstract
In this paper we study, in the relaxed context of locally convex spaces, intrinsic propertiesof monotone operators needed for the sum conjecture for maximal monotone operators tohold under classical interiority-type domain constraints.
The aim of this note is to reveal deeper properties of maximal montone operators that enjoy, undera locally convex space settings, the classical sum theorem which, in the literature, is sometimescalled, when the context is provided by Banach spaces, the Rockafellar conjecture.A breakthrough in the study of maximal monotone operators is represented by the introduction,in 2006 in [15, Theorem 2.3], of a new characterization of maximal monotonicity based on thenotions of representability and “NI-type” operator (see [18, Remark 3.5] or Remark 11 below formore details). This characterization works in locally convex spaces and, is the main argumentused after 2006 in the majority of the articles concerning the calculus rules for maximal monotoneoperators in general Banach spaces such as those in [13, 15, 16, 17, 18, 19, 20, 23].The present paper enhances the aforementioned maximality characterization by presentinglocalized versions of it together with their direct consequences.The plan of the paper is as follows. Section presents the three main notions studied in thisarticle together with their immediate properties and some variants. Section is concerned with theinterplay of these notions. Section contains the representability of the sum of two representableoperators. We conclude our article with some open problems in Section .Throughout this paper, if not otherwise explicitly mentioned, ( X, τ ) is a non-trivial (that is, X = { } ) Hausdorff separated locally convex space (LCS for short), X ∗ is its topological dualendowed with the weak-star topology ω ∗ , the topological dual of ( X ∗ , ω ∗ ) is identified with X ,and the weak topology on X is denoted by ω .We denote by V τ ( x ) the family of τ − neighborhoods of x ∈ X and the convergence of nets in ( X, τ ) by x i τ → x .The duality product or coupling of X × X ∗ is denoted by h x, x ∗ i := x ∗ ( x ) =: c ( x, x ∗ ) , for x ∈ X , x ∗ ∈ X ∗ . As usual, with respect to the dual system ( X, X ∗ ) , we denote the orthogonalof S ⊂ X by S ⊥ := { x ∗ ∈ X ∗ | h x, x ∗ i = 0 , for every x ∈ S } and the support function of S by σ S ( x ∗ ) := sup x ∈ S h x, x ∗ i , x ∗ ∈ X ∗ while for M ⊂ X ∗ , the orthogonal of M is denoted by M ⊥ := { x ∈ X | h x, x ∗ i = 0 , for every x ∈ M } and its support function is σ M ( x ) = sup x ∗ ∈ M h x, x ∗ i , x ∈ X .To a multi-valued operator T : X ⇒ X ∗ we associate its • graph : Graph T = { ( x, x ∗ ) ∈ X × X ∗ | x ∗ ∈ T x } , • inverse: T − : X ∗ ⇒ X , gph T − = { ( x ∗ , x ) | ( x, x ∗ ) ∈ Graph T } , • domain : D ( T ) := { x ∈ X | T x = ∅} = Pr X (Graph T ) , and • range : R ( T ) := { x ∗ ∈ X ∗ | x ∗ ∈ T x for some x ∈ X } = Pr X ∗ (Graph T ) .Here Pr X and Pr X ∗ are the projections of X × X ∗ onto X and X ∗ , respectively.1 direct image: T ( A ) := ∪ x ∈ A T x , A ⊂ X .When no confusion can occur, T : X ⇒ X ∗ will be identified with Graph T ⊂ X × X ∗ .In the sequel, given a locally convex space ( E, τ ) and S ⊂ E , the following notations areused: “ cl τ S = S τ ” for the τ − closure of S , “ int τ S ” for the τ − topological interior of S , “ bd τ S = S τ \ int τ S ” for the boundary of S , “ conv S ” for the convex hull of S , “ aff S ” for the affine hull of S , and “ S i = core S ” for the algebraic interior of S , “ i S ” for the relative algebraic interior of S with respect to aff S . When the topology τ is implicitly understood the use of the τ − notation isavoided.A set S ⊂ E is called algebraically open if S = core S .We denote by ι S the indicator function of S ⊂ E defined by ι S ( x ) := 0 for x ∈ S and ι S ( x ) := ∞ for x ∈ E \ S .The set [ x, y ] := { tx + (1 − t ) y | ≤ t ≤ } ⊂ E represents the closed segment with end-points x, y ∈ E .For f, g : E → R := R ∪ {−∞ , + ∞} we set [ f ≤ g ] := { x ∈ E | f ( x ) ≤ g ( x ) } , [ f = g ] , [ f < g ] and [ f > g ] are similarly defined, while e.g. f ≥ g means [ f ≥ g ] = E or, for every e ∈ E , f ( e ) ≥ g ( e ) .We consider the following classes of functions and operators on X : 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 −∞ ; Γ τ ( X ) is the class of functions f ∈ Λ( X ) that are τ – lower semicontinuous ( τ – lsc for short); whenthe topology is implicitly understood the notation Γ( X ) is used instead; M ( X ) is the class of non-void monotone operators T : X ⇒ X ∗ ( Graph T = ∅ ). Recall that T : X ⇒ X ∗ is monotone if h x − x , x ∗ − x ∗ i ≥ for all x , x ∈ D ( T ) , x ∗ ∈ T x , x ∗ ∈ T x ; M ( X ) is the class of maximal monotone operators T : X ⇒ X ∗ . The maximality is understoodin the sense of graph inclusion as subsets of X × X ∗ .Recall some notions associated to a proper function f : X → R : epi f := { ( x, t ) ∈ X × R | f ( x ) ≤ t } is the epigraph of f ; cl τ f : X → R , the τ –lsc hull of f , is the greatest τ –lsc function majorized by f ; epi(cl τ f ) =cl τ (epi f );conv f : X → R , the convex hull of f , is the greatest convex function majorized by f ; (conv f )( x ) :=inf { t ∈ R | ( x, t ) ∈ conv(epi f ) } for x ∈ X ; conv τ f : X → R , the τ − lsc convex hull of f , is the greatest τ – lsc convex function majorized by f ; epi(conv τ f ) := conv τ (epi f ) ; f ∗ : X ∗ → R is the convex conjugate of f : X → R with respect to the dual system ( X, X ∗ ) , f ∗ ( x ∗ ) := sup {h x, x ∗ i − 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 ∗ |h x ′ − x, x ∗ i + f ( x ) ≤ f ( x ′ ) , ∀ x ′ ∈ X } for x ∈ X (it follows from its definition that ∂f ( x ) := ∅ for x dom f ). Recall that N C = ∂ι C is the normal cone to C , where ι C ( x ) = 0 , if x ∈ C , ι C ( x ) = + ∞ otherwise; ι C is the indicator function of C ⊂ X .For ( X, τ ) a LCS, let Z := X × X ∗ . It is known that ( Z, τ × ω ∗ ) ∗ = Z via the coupling z · z ′ := h x, x ′∗ i + h x ′ , x ∗ i , for z = ( x, x ∗ ) , z ′ = ( x ′ , x ′∗ ) ∈ Z ;( Z, Z ) is called the natural dual system . 2or a proper function f : Z → R all the above notions are defined similarly. In addition, withrespect to the natural dual system ( Z, Z ) , the conjugate of f is given by f (cid:3) : Z → R , f (cid:3) ( z ) = sup { z · z ′ − f ( z ′ ) | z ′ ∈ Z } , and by the biconjugate formula, f (cid:3)(cid:3) = conv τ × ω ∗ f whenever f (cid:3) (or conv τ × ω ∗ f ) is proper.We introduce the following classes of functions: C : = C ( Z ) := { f ∈ Λ( Z ) | f ≥ c } , R : = R ( Z ) := Γ τ × ω ∗ ( Z ) ∩ C ( Z ) , D : = D ( Z ) := { f ∈ R ( Z ) | f (cid:3) ≥ c } . It is known that [ f = c ] ∈ M ( X ) for every f ∈ C ( Z ) (see e.g. [8, Proposition 4(h)], [18,Lemma 3.1]). Lemma 1
Let X be a LCS and let h ∈ C . Then [ h = c ] ⊂ [ h (cid:3) = c ] . If, in addition, h ∈ D then [ h = c ] = [ h (cid:3) = c ] . Proof.
Let z ∈ [ h = c ] . Then h ′ ( z ; w ) = lim t ↓ h ( z + tw ) − h ( z ) t ≥ lim t ↓ c ( z + tw ) − c ( z ) t = z · w, ∀ w ∈ Z, which shows that z ∈ ∂h ( z ) . Therefore h ( z ) + h (cid:3) ( z ) = z · z = 2 c ( z ) and so h (cid:3) ( z ) = c ( z ) .If, in addition, h ∈ D the stated equality follows from h = h (cid:3)(cid:3) and the previously showninclusion applied for h and h (cid:3) .To a multifunction T : X ⇒ X ∗ we associate the following functions: c T : Z → R , c T := c + ι Graph T , ψ T : Z → R , ψ T := cl τ × ω ∗ (conv c T ) , ϕ T : Z → R , ϕ T := c (cid:3) T = ψ (cid:3) T − the Fitzpatrickfunction of T . In expanded form ϕ T ( z ) := ϕ T ( x, x ∗ ) := sup { z · w − c ( w ) | w ∈ T } = sup {h x, u ∗ i + h u, x ∗ i − h u, u ∗ i | ( u, u ∗ ) ∈ T } , for z = ( x, x ∗ ) ∈ Z . The function ϕ T , ψ T were introduced first in [4], [15].Recall that whenever T ∈ M ( X ) , ϕ T , ψ T ∈ Γ τ × w ∗ ( Z ) .The set T + := [ ϕ T ≤ c ] describes all elements of Z that are monotonically related (m.r. forshort) to T .Let us recall several properties of these functions. Theorem 2
Let X be a LCS. (i) For every T ⊂ X × X ∗ , T ⊂ ( D ( T ) × X ∗ ) ∪ ( X × R ( T )) ⊂ [ ϕ T ≥ c ] (see [15, Theorem 1.1],[14, 17, Proposition 3.2]) , (ii) T ∈ M ( X ) iff T ⊂ [ ϕ T = c ] iff ψ T ≥ c , ([14, 17, Proposition 3.2], [23, (8)])(iii) T ∈ M ( X ) iff T ∈ M ( X ) , T = [ ψ T = c ] , and ϕ T ≥ c ([15, Theorems 2.2, 2.3], [23, Theorem1]) . For other properties of ϕ T , ψ T we refer to [15, 17, 18, 22, 23].The following set properties are frequently used in the sequel; for M, N ⊂ X × X ∗ , V, W ⊂ X • M ∩ ( V × X ∗ ) ⊂ N ⇒ (Pr X M ) ∩ V ⊂ Pr X N ; • M ∩ ( V × X ∗ ) ⊂ W × X ∗ ⇔ (Pr X M ) ∩ V ⊂ W ; and • Pr X ( M ∩ ( V × X ∗ )) = (Pr X M ) ∩ V .Throughout this article the conventions sup ∅ = −∞ and inf ∅ = ∞ are observed.3 Definitions and properties
Definition 3
Let X be a LCS. A subset V ⊂ X identifies T : X ⇒ X ∗ or T is identified by V if [ ϕ T | V ≤ c ] ∩ V × X ∗ ⊂ Graph T . Equivalently, V identifies T iff every z = ( x, x ∗ ) ∈ V × X ∗ that is m.r. to T | V belongs to T . Here T | V : X ⇒ X ∗ is defined by ( x, x ∗ ) ∈ T | V if x ∈ V and ( x, x ∗ ) ∈ T or Graph T | V = Graph T ∩ ( V × X ∗ ) .Note that the empty set identifies any operator, but if a non-empty V ⊂ X identifies T then V ∩ D ( T ) = ∅ . Indeed, if V = ∅ and V ∩ D ( T ) = ∅ then ϕ T | V = −∞ , [ ϕ T | V ≤ c ] ∩ V × X ∗ = V × X ∗ Graph T , that is, V does not identify T . Hence this notion is interesting only when V ∩ D ( T ) = ∅ .When T is non-void monotone, V identifies T iff T | V is maximal monotone in V × X ∗ , that is, Graph T | V has no proper monotone extension in V × X ∗ or [ ϕ T | V ≤ c ] ∩ V × X ∗ = Graph( T | V ) .In the context of a Banach space X , a monotone operator T is called of type (FPV) or maximalmonotone locally (notion first introduced in [12] and further studied in [3]) if for every open convex V ⊂ X either V ∩ D ( T ) = ∅ or V identifies T . For the sake of language, notation simplicity, andnotion uniformity we introduce the terminology identifiable as an extension of the type (FPV)notion to a general operator in the context of locally convex spaces. Definition 4
Let ( X, τ ) be a LCS. An operator T : X ⇒ X ∗ is ( τ − )identifiable if T is identifiedby every ( τ − )open convex subset V of X such that V ∩ D ( T ) = ∅ .Note that X identifies a monotone operator T iff T is maximal monotone. Therefore everyidentifiable monotone operator is maximal monotone.The identifiability of a maximal monotone operator T is interesting only on sets V with D ( T ) V since T ∈ M ( X ) is identified by every V that contains D ( T ) .The τ − identifiability of an operator depends explicitly on the topology τ and not only on theduality ( X, X ∗ ) .The identifiability notion unifies several other notions from the literature. For example for X a Banach space, T : X ⇒ X ∗ is locally maximal monotone (notion introduced in [2, p. 583]) iff T is monotone and T − : X ∗ ⇒ X is identified by every norm-open convex subset of X ∗ .The identifiability of a monotone operator T is intrinsically related to the sum theorem. Moreprecisely, if T + N C is maximal monotone, for every C ⊂ X closed convex with D ( T ) ∩ int C = ∅ then T is identifiable. Under a Banach space settings this implication is known for some time (seee.g. [3, Proposition 3.3]) but it also holds in a locally convex space context.The sum conjecture [SC] is true in reflexive Banach spaces (see e.g. [9, Theorem 1(a), p. 76]).Therefore every maximal monotone operator in a reflexive Banach space is identifiable.The class of open convex sets arises naturally in the identification of maximal monotone op-erators. That is not the case for the class of closed convex sets (even when they have non-emptyinteriors). Assume that X is a LCS and the closed convex C ⊂ X identifies T ∈ M ( X ) . Since T | C ⊂ T + N C ∈ M ( X ) and D ( T + N C ) ⊂ C one gets T | C = T + N C ⊂ T . The contrapositive formof this fact shows that a closed convex C ⊂ X does not identify T ∈ M ( X ) if T + N C ∈ M ( X ) (which happens for example when X is a reflexive Banach space and D ( T ) ∩ int C = ∅ ) and D ( T ) C . However, in general, C can identify T + N C (see Theorem 27 below) and that leadsto our next notion. Definition 5
Let X be a LCS. A subset V ⊂ X locates T : X ⇒ X ∗ in S ⊂ X or T is locatedby V in S if Pr X [ ϕ T | V ≤ c ] ∩ V ⊂ S (or [ ϕ T | V ≤ c ] ∩ V × X ∗ ⊂ S × X ∗ ). Equivalently, V locates T in S iff every z = ( x, x ∗ ) ∈ V × X ∗ that is m.r. to T | V has x ∈ S .An operator T : X ⇒ X ∗ is called locatable in S iff every open convex subset V of X such that V ∩ D ( T ) = ∅ locates T in S . When S = D ( T ) we simply say that V locates T or T is located by V and that T is locatable . 4s previously seen, his notion is interesting only when V ∩ D ( T ) = ∅ because if V ∩ D ( T ) = ∅ then Pr X [ ϕ T | V ≤ c ] ∩ V ⊂ S reduces to V ⊂ S .A monotone operator T is locatable if for every open convex set V ⊂ X with V ∩ D ( T ) = ∅ , T | V cannot be extended outside D ( T ) ∩ V , as a monotone operator in V × X ∗ .In the literature, for X a Banach space, an operator T is called of type weak-FPV (notionfirst introduced in [19]) if every open convex subset V ⊂ X with V ∩ D ( T ) = ∅ locates T . Theterminology locatable is used as an extension and a simplified notation of the type weak-FPVnotion to the general locally convex space settings.Every locatable operator in S is locatable in S ′ , whenever S ⊂ S ′ . Also, every identifiableoperator is locatable because V locates T whenever V identifies T . However, there exist monotoneoperators that are locatable but not identifiable. Take for example T = { } × ( X ∗ \ { } ) where X is a LCS. Note that T is not identifiable since it is not maximal monotone. By a direct verification T is locatable. Indeed, if V is open convex with ∈ V and z = ( x, x ∗ ) ∈ V × X ∗ is m.r. to T | V = T then x = 0 because { } × X ∗ is the unique maximal monotone extension of T .We will see later, in Theorem 23 below, that for a maximal monotone operator in the generalcontext of a locally convex space the locatable and identifiable notions coincide.An operator T : X ⇒ X ∗ is automatically located by every V ⊂ D ( T ) . Therefore the locationof an operator T is interesting only on sets V D ( T ) . Lemma 6
Let X be a LCS and let T : X ⇒ X ∗ . Then X locates T iff ϕ T ≥ c and Pr X [ ϕ T = c ] ⊂ D ( T ) . If, in addition, T ∈ M ( X ) then X locates T iff ϕ T ≥ c and Pr X [ ϕ T = c ] = D ( T ) . Proof.
Condition X locates T comes to [ ϕ T ≤ c ] ⊂ D ( T ) × X ∗ . The conclusion follows after wetake Theorem 2 (i), (ii) into account, that is, for every T : X ⇒ X ∗ , D ( T ) × X ∗ ⊂ [ ϕ T ≥ c ] andthat T ⊂ [ ϕ T = c ] whenever T ∈ M ( X ) .Therefore the localization of an operator T by X depends on the condition ϕ T ≥ c also knownas T is of type negative-infimum (NI for short); notion that was first used in [15] and introducedin [14, 18, Remark. 3.5]. Therefore, according to Lemma 6, every maximal monotone operator isNI because T ∈ M ( X ) ⇔ X identifies T ⇒ X locates T .It must be said that our NI notion differs fundamentally from the NI notion introduced, for X a Banach space, in X ∗ × X ∗∗ by Simons (see e.g. [11, Definition 25.5, p. 99]). The NI-operatorsin the sense of Simons coincide with those of dense-type in the sense of Gossez introduced in[5] (see [7]). Since the dense-type property is stronger and has been introduced prior to the NIclass in the sense of Simons it is our opinion that the use of NI notion in the sense of Simonsis obsolete. Another essential difference between these two notions, besides the underlying spacecontext, is that every maximal monotone operator is NI in the current sense while not everymaximal monotone operator is of dense-type (see e.g. [6, p. 89]). For more explanations oncomparing these notions see [22, p. 33] and [19, p. 662] . Since every X − locatable operator is NI, we expect in general that the localization propertydepend on a localized NI type condition. Definition 7
Let X be a LCS. An operator T : X ⇒ X ∗ is of negative-infimum type on V ⊂ X or simply V − NI if V × X ∗ ⊂ [ ϕ T | V ≥ c ] . Equivalently, T is V − NI iff [ ϕ T | V < c ] ∩ V × X ∗ = ∅ iff Pr X [ ϕ T | V < c ] ∩ V = ∅ .The operator T : X ⇒ X ∗ is called locally-NI if, for every open convex V ⊂ X such that V ∩ D ( T ) = ∅ , T is V − NI.Note that every operator is ∅− NI and if, for a certain V = ∅ , T is V − NI then V ∩ D ( T ) = ∅ ,because V ∩ D ( T ) = ∅ implies ϕ T | V = −∞ . Notice also that if T is V − NI then V ⊂ Pr X [ ϕ T | V ≥ c ] while the converse is not true. The X − NI type coincides with the NI type discussed above.5 heorem 8
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X be such that V ∩ D ( T ) = ∅ . Thefollowing are equivalent (i) T is V − NI, (iv) [ ϕ T | V ≤ c ] ∩ V × X ∗ ⊂ [ ϕ T | V ≥ c ] , (ii) ϕ T | V ≥ c in V × X ∗ , (v) [ ϕ T | V < c ] ∩ V × X ∗ ⊂ [ ϕ T | V ≥ c ] . (iii) ϕ T | V + ι V × X ∗ ≥ c , Proof.
Left to the reader.
Theorem 9
Let X be a LCS and let T ∈ M ( X ) . Assume that V open convex , V ∩ D ( T ) = ∅ = ⇒ Pr X [ ϕ T | V < c ] ∩ V ⊂ D ( T ) . (1) Then D ( T ) is convex.In particular, if T ∈ M ( X ) is locatable (in D ( T ) ) or T ∈ M ( X ) is locally-NI then D ( T ) convex. Proof.
Assume that D ( T ) is not convex. There exist x , x ∈ D ( T ) , < ρ < , and U ∈ V (0) such that ( x ρ + U ) ∩ D ( T ) = ∅ , where x t := tx + (1 − t ) x , ≤ t ≤ ; in particular x = x . Let x ∗ ∈ T x , x ∗ ∈ T x , and denote by z = ( x , x ∗ ) , z = ( x , x ∗ ) , z ρ := ρz + (1 − ρ ) z .Let V ∈ V (0) be balanced and γ > be such that V + V ⊂ U and x − x ∈ γV . Take y ∗ ∈ X ∗ with h x − x , y ∗ i ≥ γ ( ρ (1 − ρ ) c ( z − z ) + 1) > and W ∈ V (0) be open convex suchthat W ⊂ V and α := sup {|h u, y ∗ i| | u ∈ W } is finite and positive (see e.g. [10, Theorem 1.18, p.15]). Let < ǫ < min { , /α } . Let ¯ z ρ := z ρ + (0 , y ∗ ) .For every z = ( x, x ∗ ) ∈ T with x ∈ D ( T ) ∩ ([ x , x ρ ] + ǫW ) we have x − x ρ U and x − x λ ∈ ǫW for some ≥ λ > ρ . This yields that x λ − x ρ = ( λ − ρ )( x − x ) V . Hence ( λ − ρ ) ≥ /γ . Since x ρ − x = ( λ − ρ )( x − x ) + x λ − x and T ∈ M ( X ) we have c (¯ z ρ − z ) = c ( z ρ − z ) + h x ρ − x, y ∗ i = ρc ( z − z ) + (1 − ρ ) c ( z − z ) − ρ (1 − ρ ) c ( z − z ) + h x ρ − x, y ∗ i≥ ( λ − ρ ) h x − x , y ∗ i + h x λ − x, y ∗ i− ρ (1 − ρ ) c ( z − z ) ≥ γ h x − x , y ∗ i− αǫ − ρ (1 − ρ ) c ( z − z ) > . This yields ¯ z ρ ∈ [ ϕ T | [ x ,xρ ]+ ǫW < c ] ∩ ([ x , x ρ ]+ ǫW ) ⊂ D ( T ) × X ∗ and the contradiction x ρ ∈ D ( T ) . Remark 10 ( T is V − NI versus T | V is NI) First note that T is V − NI whenever V ⊂ D ( T ) since in this case V × X ∗ = D ( T | V ) × X ∗ ⊂ [ ϕ T | V ≥ c ] (see Theorem 2 (i)) or because, in thiscase, V locates T (see Theorem 13 below). Also, it is straightforward that T is V − NI whenever T | V is ( X − ) NI. The converse of this fact, namely, whether T | V is NI whenever T is V − NI fails tobe true in any LCS X even when T is maximal monotone, V is convex, and V is open or closedwith empty or non-empty interior.We base our following examples on the fact that every monotone NI operator admits a uniquemaximal monotone extension (see Theorem 22 below or [22, Proposition 4 (iii)]).For example, for every T ∈ M ( X ) with a non-singleton domain and for every x ∈ D ( T ) , T is { x }− NI while T | { x } = { x } × T x is not NI because { x } × X and any maximal monotone extensionof T are different maximal monotone extensions of T | { x } .Similar considerations can be made for a non-NI operator T which is D ( T ) − NI but T | D ( T ) = T is not NI.Let C X be closed convex with int C = ∅ . Then N C is int C − NI (this fact can also bechecked directly from N C | int C = int C × { } and ϕ int C ×{ } ( x, x ∗ ) = σ C ( x ∗ ) , ( x, x ∗ ) ∈ X × X ∗ ).But N C | int C is not NI since it admits two distinct (maximal) monotone extensions: N C and X × { } . Similarly, for every closed convex set D ⊂ int C (with possible empty interior) we havethat N C is D − NI and N C | D is not NI because N C | int C is not NI.6 emark 11 (The NI method) In general it is hard to verify the NI condition directly, evenwhen T is monotone, since the closed forms of ϕ T , ψ T are known only for few types of operators(see e.g. [1, 22]) and, when X is a non-reflexive Banach space, the coupling c is not continuouswith respect to any topology on X × X ∗ compatible with the natural duality ( X × X ∗ , X ∗ × X ) (see [23, Appendix]).Given a LCS X , the first direct method to prove that an operator T : X ⇒ X ∗ is of NI typehas been developed in [15, Theorem. 1.1] and is summarized as follows ( z = ( x, x ∗ ) is m . r . to T ⇒ x ∈ D ( T )) = ⇒ T is NI , (2)or, equivalently, X locates T = ⇒ T is NI . The reader recognizes that this NI method is containedin Lemma 6 and that its converse holds under the additional condition Pr X [ ϕ T = c ] ⊂ D ( T ) .The following is a slightly improved version of (2), namely (Pr X [ ϕ T < c ] ⊂ D ( T )) = ⇒ T is NI . (3)Indeed, if z = ( x, x ∗ ) ∈ [ ϕ T < c ] then x ∈ Pr X [ ϕ T < c ] ⊂ D ( T ) ; whence, according to Theorem 2(i), z ∈ [ ϕ T ≥ c ] a contradiction. Therefore [ ϕ T < c ] is empty, that is, T is NI.Similar considerations for a V − NI method are contained in the following result.
Theorem 12 (The V − NI method)
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X . Then T is V − NI iff Pr X [ ϕ T | V < c ] ∩ V ⊂ D ( T ) . Proof.
It suffices to note that, in general, Pr X [ ϕ T | V < c ] ∩ V ∩ D ( T ) = ∅ due to D ( T | V ) × X ∗ =( D ( T ) ∩ V ) × X ∗ ⊂ [ ϕ T | V ≥ c ] . Theorem 13
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X be such that V ∩ D ( T ) = ∅ . Thefollowing are equivalent (i) V locates T , (ii) T is V − NI and Pr X [ ϕ T | V = c ] ∩ V ⊂ D ( T ) , (iii) Pr X dom ϕ T | V ∩ V ⊂ Pr X ([ ϕ T | V ≥ c ] ∩ dom ϕ T | V ) and Pr X [ ϕ T | V = c ] ∩ V ⊂ D ( T ) .If, in addition, T | V ∈ M ( X ) then V locates T iff T is V − NI and Pr X [ ϕ T | V = c ] ∩ V = D ( T ) ∩ V . Proof. (i) ⇒ (ii) Recall that V locates T means [ ϕ T | V ≤ c ] ∩ V × X ∗ ⊂ D ( T ) × X ∗ from which [ ϕ T | V ≤ c ] ∩ V × X ∗ ⊂ D ( T | V ) × X ∗ ⊂ [ ϕ T | V ≥ c ] ; whence T is V − NI and Pr X [ ϕ T | V ≤ c ] ∩ V =Pr X [ ϕ T | V = c ] ∩ V ⊂ D ( T ) .(ii) ⇒ (iii) Because T is V − NI we have V × X ∗ ⊂ [ ϕ T | V ≥ c ] followed by dom ϕ T | V ∩ V × X ∗ ⊂ dom ϕ T | V ∩ [ ϕ T | V ≥ c ] .(iii) ⇒ (i) Let x ∈ Pr X [ ϕ T | V ≤ c ] ∩ V . Take x ∗ ∈ X ∗ such that ( x, x ∗ ) ∈ [ ϕ T | V ≤ c ] ⊂ dom ϕ T | V .Then x ∈ Pr X (dom ϕ T | V ) ∩ V ⊂ Pr X ([ ϕ T | V ≥ c ] ∩ dom ϕ T | V ) , so ( x, x ∗ ) ∈ [ ϕ T | V ≥ c ] ∩ dom ϕ T | V forsome x ∗ ∈ X ∗ . The function f : [0 , → R , f ( t ) = ( ϕ T | V − c )( x, tx ∗ + (1 − t ) x ∗ ) is continuous and f (0) ≥ , f (1) ≤ . Therefore there is s ∈ [0 , such that f ( s ) = 0 , that is, ( x, sx ∗ + (1 − s ) x ∗ ) ∈ [ ϕ T | V = c ] . Therefore x ∈ Pr X [ ϕ T | V = c ] ∩ V ⊂ D ( T ) and so V locates T .If in addition T | V ∈ M ( X ) then T | V ⊂ [ ϕ T | V = c ] , D ( T ) ∩ V = D ( T | V ) ⊂ Pr X [ ϕ T | V = c ] , andthe last part of the conclusion follows from (i) ⇔ (ii).In Theorem 13 we saw that T being of V − NI type is an important part of V locating T andas a consequence every locatable operator is locally-NI. As previously stated, the other conditionin Theorem 13, namely, Pr X [ ϕ T | V = c ] ∩ V ⊂ D ( T ) is hard to verify directly due to the unwieldynature of ϕ T . Fortunately, this latter condition can be replaced by representability.7 efinition 14 Let ( X, τ ) be a LCS. An operator T : X ⇒ X ∗ is representable in V ⊂ X or V − representable if V ∩ D ( T ) = ∅ and there is h ∈ R (that is, h ≥ c and h ∈ Γ τ × w ∗ ( X × X ∗ ) )such that [ h = c ] ∩ V × X ∗ = Graph( T | V ) . The function h is called a V − representative of T . Theclass of V − representatives of T is denoted by R VT .As previously seen, the condition V ∩ D ( T ) = ∅ can be avoided but its presence makes theprevious definition meaningful.An X − representable operator T : X ⇒ X ∗ is simply called representable and the class ofits representatives is denoted by R T , notion that was first considered in this form in [15]. Forproperties of representable operators see [17, 18, 21, 22, 23].In other words, T is V − representable if T | V is the trace of the representable operator [ h = c ] on V × X ∗ , where h ∈ R . Remark 15
Note that • T | V is monotone whenever T is V − representable since [ h = c ] ∈ M ( X ) for every h ∈ C (seee.g. [8, Proposition 4] or [18, Lemma 3.1]); • T is W − representable whenever T is V − representable and V ⊃ W ; in this case every V − representative is a W − representative of T , that is, R VT ⊂ R WT . In particular, if T is rep-resentable then, for every V ⊂ X , T is V − representable. Conversely, if T is V − representablethen T need not be representable because we can modify T outside V × X ∗ . For example, forevery V X , x V , T = ( X \{ x } ) ×{ } is V − representable since ϕ T = ψ T = ι X ×{ } ∈ R VT and T is not representable because T is not closed or because T [ ψ T = c ] (see [15, Theorem2.2] or Theorem 16 below); • If T | V is representable then T is V − representable; in this case every representative of T | V is a V − representative of T , i.e., R T | V ⊂ R VT . Indeed, if h ∈ R has T | V = [ h = c ] then V × X ∗ ∩ [ h = c ] = T | V . Conversely, if T is V − representable with h a V − representativeof T such that Pr X [ h = c ] ⊂ V then T | V is representable with representative h . In general,without the additional condition, the converse is not true, in any LCS X , even if we workwith T ∈ M ( X ) and V open convex. Indeed, take C X closed convex with int C = ∅ , T = N C , V = int C . Then T is representable, since it is maximal monotone (see e.g. [15,Theorem 2.3] or Theorem 20 below), while T | V = int C × { } is not, for example because ψ T | V = ι C ×{ } and T | V = int C × { } ( C × { } = [ ψ T | V = c ] (see [15, Theorem 2.2] orTheorem 16 below); • However, when C is closed convex, T is C − representable iff T | C is representable. Indeed, if h is a C − representative of T then h + ι C × X ∗ is a representative of T | C . • An operator T ∈ M ( X ) is D ( T ) − representable whenever D ( T ) identifies T . Indeed, let T bea representable extension of T (such as [ ψ T = c ] ), and let h ∈ R T , in particular, T = [ h = c ] .Hence [ h = c ] ∩ D ( T ) × X ∗ = T | D ( T ) = T because T is maximal monotone in D ( T ) × X ∗ .The following result is a generalization of [15, Theorem 2.2], [23, Theorem 1(ii)]. Theorem 16
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X be such that V ∩ D ( T ) = ∅ . Thefollowing are equivalent (i) T is V − representable, (ii) T | V ∈ M ( X ) and [ ψ T | V = c ] ∩ V × X ∗ ⊂ T | V , (iii) T | V ∈ M ( X ) and [ ψ T | V = c ] ∩ V × X ∗ = T | V , (iv) ψ T | V is a V − representative of T | V , i.e., ψ T | V ∈ R VT | V . roof. (i) ⇒ (ii) Let h ∈ R be such that [ h = c ] ∩ V × X ∗ = T | V . Then h ≤ c T | V followed by c ≤ h ≤ ψ T | V since h ∈ R . Therefore [ ψ T | V = c ] ⊂ [ h = c ] and so [ ψ T | V = c ] ∩ V × X ∗ ⊂ T | V .(ii) ⇒ (iii) From T | V ∈ M ( X ) we know that T | V ⊂ [ ψ T | V = c ] (see [14, 17, Proposition 3.2(viii)] or [23, (9)]).For (iii) ⇒ (iv) it suffices to notice that ψ T | V ∈ R , since T | V ∈ M ( X ) (see Theorem 2(ii)).The implication (iv) ⇒ (i) is trivial. Remark 17
In case V is closed convex we have dom ψ T | V ⊂ V × X ∗ so [ ψ T | V = c ] ⊂ V × X ∗ , [ ψ T | V = c ] ∩ V × X ∗ = [ ψ T | V = c ] , and Theorem 16 says again that T is V − representable iff T | V is representable. Remark 18
Given X a LCS, T : X ⇒ X ∗ , and V ⊂ X such that T | V ∈ M ( X ) , the operator R := [ ψ T | V = c ] ∩ V × X ∗ is the smallest V − representable extension of T | V in V × X ∗ . Indeed,for every h ∈ R such that [ h = c ] ∩ V × X ∗ ⊃ T | V we have h ≤ c T | V , c ≤ h ≤ ψ T | V and so [ ψ T | V = c ] ∩ V × X ∗ ⊂ [ h = c ] ∩ V × X ∗ .Also, ϕ R = ϕ T | V , ψ R = ψ T | V because T | V ⊂ R ⊂ [ ψ T | V = c ] and ϕ T | V = ϕ [ ψ T | V = c ] (see [22,Proposition 4, p. 35]). Theorem 19
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X be such that V ∩ D ( T ) = ∅ .Consider the conditions (i) T | V ∈ M ( X ) and V identifies T , (ii) T is V − representable and V locates T , (iii) T is V − representable and V − NI.Then (i) ⇒ (ii) ⇒ (iii) . If, in addition, V is algebraically open then (i) ⇔ (ii) ⇔ (iii) . Proof.
We adapt the proof in [15, Theorem 2.3] and we refer to [17, 18, 15] for other differentarguments. The implication (ii) ⇒ (iii) is contained in Theorem 13.(i) ⇒ (ii) Since V identifies it also locates T . Hence, according to Theorem 13, T is V − NI.We know that ψ T | V ≥ max { ϕ T | V , c } since T | V ∈ M ( X ) (see [14, 17, Proposition 3.2 (vii)]). Thisyields that [ ψ T | V = c ] ∩ V × X ∗ ⊂ [ ϕ T | V = c ] ∩ V × X ∗ ⊂ T from which, according to Theorem16, it follows that T is V − representable.Assume that V is algebraically open, i.e., V = core V .(iii) ⇒ (i) Since T is V − NI we have [ ϕ T | V ≤ c ] ∩ V × X ∗ = [ ϕ T | V = c ] ∩ V × X ∗ and from T being V − representable we know that T | V ∈ M ( X ) so, according to Theorem 2 (ii), ψ T | V ≥ c ;whence [ ψ T | V = c ] ⊂ [ ϕ T | V = c ] (see Lemma 1). This yields T | V = [ ψ T | V = c ] ∩ V × X ∗ ⊂ [ ϕ T | V = c ] ∩ V × X ∗ since T is V − representable. To conclude it suffices to show that [ ϕ T | V = c ] ∩ V × X ∗ ⊂ [ ψ T | V = c ] ∩ V × X ∗ . Let z ∈ [ ϕ T | V = c ] ∩ V × X ∗ . Because V is algebraically open, for every v ∈ X × X ∗ there is t v > such that z + tv ∈ V × X ∗ , for every < t < t v . Hence, since T is V − NI, ϕ T | V ( z + tv ) ≥ c ( z + tv ) ,for every < t < t v . The directional derivative of ϕ T | V at z in the direction of v satisfies ∀ v ∈ Z, ϕ ′ T | V ( z ; v ) := lim t ↓ ϕ T | V ( z + tv ) − ϕ T | V ( z ) t ≥ lim t ↓ c ( z + tv ) − c ( z ) t = z · v. This shows that z ∈ ∂ϕ T | V ( z ) , where “ ∂ ” is considered under the natural duality ( Z, Z ) . Therefore ψ T | V ( z )+ ϕ T | V ( z ) = z · z = 2 c ( z ) which, together with ϕ T | V ( z ) = c ( z ) , implies that ψ T | V ( z ) = c ( z ) .In particular, for V = X we recover the following maximal monotonicity characterization.9 heorem 20 ([15, Theorem 2.3], [23, Theorem 1 (ii)]) Let X be a LCS. Then T ∈ M ( X ) iff T is representable and NI. Theorem 21
Let X be a LCS, let T : X ⇒ X ∗ , and let V be a class of algebraically open subsetsof X such that X ∈ V . Then T ∈ M ( X ) and T is identified by every V ∈ V iff T is representableand, for every V ∈ V , T is V − NI.
Proof.
Since every representable operator is V − representable (for every V ⊂ X ) and monotonethe converse implication is straightforward from Theorem 19. For the direct implication one getsthat T is maximal monotone (and implicitly representable) because it is identified by X ∈ V .Again Theorem 19 completes the argument. Theorem 22
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X be non-empty algebraically openand convex such that T | V ∈ M ( X ) and T is V − NI. Then [ ψ T | V = c ] ∩ V × X ∗ = [ ϕ T | V = c ] ∩ V × X ∗ = [ ϕ T | V ≤ c ] ∩ V × X ∗ (4) is the unique V − representable extension and the unique maximal monotone extension in V × X ∗ of T | V .If, in addition, T ∈ M ( X ) then the string of equalities in (4) can be completed to [ ψ T | V = c ] ∩ V × X ∗ = [ ϕ T = c ] ∩ V × X ∗ = [ ψ T = c ] ∩ V × X ∗ . (5) If, in addition, T ∈ M ( X ) and T is V − representable then V identifies T and [ ψ T | V = c ] ∩ V × X ∗ = [ ϕ T | V = c ] ∩ V × X ∗ = [ ϕ T | V ≤ c ] ∩ V × X ∗ = [ ϕ T = c ] ∩ V × X ∗ = [ ψ T = c ] ∩ V × X ∗ = Graph( T | V ) . (6) Proof.
Let R := [ ψ T | V = c ] ∩ V × X ∗ . Then R is V − representable and V − NI since T | V ∈ M ( X ) , T is V − NI, and ϕ R = ϕ T | V . According to Theorem 19, V identifies R , i.e., R is maximal monotonein V × X ∗ . From ψ T | V ≥ ϕ T | V and ϕ T | V ≥ c in V × X ∗ we know that R ⊂ [ ϕ T | V = c ] ∩ V × X ∗ =[ ϕ T | V ≤ c ] ∩ V × X ∗ so R = [ ϕ T | V = c ] ∩ V × X ∗ since [ ϕ T | V = c ] ∩ V × X ∗ ∈ M ( X ) . Taking intoconsideration that R is the smallest V − representable extension of T | V the conclusion follows.If, in addition, T ∈ M ( X ) then, due to the facts that T is V − NI and T | V ⊂ T , we have thatfor every z ∈ V × X ∗ c ( z ) ≤ ϕ T | V ( z ) ≤ ϕ T ( z ) ≤ ψ T ( z ) ≤ ψ T | V ( z ) , whence [ ψ T | V = c ] ∩ V × X ∗ ⊂ [ ψ T = c ] ∩ V × X ∗ ⊂ [ ϕ T = c ] ∩ V × X ∗ ⊂ [ ϕ T | V = c ] ∩ V × X ∗ .Relation (4) completes the proof of (5).Relation (6) follows from Theorems 16, 19 and relations (4), (5).We are ready to prove that for a representable (and implicitly for a maximal monotone) oper-ator the locatable and identifiable notions coincide. Theorem 23
Let X be a LCS and let T : X ⇒ X ∗ . The following are equivalent (i) T ∈ M ( X ) and T is identifiable, (ii) T is representable and locatable, (iii) T is representable and locally-NI.In particular, every representable and locatable operator is maximal monotone. Proof. (i) ⇔ (iii) is a particular case of Theorem 21 for V = { V ⊂ X | V is open and convex , V ∩ D ( T ) = ∅} .(i) ⇒ (ii) is true since every identifiable monotone operator is locatable and maximal monotone.(ii) ⇒ (iii) is straightforward since every locatable operator is locally-NI.The global representability condition in the previous theorem can be replaced by a weaker localform of it. 10 efinition 24 Let X be a LCS. An operator T : X ⇒ X ∗ is low-representable if, for every z = ( x, x ∗ ) ∈ [ ψ T = c ] , there is V ∈ V ( x ) such that T is V − representable.Every representable operator is low-representable (just take V = X ). Theorem 25
Let X be a LCS and let T : X ⇒ X ∗ . Consider the conditions (iv) T is monotone, low-representable, and locatable; (v) T is monotone, low-representable, and locally-NI.Conditions (i) – (iii) being those from Theorem 23, we have (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) ⇔ (v). Proof.
The implications (ii) ⇒ (iv), (iii) ⇒ (v), (iv) ⇒ (v) are plain.For (v) ⇒ (iii) we prove that T is representable, i.e., [ ψ T = c ] ⊂ T . For every z = ( x, x ∗ ) ∈ [ ψ T = c ] let V ∈ V ( x ) be open convex and such that T is V − representable. Then, according toTheorem 22, z ∈ [ ψ T = c ] ∩ V × X ∗ ⊂ T . Theorem 26
Let ( X, τ ) be a LCS and let T ∈ M ( X ) be locally-NI. Then [ ϕ T = c ] = [ ϕ T ≤ c ] =[ ψ T = c ] is the unique identifiable extension of T . Proof.
Because T is NI, from (4), S := [ ϕ T = c ] = [ ϕ T ≤ c ] = [ ψ T = c ] is the unique maximalmonotone extension of T . Since every identifiable operator is maximal monotone, it suffices toprove that S is locally-NI to get that S is the unique identifiable extension of T .But, if an open convex V ⊂ X has V ∩ D ( S ) = ∅ then V ∩ D ( T ) = ∅ .Indeed, if x ∈ V ∩ D ( S ) pick any x ∗ ∈ S ( x ) and set z = ( x, x ∗ ) ∈ S . Then, since T is V − NI, c ( z ) ≤ ϕ T | V ( z ) ≤ ϕ T ( z ) = c ( z ) so, according to Theorem 22, z ∈ [ ϕ T | V = c ] ∩ V × X ∗ =[ ψ T | V = c ] ∩ V × X ∗ ⊂ dom ψ T | V ⊂ cl τ × w ∗ (Graph( T | V )) followed by x ∈ Pr X (dom ψ T | V ) ⊂ conv( D ( T ) ∩ V ) ⊂ D ( T ) due to the convexity of D ( T ) (see Theorem 9). Hence x ∈ V ∩ D ( T ) = ∅ from which V ∩ D ( T ) = ∅ since V is open.Hence, for every open convex V ⊂ X such that V ∩ D ( S ) = ∅ , ϕ S | V ≥ ϕ T | V ≥ c in V × X ∗ ,because T is V − NI, i.e., S is V − NI.Therefore S is locally-NI.The next result is a version of Theorem 19 for closed convex sets with non-empty interior.First note that for every T : X ⇒ X ∗ and C ⊂ X we have [ ϕ T + N C ≤ c ] ∩ C × X ∗ = [ ϕ T | C ≤ c ] ∩ C × X ∗ . (7)Indeed, the direct inclusion follows from T | C ⊂ T + N C . Conversely, if z = ( x, x ∗ ) ∈ C × X ∗ is m.r. to T | C and ( a, a ∗ ) ∈ T | C , n ∗ ∈ N C ( a ) then h x − a, n ∗ i ≤ , h x − a, x ∗ − a ∗ i ≥ , and h x − a, x ∗ − a ∗ − n ∗ i ≥ , that is, z is m.r. to T + N C . Theorem 27
Let X be a LCS, let T : X ⇒ X ∗ , and let C ⊂ X be closed convex such that D ( T ) ∩ int C = ∅ . If T is C − representable then the following are equivalent (i) C locates T , (ii) T is C − NI, (iii) [ ϕ T | C ≤ c ] ∩ C × X ∗ ⊂ T + N C , (iv) C identifies T + N C . Proof.
The implication (i) ⇒ (ii) is part of Theorem 13 while (iii) ⇒ (i) is plain.(ii) ⇒ (iii) Since T is C − NI, we have h := ϕ T | C + ι C × X ∗ ∈ R and [ h = c ] = [ ϕ T | C = c ] ∩ C × X ∗ = [ ϕ T | C ≤ c ] ∩ C × X ∗ . According to [24, Theorem 2.8.7 (iii), p. 127] h (cid:3) ( x, x ∗ ) = min { ψ T | C ( x, u ∗ ) + σ C ( x ∗ − u ∗ ) | u ∗ ∈ X ∗ } , ( x, x ∗ ) ∈ X × X ∗ . (8)11ere “ min ” stands for an infimum that is attained when finite.For every z = ( x, x ∗ ) ∈ [ h (cid:3) = c ] ∩ C × X ∗ there is v ∗ ∈ X ∗ such that ψ T | C ( x, v ∗ )+ σ C ( x ∗ − v ∗ ) = h x, x ∗ i . Since ψ T | C ≥ c , σ C ( x ∗ − v ∗ ) ≥ h x, x ∗ − v ∗ i , this implies that ( x, v ∗ ) ∈ [ ψ T | C = c ] ∩ C × X ∗ = T | C , x ∗ − v ∗ ∈ N C ( x ) , and z ∈ T + N C . Therefore [ h (cid:3) = c ] ∩ C × X ∗ ⊂ T + N C . The inclusion [ h = c ] ⊂ [ h (cid:3) = c ] completes the proof of this implication (see Lemma 1).(iii) ⇔ (iv) Since ( T + N C ) | C = T + N C , this equivalence follows from (7).In the absence of the C − representability of T the previous result still holds with T replacedby R = [ ψ T | C = c ] ∩ C × X ∗ which is the smallest C − representable extension of T in the followingstring of implications: C locates T ⇒ T is C − NI ⇔ R is C − NI ⇔ C locates R ⇔ C identifies R + N C . The converse of the first implication is false as seen from Remark 31 below for C = V . Corollary 28
Let X be a LCS, let T : X ⇒ X ∗ , and let C ⊂ X be closed convex such that D ( T ) ∩ int C = ∅ and T is C − representable. Then T + N C ∈ M ( X ) iff C locates T and [ ϕ T + N C ≤ c ] ⊂ C × X ∗ iff T is C − NI and [ ϕ T + N C ≤ c ] ⊂ C × X ∗ . Proposition 29
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X be open convex such that D ( T ) ∩ V = ∅ and T | V ∈ M ( X ) . If T is V − NI then ϕ T | V ≥ c, on V × X ∗ . In particular, for every V ⊂ S ⊂ V , T is S − NI.If, in addition, T is V − representable, then [ ϕ T | V ≤ c ] ∩ V × X ∗ ⊂ T + N V ⊂ D ( T ) × X ∗ . In particular, for every V ⊂ S ⊂ V , S locates T and identifies T + N V . Proof.
Seeking a contradiction assume that there is z = ( x, x ∗ ) ∈ [ ϕ T | V < c ] ∩ V × X ∗ . Since T is V − NI we know that x ∈ bd V .Let y ∈ V ∩ D ( T ) , w = ( y, y ∗ ) ∈ T , and h : [0 , → R , h ( t ) = ( ϕ T | V − c )( tw + (1 − t ) z ) . Notethat h (0) < , h (1) = 0 , since w ∈ T | V ∈ M ( X ) , and h is continuous. Hence there is δ ∈ (0 , such that h ( δ ) < . That provides the contradiction δw + (1 − δ ) z ∈ [ ϕ T | V < c ] ∩ V × X ∗ .Note that h := ϕ T | V + ι V × X ∗ ∈ R , [ h = c ] = [ ϕ T | V ≤ c ] ∩ V × X ∗ , and h (cid:3) ( x, x ∗ ) = min { ψ T | V ( x, u ∗ ) + σ V ( x ∗ − u ∗ ) | u ∗ ∈ X ∗ } , ( x, x ∗ ) ∈ X × X ∗ . For every z = ( x, x ∗ ) ∈ [ h (cid:3) = c ] ∩ V × X ∗ there is v ∗ ∈ X ∗ such that ψ T | V ( x, v ∗ ) + σ V ( x ∗ − v ∗ ) = h x, x ∗ i . This implies that ( x, v ∗ ) ∈ [ ψ T | V = c ] ∩ V × X ∗ ⊂ [ ψ T | V = c ] ∩ V × X ∗ and x ∗ − v ∗ ∈ N V ( x ) .If, in addition, T is V − representable then [ ψ T | V = c ] ∩ V × X ∗ = T | V so z ∈ T + N V . Hence [ ϕ T | V ≤ c ] ∩ V × X ∗ = [ h = c ] ⊂ [ h (cid:3) = c ] ∩ V × X ∗ ⊂ T + N V ⊂ D ( T ) × X ∗ . Proposition 30
Let X be a LCS, let T : X ⇒ X ∗ , and let V ⊂ X be open convex such that D ( T ) ∩ V = ∅ and T is V − representable. If V locates T then, for every V ⊂ S ⊂ V , S locates T and identifies T + N V . Remark 31
Let T = (0 , × { } ⊂ R . Then V = (0 ,
1) = D ( T ) is open convex and identifies T (and, according to Remark 15, that makes T become V − representable) while V = [0 , does notlocate T since z = (0 , is m.r. to T and ∈ V \ D ( T ) .This example shows the necessity of the V − representability condition in the previous twopropositions and also, that this condition cannot be replaced by V − representability.12 heorem 32 Let X be a LCS and let T : X ⇒ X ∗ . (i) T is locally-NI iff, for every closed convex C ⊂ X such that D ( T ) ∩ int C = ∅ , T is C − NI. (ii) T is locatable iff, for every closed convex C ⊂ X such that D ( T ) ∩ int C = ∅ , [ ϕ T | C ≤ c ] ∩ int C × X ∗ ⊂ D ( T ) × X ∗ . In particular if, for every closed convex C ⊂ X such that D ( T ) ∩ int C = ∅ , C locates T then T islocatable. (iii) T is identifiable iff, for every closed convex C ⊂ X such that D ( T ) ∩ int C = ∅ , [ ϕ T | C ≤ c ] ∩ int C × X ∗ ⊂ Graph( T ) , In particular if, for every closed convex C ⊂ X such that D ( T ) ∩ int C = ∅ , C identifies T + N C then T is identifiable. (iv) T is monotone and identifiable iff T is representable and, for every closed convex C ⊂ X with D ( T ) ∩ int C = ∅ , C locates T iff T is monotone low-representable and, for every closedconvex C ⊂ X with D ( T ) ∩ int C = ∅ , T is C − NI.
Proof.
First we prove that for every open convex V ⊂ X such that D ( T ) ∩ V = ∅ and forevery x ∈ V there is a closed convex C ⊂ V such that D ( T ) ∩ int C = ∅ and x ∈ int C . Indeed,take y ∈ D ( T ) ∩ V and a closed convex U ∈ V (0) such that C := [ x, y ] + U ⊂ V . Note that C is closed convex and y ∈ D ( T ) ∩ int C . The last inclusion is possible since if we assume theopposite, namely that for every U ∈ V (0) there is x U ∈ ([ x, y ] + U ) \ V , that is, x U − y U ∈ U forsome y U ∈ [ x, y ] , because [ x, y ] is compact, on a subnet, denoted by the same index for notationsimplicity, y U → y ∈ [ x, y ] ⊂ V and so we reach the contradiction x U → y V .(i) ( ⇒ ) For every closed convex C ⊂ X such that D ( T ) ∩ int C = ∅ , T is int C − NI. Accordingto Proposition 29, T is also C = cl(int C ) − NI.( ⇐ ) For every V ⊂ X open convex such that D ( T ) ∩ V = ∅ and every z = ( x, x ∗ ) ∈ V × X ∗ let C ⊂ V be closed convex such that D ( T ) ∩ int C = ∅ and x ∈ C . Since T is C − NI and T | C ⊂ T | V we get ϕ T | V ( z ) ≥ ϕ T | C ( z ) ≥ c ( z ) , i.e., T is locally-NI.(ii) ( ⇒ ) For every closed convex C ⊂ X such that D ( T ) ∩ int C = ∅ , we have [ ϕ T | C ≤ c ] ∩ int C × X ∗ ⊂ [ ϕ T | int C ≤ c ] ∩ int C × X ∗ ⊂ D ( T ) × X ∗ since int C locates T .( ⇐ ) For every V ⊂ X open convex such that D ( T ) ∩ V = ∅ and every z = ( x, x ∗ ) ∈ [ ϕ T | V ≤ c ] ∩ V × X ∗ let C ⊂ V be closed convex such that D ( T ) ∩ int C = ∅ and x ∈ int C . Then z ∈ [ ϕ T | C ≤ c ] ∩ int C × X ∗ ⊂ D ( T ) × X ∗ . This yields that [ ϕ T | V ≤ c ] ∩ V × X ∗ ⊂ D ( T ) × X ∗ ,that is, V locates T .The proof of (iii) is similar to the argument used for (ii). In particular if, for every closedconvex C ⊂ X such that D ( T ) ∩ int C = ∅ , C identifies T + N C then for every V ⊂ X openconvex such that D ( T ) ∩ V = ∅ and every z = ( x, x ∗ ) ∈ [ ϕ T | V ≤ c ] ∩ V × X ∗ let C ⊂ V be closedconvex such that D ( T ) ∩ int C = ∅ and x ∈ int C . Hence z ∈ [ ϕ T | C ≤ c ] ∩ int C × X ∗ = [ ϕ T + N C ≤ c ] ∩ int C × X ∗ ⊂ Graph( T ) .Subpoint (iv) is a direct consequence of (i) and Theorems 23, 25, 27.13 Representability via the convolution operation
The goal of this section is to study, in the context of locally convex spaces, the representabilityof the sum A + B of two representable operators A , B under classical qualification constraints.Under a Banach space settings, the calculus rules of representable operators can be found in [18,Section 5]. The following result holds Proposition 33 (Zălinescu [25, Proposition 1]) Let X , X be LCS’s and let f , f : X = X × X → R be proper convex functions. If there exists ( x , x ) ∈ dom f ∩ dom f such that f ( · , x ) is continuous at x and f ( x , · ) is continuous at x then, for every x ∗ ∈ X ∗ = X ∗ × X ∗ ( f + f ) ∗ ( x ∗ ) = min { f ∗ ( u ∗ ) + f ∗ ( x ∗ − u ∗ ) | u ∗ ∈ X ∗ } . Here “ min ” stands for an infimum that is attained when finite.
Theorem 34
Let E , F be LCS’s and let φ , φ : E × F → R be proper convex functions. Consider ρ : E × F → R defined by ρ ( x, y ) := inf { φ ( x, y ) + φ ( x, y ) | y + y = y } . Assume that there exists ( x , y ) ∈ dom φ such that x ∈ Pr E (dom φ ) and φ ( · , y ) is con-tinuous at x . Then, for every x ∗ ∈ E ∗ , y ∗ ∈ F ∗ ρ ∗ ( x ∗ , y ∗ ) = min { φ ∗ ( x ∗ , y ∗ ) + φ ∗ ( x ∗ , y ∗ ) | x ∗ + x ∗ = x ∗ ∈ X ∗ } . Proof.
Note first that dom ρ = ∅ because there is ˜ y ∈ F such that ( x , ˜ y ) ∈ dom φ , so ρ ( x , ˜ y + y ) ≤ φ ( x , ˜ y ) + φ ( x , y ) < + ∞ ; whence ρ ∗ does not take the value −∞ .For every x ∈ E , y , y ∈ F , x ∗ , x ∗ ∈ E ∗ , y ∗ ∈ F ∗ we have φ ( x, y ) + φ ( x, y ) + φ ∗ ( x ∗ , y ∗ ) + φ ∗ ( x ∗ , y ∗ ) ≥ h x, x ∗ + x ∗ i + h y + y , y ∗ i , so, for every ( x, y ) ∈ E × F , x ∗ , x ∗ ∈ E ∗ , y ∗ ∈ F ∗ , ρ ( x, y ) + φ ∗ ( x ∗ , y ∗ ) + φ ∗ ( x ∗ , y ∗ ) ≥ h x, x ∗ + x ∗ i + h y, y ∗ i , from which, for every x ∗ , x ∗ ∈ E ∗ , y ∗ ∈ F ∗ , φ ∗ ( x ∗ , y ∗ ) + φ ∗ ( x ∗ , y ∗ ) ≥ ρ ∗ ( x ∗ + x ∗ , y ∗ ) . Hence ∀ ( x ∗ , y ∗ ) ∈ E ∗ × F ∗ , ρ ∗ ( x ∗ , y ∗ ) ≤ inf { φ ∗ ( x ∗ , y ∗ ) + φ ∗ ( x ∗ , y ∗ ) | x ∗ + x ∗ = x ∗ ∈ X ∗ } . If ρ ∗ ( x ∗ , y ∗ ) = + ∞ the conclusion holds.If ρ ∗ ( x ∗ , y ∗ ) ∈ R consider f , f : ( E × F ) × F → R given by f ( x, y ; z ) = φ ( x, z ) , f ( x, y ; z ) = φ ( x, y ) − h x, x ∗ i − h y + z, y ∗ i . Notice that, for every u ∗ ∈ E ∗ , v ∗ , z ∗ ∈ F ∗ f ∗ ( u ∗ , v ∗ ; z ∗ ) = φ ( u ∗ , z ∗ ) + ι { } ( v ∗ ) f ∗ ( u ∗ , v ∗ ; z ∗ ) = φ ( x ∗ + u ∗ , y ∗ + v ∗ ) + ι { } ( y ∗ + z ∗ ) , ( f + f ) ∗ (0) = − inf { φ ( x, y ) + φ ( x, z ) − h x, x ∗ i − h y + z, y ∗ i | x ∈ E, y, z ∈ F } = − inf { ρ ( x, v ) − h x, x ∗ i − h v, y ∗ i | x ∈ E, v ∈ F } = ρ ∗ ( x ∗ , y ∗ ) ∈ R . Let ζ := ( x , ˜ y ) ∈ dom φ and let ζ = y . Then ( ζ , ζ ) ∈ dom f ∩ dom f , ζ = ( x, y ) → f ( x, y ; ζ ) = φ ( x, y ) is continuous at ζ , and f ( x , ˜ y ; · ) is continuous at ζ . From Proposition33 we obtain u ∗ ∈ E ∗ , v ∗ , z ∗ ∈ F ∗ such that ρ ∗ ( x ∗ , y ∗ ) = f ∗ ( u ∗ , v ∗ ; z ∗ ) + f ∗ ( − u ∗ , − v ∗ ; − z ∗ ) , i.e., v ∗ = 0 , z ∗ = y ∗ and ρ ∗ ( x ∗ , y ∗ ) = φ ( x ∗ − u ∗ , y ∗ ) + φ ( u ∗ , y ∗ ) . Theorem 35
Let X be a LCS, let A : X ⇒ X ∗ be representable, and let C ⊂ X be closed convex.If D ( A ) ∩ int C = ∅ then A + N C is representable. roof. Let x ∈ D ( A ) ∩ int C , a ∗ ∈ Ax . We apply the previous theorem for E = X , F =( X ∗ , w ∗ ) , φ = ϕ A , φ ( x, x ∗ ) = ι C ( x ) + σ C ( x ∗ ) , ( x, x ∗ ) ∈ X × X ∗ , and y = 0 ∈ dom σ C to getthat ρ ( x, x ∗ ) := inf { ϕ A ( x, x ∗ ) + ι C ( x ) + σ C ( x ∗ ) | x ∗ + x ∗ = x ∗ } , ( x, x ∗ ) ∈ X × X ∗ , has ρ (cid:3) ( x, x ∗ ) := min { ψ A ( x, x ∗ ) + ι C ( x ) + σ C ( x ∗ ) | x ∗ + x ∗ = x ∗ } , ( x, x ∗ ) ∈ X × X ∗ . which is a representative of A + N C . Theorem 36
Let X be a barreled LCS and let A, B : X ⇒ X ∗ be representable such that D ( A ) ∩ int D ( B ) = ∅ . For every x ∈ X , there exist V ∈ V ( x ) , such that A + B is V − representable. Inparticular, A + B is low-representable. Proof.
Fix x ∈ D ( A ) ∩ int D ( B ) , a ∗ ∈ Ax , b ∗ ∈ Bx , x ∗ = a ∗ + b ∗ , z := ( x , x ∗ ) ∈ A + B, and symmetric open convex U , U ∈ V (0) such that U + U ⊂ U , x + U ⊂ D ( B ) , and B ( x + U ) is equicontinuous; for simplicity B ( x + U ) ⊂ U ◦ .Take an arbitrary x ∈ X and denote by V = [ x , x ] + U ∈ V ( x ) .Notice that ( x , a ∗ ) ∈ A | V so x ∈ Pr X (dom ϕ A | V ) and ϕ B | V ( · , is continuous at x since ϕ B | V ( · , is bounded from above on x + U . Indeed, for every y ∈ x + U ⊂ D ( B ) , y ∗ ∈ B ( y ) , b ∈ V ∩ D ( B ) , b ∗ ∈ B ( b ) , we have y − b ∈ [0 , x − x ] + U and h y − b, b ∗ i ≤ h y − b, y ∗ i . Hence, forevery y ∈ x + U , y ∗ ∈ B ( y ) ϕ B | V ( y,
0) = sup {h y − b, b ∗ i | b ∈ V ∩ D ( B ) , b ∗ ∈ B ( b ) }≤ sup {h y − b, y ∗ i | b ∈ V ∩ D ( B ) , b ∗ ∈ B ( b ) }≤ sup {|h x − x, u ∗ i| + 1 | u ∗ ∈ B ( x + U ) } < + ∞ . Consider ρ : X × X ∗ → R , ρ ( y, y ∗ ) = inf { ϕ A | V ( y, u ∗ ) + ϕ B | V ( y, v ∗ ) | u ∗ + v ∗ = y ∗ } . We apply Proposition 34 for E = X , F = ( X ∗ , w ∗ ) to get ρ (cid:3) ( y, y ∗ ) = min { ψ A | V ( y, u ∗ ) + ψ B | V ( y, v ∗ ) | u ∗ + v ∗ = y ∗ } . Note that ϕ A + B | V ≤ ρ , so, ρ (cid:3) ≤ ψ A + B | V . Therefore, for every w = ( y, y ∗ ) ∈ [ ψ A + B | V = c ] ∩ V × X ∗ there exists u ∗ ∈ X ∗ such that ( y, u ∗ ) ∈ [ ψ A | V = c ] , ( y, x ∗ − u ∗ ) ∈ [ ψ B | V = c ] , since ρ (cid:3) ≥ c . Because y ∈ V and A, B are V − representable, we get that w ∈ A + B , that is, A + B is V − representable.
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