Keywords monotone operator, dual system, Fitzpatrick function

Mathematics Subject Classification (2020) 47H05, 46N99.

1 Introduction↩︎

In [1] Gossez presents his famous example of a monotone operator which exhibits strange range properties (see also [2], [3]), namely, \(G:\ell_{1}\to\ell_{\infty}=\ell_{1}^{*}\) given by \[(Gx)_{n}:=-\sum_{k<n}x_{k}+\sum_{k>n}x_{k}=\sum_{k\in\mathbb{N}}x_{k}\alpha(k,n),\;x=(x_{k})_{k\in\mathbb{N}}\in\ell_{1},\;n\in\mathbb{N},\label{eq:}\tag{1}\] where \(\alpha(k,n)=\left\{ \begin{array}{lll} -1, & \mathrm{if} & k<n\\ 0, & \mathrm{if} & k=n\\ 1, & \mathrm{if} & k>n \end{array}\right.\), \(k,n\in\mathbb{N}\).

Here \[\ell_{1}:=\{x=(x_{n})_{n\in\mathbb{N}}\mid\|x\|_{\ell_{1}}:=\sum_{n\in\mathbb{N}}|x_{n}|<+\infty\},\;\ell_{\infty}:=\{x=(x_{n})_{n\in\mathbb{N}}\mid\|x\|_{\ell_{\infty}}:=\sup_{n\in\mathbb{N}}|x_{n}|<+\infty\}\] \[\mathfrak{c}:=\{y=(y_{n})_{n\in\mathbb{N}}\mid\lim_{n\to\infty}y_{n}\in\mathbb{R}\}.\]

The linear operator \(G\) is well-defined and bounded since, for every \(x=(x_{n})_{n}\in\ell_{1}\), \[\|Gx\|_{\ell_{\infty}}=\sup_{n\in\mathbb{N}}|(Gx)_{n}|\le\sum_{n\in\mathbb{N}}|x_{n}|=\|x\|_{\ell_{1}}.\]

For a topologically vector space \((E,\mu)\), \(A\subset E\), and \(f,g:E\rightarrow\overline{\mathbb{R}}\) we denote by \(E^{*}\) the topological dual of \(E\) and by \([f\le g]\) \(:=\{x\in E\mid f(x)\leq g(x)\}\); the sets \([f=g]\), \([f<g]\), \([f>g]\), \([f\ge g]\) are similarly defined.

A dual system is a triple \((X,Y,c)\) consisting of two vector spaces \(X\), \(Y\) and a bilinear map \(c=\langle\cdot,\cdot\rangle:X\times Y\to\mathbb{R}\). The weakest topology on \(X\) for which all linear forms \(\{X\ni x\to\langle x,y\rangle\mid y\in Y\}\) are continuous is denoted by \(\sigma(X,Y)\). This topology is locally convex and called the weak topology of \(X\) with respect to the duality \((X,Y,c)\). Similarly one defines the weak topology \(\sigma(Y,X)\) on \(Y\).

For \(S\subset X\), \[S^{\perp}:=\{y\in Y\mid\forall x\in S,\;\langle x,y\rangle=0\}\] denotes the orthogonal of \(S\). Similarly, for \(A\subset Y\), \(A^{\perp}:=\{x\in X\mid\forall y\in A,\;\langle x,y\rangle=0\}\) is the orthogonal of \(A\).

A template for dual systems \((X,Y,\langle\cdot,\cdot\rangle)\) is given by \((X,X^{*},\langle\cdot,\cdot\rangle)\) where \((X,\tau)\) is a locally convex space, \(X^{*}\) is the topological dual of \((X,\tau)\), and \(\langle x,x^{*}\rangle=x^{*}(x)\), \(x\in X\), \(x^{*}\in X^{*}\).

In this paper we study several operators and sets defined in special dual systems, e.g., \((\ell_{1},\ell_{\infty})\), \((\ell_{\infty},\ell_{\infty}^{*})\), \((\ell_{1}\times\ell_{\infty},\ell_{\infty}\times\ell_{1})\), or \((\ell_{1}\times\ell_{\infty},\ell_{\infty}\times\ell_{\infty}^{*})\); \(\ell_{1}^{*}\) is identified with \(\ell_{\infty}\), and recall that the coupling for the duality \((\ell_{1},\ell_{\infty})\) is \[\langle x,y\rangle_{\ell_{1}\times\ell_{\infty}}:=\langle x,y\rangle:=\sum_{n\in\mathbb{N}}x_{n}y_{n},\;x=(x_{n})_{n\in\mathbb{N}}\in\ell_{1},y=(x_{n})_{n\in\mathbb{N}}\in\ell_{\infty},\] and it coincides with the coupling of \((\ell_{\infty}^{*},\ell_{\infty})\), that is, \[\forall x\in\ell_{1},\;\forall y\in\ell_{\infty},\;\langle x,y\rangle_{\ell_{\infty}^{*}\times\ell_{\infty}}=\langle x,y\rangle_{\ell_{1}\times\ell_{\infty}}\] which allows us to use the simplified notation of \(\langle\cdot,\cdot\rangle\) for both couplings.

Given a dual system \((X,Y,c=\langle\cdot,\cdot\rangle)\), \(Z:=X\times Y\) forms naturally a dual system \((Z,Z,\cdot)\) where, for \(z=(x,y),\;w=(u,v)\in Z\), \[z\cdot w:=c(x,v)+c(u,y).\label{eq-161}\tag{2}\] The space \(Z\) is endowed with a locally convex topology \(\mu\) compatible with the natural duality \((Z,Z,\cdot)\), that is, for which \((Z,\mu)^{*}=Z\); for example \(\mu=\sigma(Z,Z)=\sigma(X,Y)\times\sigma(Y,X)\). With respect to the natural dual system \((Z,Z,\cdot)\), the conjugate of \(f\) is denoted by \[f^{\square}:Z\rightarrow\overline{\mathbb{R}},\quad f^{\square}(z)=\sup\{z\cdot z^{\prime}-f(z^{\prime})\mid z^{\prime}\in Z\}.\label{eq-160}\tag{3}\] By the biconjugate formula, \(f^{\square\square}=\operatorname*{cl}\operatorname*{conv}f\) whenever \(f^{\square}\) (or \(\operatorname*{cl}\operatorname*{conv}f\)) is proper.

To \(A\subset Z\) we associate \(c_{A}:Z\rightarrow\overline{\mathbb{R}}\), \(c_{A}:=c+\iota_{A}\), where \(\iota_{A}(z)=0\), for \(z\in A\), \(\iota_{A}(z)=+\infty\), otherwise; the Fitzpatrick function of \(A\) \[\varphi_{A}:Z\rightarrow\overline{\mathbb{R}},\;\varphi_{A}(z):=c_{A}^{\square}(z)=\sup\{z\cdot w-c(w)\mid w\in A\},\label{eq-9}\tag{4}\] and \(\psi_{A}:Z\rightarrow\overline{\mathbb{R}}\), \(\psi_{A}:=\varphi_{A}^{\square}=c_{A}^{\square\square}\).

In particular when \(\varphi_{A}\) is proper (or \(\operatorname*{cl}\operatorname*{conv}c_{A}\) is proper), \[\psi_{A}=\operatorname*{cl}\operatorname*{conv}c_{A},\;\varphi_{A}=\psi_{A}^{\square}.\] Note that \(\varphi_{A}\) is improper iff \(\varphi_{A}\) is identically \(-\infty\) (when \(A=\emptyset\)) or \(\varphi_{A}\) is identically \(+\infty\).

Similarly, for a multifunction \(T:X\rightrightarrows Y\) we define \(c_{T}:=c_{\operatorname*{Graph}T}\), \(\psi_{T}:=\psi_{\operatorname*{Graph}T}\), and the Fitzpatrick function of \(T\), \(\varphi_{T}:=\varphi_{\operatorname*{Graph}T}\). By convention \(\varphi_{\emptyset}=-\infty\), \(c_{\emptyset}=\operatorname*{conv}c_{\emptyset}=\psi_{\emptyset}=+\infty\) in agreement with the usual conventions of \(\inf_{\emptyset}=+\infty\), \(\sup_{\emptyset}=-\infty\).

In expanded form, for \(T:X\rightrightarrows Y\) and \(z=(x,y)\in Z\), \((x,y)\in X\times Y\)

\[\begin{align} \varphi_{T}(z) & =\sup\{z\cdot w-c(w)\mid w\in\operatorname*{Graph}T\},\nonumber \\ \varphi_{T}(x,y) & =\sup\{\langle x,v\rangle+\langle u,y\rangle-\langle u,v\rangle\mid(u,v)\in\operatorname*{Graph}T\}.\label{def-Ff} \end{align}\tag{5}\]

In the sequel, for \(A\subset X\times Y\) we denote by \[\neg A:=\{(x,y)\in X\times Y\mid(x,-y)\in A\}.\label{eq-2}\tag{6}\]

For a multi-valued operator \(T:X\rightrightarrows Y\) we define \(\neg T:X\rightrightarrows Y\) via \(\operatorname*{Graph}(\neg T):=\neg\operatorname*{Graph}(T)\), or, equivalently, \[(\neg T)(x)=-T(x),\;x\in D(\neg T):=D(T).\]

Consider the following classes of functions on \(Z=X\times Y\):

\(\mathscr{C}:=\mathscr{C}(Z):=\{f\in\Lambda(Z)\mid f\geq c\}\) the class of proper convex functions that are greater than the coupling \(c\) on \(Z\);

\(\mathscr{R}:=\mathscr{R}(Z):=\Gamma(Z)\cap\mathscr{C}(Z)\) the class of proper convex lower semicontinuous functions that are greater than the coupling \(c\) on \(Z\);

\(\mathscr{D}:=\mathscr{D}(Z):=\{f\in\mathscr{R}(Z)\mid f^{\square}\geq c\}\) the class of proper convex lower semicontinuous functions \(f\) such that \(f\), \(f^{\square}\) are greater than the coupling \(c\) on \(Z\);

A multi-function \(T:X\rightrightarrows Y\) is

• monotone if, for all \(y_{1}\in T(x_{1})\), \(y_{2}\in T(x_{2})\), \(\left\langle x_{1}-x_{2},y_{1}-y_{2}\right\rangle \geq0\);

• maximal monotone if \(\operatorname*{Graph}T\) is maximal in the sense of graph inclusion as monotone subsets of \(X\times Y\);

• unique in \(Z=X\times Y\) if \(T\) is monotone and admits a unique maximal monotone extension in \(Z\).

• representable in \(Z=X\times Y\) if \(\operatorname*{Graph}T=[f=c]\) for some \(f\in\mathscr{R}(Z)\); in this case \(f\) is called a representative of \(T\). We denote by \(\mathscr{R}_{T}:=\mathscr{R}_{T}(Z)\) the class of representatives of \(T\);

• dual–representable in \(Z\) if \(\operatorname*{Graph}T=[f=c]\) for some \(f\in\mathscr{D}(Z)\); in this case \(f\) is called a d–representative of \(T\). We denote by \(\mathscr{D}_{T}:=\mathscr{D}_{T}(Z)\) the class of d–representatives of \(T\);

• of negative infimum type (NI for short) in \(Z\) if \(\varphi_{T}\geq c\) (in \(Z\));

A subset \(A\subset Z\) is said to have a certain property if \(A=\operatorname*{Graph}T\) for some muti-valued \(T:X\rightrightarrows Y\) which has that same property.

Gossez’s example in (1 ) distinguishes these notions in the sense seen in Theorem 4 below.

Recall that \(T:X\rightrightarrows Y\) is maximal monotone iff it is representable and NI iff it is dual-representable and unique (see e.g. [4]–[9]).

2 Preliminaries↩︎

Consider \(G^{*}:\ell_{\infty}^{*}\to\ell_{\infty}\) the adjoint of \(G:\ell_{1}\to\ell_{\infty}\), given by \[\langle y,G^{*}\mu\rangle=\langle\mu,Gy\rangle,\;y\in\ell_{1},\;\mu\in\ell_{\infty}^{*}.\]

In order to describe \(G^{*}\), in this article \(\ell_{\infty}\) is identified with \(C(\beta\mathbb{N})\) the space of continuous functions over the Hausdorff separated compact space \(\beta\mathbb{N}\) which is the Stone-Čech compactification of \(\mathbb{N}\) in the sense that every \(x\in\ell_{\infty}\) is identified with and uniquely extended to \(\beta x\in C(\beta\mathbb{N})\) such that \(\beta x|_{\mathbb{N}}=x\); \(\ell_{\infty}^{*}\) is identified with \(C(\beta\mathbb{N})^{*}\) which, by the Riesz’s Representation Theorem, is identified with \(M(\beta\mathbb{N})\) the space of signed regular Borel measures of finite total variation on \(\beta\mathbb{N}\) in the sense that for every \(F\in C(\beta\mathbb{N})^{*}\) there is a unique \(\mu\in M(\beta\mathbb{N})\) such that \[\forall f\in C(\beta\mathbb{N}),\;F(f)=\int_{\beta\mathbb{N}}fd\mu,\] \(F\) is denoted by \(\mu\) and, for \(\mu\in\ell_{\infty}^{*}\), \(x\in\ell_{\infty}\), we write \(\mu(x)=:\langle\mu,x\rangle={\displaystyle \int_{\beta\mathbb{N}}}\beta xd\mu\).

Dual-system results↩︎

References↩︎