On Special Properties for Continuous Convex Operators and Related Linear Operators

Abstract

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above.

1. Introduction

This paper provides an overview on a few basic topics in functional analysis, joined together by the notion of convexity and its applications. The references partially illustrate old and recent research in this area and relationships between them. The motivation of this paper consists of pointing out two different main aspects of convexity: convex operators and their properties, and Hahn–Banach type theorems applied to the Moment Problem. Concerning the second aspect, a related optimization problem with infinitely many linear constraints is solved. For basic notions in analysis and functional analysis related to this work, see references [1,2,3,4,5,6,7,8,9]. First, we prove a uniform boundedness theorem for a class of convex continuous operators. The corresponding result for classes of bounded linear operators is the well-known Banach–Steinhaus theorem, whose proof is based on Baire’s theorem. We assume that the domain space, which is a topological vector space, cannot be written as a union of a sequence of closed subsets, each of them having an empty interior. Similar results to our Theorem 1 proved below concerning classes of continuous convex operators were published in [9,10,11]. Notably, in [9], the case of sublinear operators is under attention. Following the idea of [10], we prove the existence of a common convex neighborhood of the origin $W_0$ in the domain space, for all involved convex operators, without assuming that the domain space is locally convex. The convexity of $W_0$ is a consequence of the properties of the codomain and of the convex continuous operators in the given class. The important case of classes of continuous sublinear operators is under attention. We study the classes of sublinear operators $P$ satisfying the symmetry condition $P\left(x\right)=P\left(-x\right)$ for all $x$ in the domain space $X$. We point out an example related to this first part. The relevance consists not only in reviewing the result from [10] but also completing it with some consequences and remarks, discussed in the end of Section 3.1. Such theorems and their consequences are published in [11]. From the point of view of uniform boundedness, references [12,13] discuss the collections of linear operators more. In the papers [14,15,16,17], the interested reader could find similar properties formulated in the physics setting and possible interactions, especially concerning new results in the Jensen-type inequalities.

The second part of the results section is first motivated by solving the existence problems related to the moment problem. Basic results on this subject are outlined in [1,2,3,4] and [18]. Second, we continue with results on the extension of linear functionals and linear operators, most of them being related to the moment problem. The classical moment problem is formulated as follows: given a sequence ${\left(y_n\right)}_{n\in {\mathbb{N}}^n}$ of real numbers, and a non-empty closed subset $F\subseteq {ℝ}^n$, find a positive regular Borel measure $\nu$ on $F$ such that the interpolation moment conditions hold.

$${{\displaystyle \int }}_F^{}{\phi }_j\left(t\right)\mathrm{d}\nu ={{\displaystyle \int }}_F^{}{t}^j\mathrm{d}\nu =y_j, j\in {\mathbb{N}}^n,$$

(1)

Here, we use the notations:

$$\begin{array}{c}\mathbb{N}=\{0,1,2,\dots \},{\phi }_j\left(t\right)=t^j=t_1^{j_1}\cdots t_n^{j_n},j=\left(j_1,\dots ,j_n\right)\in {\mathbb{N}}^n,\\ t=\left(t_1,\dots ,t_n\right)\in F\subseteq {\mathbb{R}}^n,\mathcal{P}=\mathbb{R}\left[t_1,\dots ,t_n\right],n\in \mathbb{N},n\ge 1.\end{array}$$

(2)

If $n=1$, we have a one-dimensional moment problem, while for $n\ge 2$, the corresponding moment problem is called a multidimensional moment problem. From the scalar moment problem (1), many authors studied the vector valued (or operator valued, or matrix valued) moment problems, when the $y_j, j\in {\mathbb{N}}^n$ are elements of an ordered vector space $Y$ with additional properties, whose elements are vectors, functions, self-adjoint operators or symmetric matrices with real entries. The moment problem is an inverse problem, since we are looking for an unknown positive measure $\nu$ which satisfies the moment conditions (1), knowing only his known (given) moments ${{\displaystyle \int }}_F^{}{t}^{j}d\nu ,$ $j\in {\mathbb{N}}^n.$ Finding the measure means studying its existence, uniqueness, and construction. In case of the vector-valued moment problem, the codomain $Y$ is assumed to be an order complete vector space. This condition is required since we need to extend the linear operator

$$T_0:\mathcal{P}\to Y,T_0\left({\displaystyle \sum }_{j\in J_0}{\alpha }_j{\phi }_j\right)={\displaystyle \sum }_{j\in J_0}{\alpha }_j{y}_j.$$

(3)

from the vector space of all polynomials with real coefficients to an ordered Banach function space $X$ which contains $\mathcal{P}$ and the vector space $C_c\left(F\right)$ of all real valued continuous compactly supported functions defined on $F.$ In Equation (3), $J_0\subset {\mathbb{N}}^n$ is a finite subset, ${\alpha }_j\in ℝ.$ To ensure the existence of a linear positive extension $T:X\to Y$ of $T_0,$ we need a Hahn–Banach type extension result, which requires the order completeness of $Y.$ From (3), it results

$$T\left({\phi }_j\right)=T_0\left({\phi }_j\right)=y_j, j\in {\mathbb{N}}^n,$$

which is the vector-valued variant of (1). There are moment problems when, besides the positivity of the solution $T,$ we naturally obtain, from the proof of its existence, the property

$$T\left(x\right)\le P\left(x\right),$$

(4)

for all $x\in X,$ where $X,Y$ are Banach lattices, $Y$ is order complete, and $P:X\to Y$ is a continuous convex or sublinear operator. Such problems are Markov moment problems. Sometimes, the constraints on the solution $T$ are $T_1\le T\le T_2$ on the positive cone of the domain space $X$, where $T_i, i=1,2$ are two given bounded linear operators from $X$ to $Y.$

The moment problems mentioned up to now are called full moment problems, because they involve the moment conditions $T\left({\phi }_j\right)=y_j$ for all $j\in {\mathbb{N}}^n.$ The reduced (or truncated) moment problem requires the conditions $T\left({\phi }_j\right)=y_j$ only for

$$j=\left(j_1,\dots ,j_n\right), j_k\in \left\{0,1,\dots ,d\right\}, k=1,\dots ,n,$$

where $d$ is a fixed natural number. For a basic result on the extension of linear positive operators, see [19]. Other extension results of linear operators, with two constraints, were published in [20,21,22]. Such old theorems found new applications in characterizing the isotonicity of continuous convex operators on a convex cone, recently published in [23]. We recall that an operator $P:X_{+}\to Y$ defined on the positive cone $X_{+}$ of the ordered vector space $X,$ to the ordered vector space $Y$ is called isotone (monotone increasing) if:

$$\mathbf{0}\le x_1\le x_2 \mathrm{in} X \mathrm{implies} P\left(x_1\right)\le P\left(x_2\right).$$

Various aspects of the full and reduced moment problem are discussed in [24,25,26,27,28,29,30,31,32,33,34]. These results include the existence, the uniqueness, and the construction of the solution. Obviously, the uniqueness of the solution makes sense only for the full moment problem. In the end of the article [34], a minimization problem related to a Markov moment problem is discussed. Here, we start from an idea appearing in the PhD thesis [28], also using some other methods. This is the second purpose of the paper. Optimization problems are studied in the articles [35,36,37,38,39], from which the last three are providing corresponding iterative methods and algorithms. As is well known, in any reflexive Banach space, for a non-empty closed convex subset not containing the origin, there exists at least one element of minimum norm in that subset. The point of this work is to discuss the case when the convex subset under attention appears from natural constraints related to a Markov moment problem.

Thus, the points of the first part of this paper are recalling and mainly completing the uniformly boundedness of some classes of convex operators, a subject which is not very well covered in the literature, except the references cited here. The significance of the second part consists in pointing out a necessary and sufficient condition for the existence of a solution of a Markov moment problem (an interpolation problem with two constraints), accompanied by a related minimization problem with infinitely many constraints. One characterizes the non-emptiness of the set of feasible solutions, and the existence of at least one minimum point is also proved (see Theorem 4). The uniqueness of such a point is briefly discussed (see Remark 7). The reader can find details and completions to the second part of this work by means of our references.

The rest of the paper is organized as follows. In Section 2, the main methods used in the sequel are pointed out. Section 3 contains the results on the subjects briefly mentioned above and is divided into two subsections. The common point is the notion of convexity for operators and for real valued functions, and its relationships with linear operators. Section 4 discusses the relevant results and concludes the paper.

2. Methods

The main methods used in what follows are:

(1)

The general notions and results in algebra and topology, Baire categories, Baire spaces, Banach spaces, Banach lattices, and the Banach–Steinhaus theorem (see [5,9,10,11]).

(2)

General knowledge on convex functions and convex operators (see [7,10,11,13,19,20,21,22,23,24,25,26,28,29,30,34,35,36,37,38,39].

(3)

A Hahn–Banach-type theorem formulated in terms of a Markov moment problem, recalled in the second subsection of Section 3 (see [11,22,24,26]).

(4)

Weak compactness and a related property of weakly lower semi-continuous real function on a weak compact subset (see [5,34,36]).

(5)

Giving supporting examples for the theoretical results (see [5,11,23]).

3. Results

3.1. Uniform Boundedness for Families of Convex Operators and Related Consequences

In the sequel, $X$ will be a (not necessarily locally convex) topological vector space which cannot be expressible as the countable union of closed subsets having empty interiors, and $Y$ will be a locally convex vector lattice (on which the lattice operations are continuous and there exists a fundamental system $\mathcal{V}$ of neighborhoods $V$ of $0_Y$ which are convex, closed, and solid subsets, i.e.,

$$|y_1|\le |y_2|, y_2\in V\Rightarrow y_1\in V).$$

Both spaces $X,Y$ are vector spaces over the real field. Consider a class $\mathcal{C}$ of convex continuous operators $P:X\to Y,P\left(\mathbf{0}\right)=\mathbf{0}$. Recall that we can always reduce the problem of proving the equicontinuity of a family of convex operators at a point $x_0\in X$ to the equicontinuity of a corresponding family of convex operators at $\mathbf{0},$ where each element $P$ of the latter family satisfies the condition $P\left(\mathbf{0}\right)=\mathbf{0}$ (cf. [10], the proof of Theorem 3.1). The next result was published in [11].

Theorem 1.

Additionally assume that for each$V\in \mathcal{V},$and any$x\in X,$there exists a small enough positive number$r$such that

$$rP\left(x\right)\in V \forall P\in \mathcal{C}.$$

Then, for any$V_0\in \mathcal{V}$, there exists a closed convex neighborhood$W_0$of$0_X$such that

$${\displaystyle \bigcup }_{P\in \mathcal{C}}P\left( W_0\right)\subset V_0.$$

One writes$\underset{x\to 0_X}{lim}P\left(x\right)={\mathbf{0}}_Y$uniformly in$P\in \mathcal{C}.$

Proof. 

For any $V_0\in \mathcal{V}$ and any $P\in \mathcal{C}$, define $P_1:X\to Y,$ $P_1\left(x\right)≔sup\left\{P\left(x\right),P\left(-x\right)\right\}, x\in X.$The operator $P_1$ is obviously convex. An additional property of $P_1$ is $P_1\left(x\right)=P_1\left(-x\right),x\in X.$ Consequently, the codomain of $P_1$is $Y_{+}$, since ${\mathbf{0}}_Y=P_1\left({\mathbf{0}}_X\right)=P_1\left(\frac{1}{2}x+\frac{1}{2}\left(-x\right)\right)\le \frac{1}{2}2P_1\left(x\right)=P_1\left(x\right),x\in X.$ The operator $P_1$ is also continuous, as the least upper bound of two continuous operators, thanks to the continuity of “sup” operation from $Y\times Y$ to $Y.$ The subset $P_1^{-1}\left(V_0\right)$ is closed, due to the continuity of $P_1.$ Now, we prove that it is also convex. Indeed, for $x_1,x_2\in P_1^{-1}\left(V_0\right)$, $t\in \left[0,1\right],$ the following relations hold:

$$P_1\left(\left(1-t\right)x_1+t{x}_2\right)\le \left(1-t\right)P_1\left(x_1\right)+t{P}_1\left(x_2\right)\in V_0,$$

since $V_0$ is convex and $P_1$ is convex too. Now, using the assumption on $V_0$ of being solid, it results

$$P_1\left(\left(1-t\right)x_1+t{x}_2\right)\in V_0\left(\iff \left(\left(1-t\right)x_1+t{x}_2\right)\in P_1^{-1}\left(V_0\right)\right).$$

We define

$$W_0≔{{{\displaystyle \bigcap }}^{\hspace{0.25em}}}_{P\in \mathcal{C}}{P}_1^{-1}\left(V_0\right).$$

The subset $W_0$ is closed and convex, as an intersection of such subsets. Clearly, ${\displaystyle \bigcup }_{P\in \mathcal{C}}{P}_1\left( W_0\right)\subset V_0$. For any $x\in W_0$ and any $P\in \mathcal{P},$ it results

$$|P\left(x\right)|\le sup\left\{P\left(x\right),P\left(-x\right)\right\}=P_1\left(x\right)\in V_0,$$

because of $-P\left(x\right)\le P\left(-x\right), x\in X.$ Indeed, $0_Y=P\left(0_X\right)\le \frac{1}{2}\left(P\left(x\right)+P\left(-x\right)\right), x\in X.$ Having in mind the property of $V_0,$ we infer that $P\left(x\right)\in V_0, \forall x\in W_0, \forall P\in \mathcal{C}.$ The first conclusion is ${{\displaystyle \bigcup }}_{P\in \mathcal{C}}P\left( W_0\right)\subset V_0$. To finish the proof, we have to show that $W_0$ is a neighborhood of ${\mathbf{0}}_X$. For any $x\in X$ and for any $V_0\in \mathcal{V},$ there exists a sufficiently small $r_0>0$ such that $\alpha P_1\left(x\right)\in V_0 \forall \alpha \in ℝ, |\alpha |\le r_0, \forall P\in \mathcal{C}.$ We can suppose that $r_0\le 1$. From the preceding considerations, it results

$$\alpha \in \left[0,r_0\right]\subset \left[0,1\right]\Rightarrow P_1\left(\alpha x\right)=P_1\left(\left(1-\alpha \right){\mathbf{0}}_X+\alpha x\right)\le$$

$$\alpha P_1\left(x\right)\in V_0\Rightarrow P_1\left(\alpha x\right)\in V_0,$$

$$\alpha \in \left[-r_0,0\right]\Rightarrow P_1\left(\alpha x\right)=P_1\left(\left(-\alpha \right)\left(-x\right)\right)\le \left(-\alpha \right)P_1\left(-x\right)\le r_0{P}_1\left(x\right)\in V_0, P\in \mathcal{C}.$$

These relations lead to $x\in X, |\alpha |\le r_0\Rightarrow \alpha x\in W_0\Rightarrow x\in \frac{1}{|\alpha |}{W}_0\subset n{W}_0$ for a sufficiently large $n\in \mathbb{N}$. Consequently, the following basic relation holds true: $X={\displaystyle \bigcup }_{n\in \mathbb{N}}n{W}_0$. Now, recall that $W_0$ is closed, convex, and our assumption on $X$ yields $int\left(W_0\right)\ne \varnothing ,$ so that there exists $x_0\in int\left(W_0\right)\Rightarrow {\mathbf{0}}_X=\frac{1}{2}\left(x_0+\left(-x_0\right)\right)\in int\left(W_0\right).$ This concludes the proof. □

Corollary 1.

Let$X$be a Banach space,$Y$a Banach lattice,$\mathcal{C}$a collection of continuous convex operators$P:X\to Y,P\left(\mathbf{0}\right)=\mathbf{0},$such that for any$x\in X,$we have$\underset{P\in \mathcal{C}}{\mathit{sup}}||P{\left(x\right)||}_Y<\infty .$Then the following relation holds:$\underset{P\in \mathcal{C}, ||x||\le 1}{\mathit{sup}}||P{\left(x\right)||}_Y<\infty .$

In the sequel, $X$ will be an (F) space, i.e., a metrizable complete (not necessarily locally convex) topological vector space, $Y$ will be a normed vector lattice (in particular, its norm is monotone on $Y_{+}: \left({\mathbf{0}}_Y\le y_1\le y_2\Rightarrow ||y_1||{}_Y\le ||y_2||{}_Y\right)$ and the multiplication with scalars is continuous). Recall that a normed vector lattice $Y$ is a vector lattice endowed with a solid norm ($|y_1|\le |y_2|\Rightarrow$ $||y_1||\le ||y_2||),$ so the lattice operations are continuous. Consider a class $\mathcal{S}$ of sublinear operators $\mathsf{\Phi }:X\to Y_{+}$ such that $\mathsf{\Phi }\left(x\right)=\mathsf{\Phi }\left(-x\right) \forall x\in X, \forall \mathsf{\Phi }\in \mathcal{S}$.

Corollary 2.

Let X$,Y,\mathcal{S}$be as above. Assume that$\mathsf{\Phi }$is continuous$\forall \mathsf{\Phi }\in \mathcal{S}$and$\underset{\mathsf{\Phi }\in \mathcal{S}}{\mathit{sup}}||\mathsf{\Phi }{\left(x\right)||}_Y<\infty \forall x\in X.$Then there exists a convex closed neighborhood$U$of$0_X$such that${\displaystyle \bigcup }_{\mathsf{\Phi }\in \mathcal{S}}\mathsf{\Phi }\left(U\right)\subset B_{1,Y}≔B_1\left({\mathbf{0}}_Y\right),$where$B_1\left({\mathbf{0}}_Y\right)$is the closed unit ball centered at the origin of the space$Y.$

The poof follows the ideas from that of Theorem 1, also applying Baire’s theorem.

Remark 1.

Under previous conditions, assuming that$Y$is a normed vector lattice (the norm on$Y$is solid and the lattice operations are continuous), Corollary 2 says that

$$x_1-x_2\in U\Rightarrow |\mathsf{\Phi }\left(x_1\right)-\mathsf{\Phi }\left(x_2\right)|\le \mathsf{\Phi }\left(x_1-x_2\right)\in B_{1,Y}\Rightarrow$$

$$\mathsf{\Phi }\left(x_1\right)-\mathsf{\Phi }\left(x_2\right)\in B_{1,Y} \forall \mathsf{\Phi }\in \mathcal{S}.$$

It results that$\mathcal{S}$is equicontinuous.

Example 1.

Using the above notations, let$ℒ$be a family of linear continuous operators from$X$to$Y$such that $\underset{T\in ℒ}{\mathit{sup}}||T{\left(x\right)||}_Y<\infty \forall x\in X.$ Define $\mathsf{\Phi }\left(x\right)={\mathsf{\Phi }}_T\left(x\right)≔|T\left(x\right)|, x\in X, T\in ℒ.$ Then, the family $\mathcal{S}={\left\{{\mathsf{\Phi }}_T\right\}}_{T\in ℒ}$ verifies the condition $\underset{T\in ℒ}{\mathit{sup}}||{\mathsf{\Phi }}_T{\left(x\right)||}_Y<\infty \forall x\in X.$

Remark 2.

Theorem 1 holds true when$X$is a Banach space,$Y$is a normed vector lattice, and the other conditions of Theorem 1 are accomplished. It is possible that a similar result be true for more general spaces$X$(involving the notion of a barreled TVS). However, only for a few spaces can it be easily proved that they are barreled spaces, without using Baire’s theorem. On the other side, for applications, the most important spaces are Banach spaces, especially Banach lattices.

Theorem 2.

Let$X$be a Banach space and$Y$an order complete normed vector lattice with strong order unit$u_0,$such that$B_{1,Y}=\left[-u_0,u_0\right].$Let$\mathcal{S}$be a class of sublinear operators with the properties mentioned in Corollary 2. Additionally, assume that$\mathsf{\Phi }\left(x\right)=\mathsf{\Phi }\left(-x\right) \forall x\in X, \forall \mathsf{\Phi }\in \mathcal{S}.$Then, the relation

$$\tilde{\mathsf{\Phi }}\left(x\right)=\underset{\mathsf{\Phi }\in \mathcal{S}}{\mathit{sup}}\mathsf{\Phi }\left(x\right) \forall x\in X,$$

defines a sublinear Lipschitz operator$\tilde{\mathsf{\Phi }},$such that$\tilde{\mathsf{\Phi }}\left(x\right)=\tilde{\mathsf{\Phi }}\left(-x\right)\in Y_{+} \forall x\in X.$

Proof. 

Application of Corollary 2 leads to the existence of a closed ball of sufficiently small radius $r>0$ such that

$${||x||}_X\le r\Rightarrow \mathsf{\Phi }\left(x\right)\in B_{1,Y}=\left[-u_0,u_0\right] \forall \mathsf{\Phi }\in \mathcal{S}.$$

It results

$$\mathsf{\Phi }\left(r\frac{x}{{||x||}_X}\right)\le u_0 \iff \mathsf{\Phi }\left(x\right)\le \frac{{||x||}_X}{r}{u}_0 \forall x\in X\backslash \left\{0_X\right\}, \forall \mathsf{\Phi }\in \mathcal{S}.$$

(5)

Thus, according to (5), for any fixed $x\in X$, the set $\left\{\mathsf{\Phi }\left(x\right);\mathsf{\Phi }\in \mathcal{S} \right\}$ is bounded from above in $Y.$ Thanks to the hypothesis on order completeness of $Y,$ there exists

$$\tilde{\mathsf{\Phi }}\left(x\right)≔\underset{\mathsf{\Phi }\in \mathcal{S}}{\mathit{sup}}\mathsf{\Phi }\left(x\right)\le \frac{{||x||}_X}{r}{u}_0 \forall x\in X.$$

(6)

It is easy to see that $\tilde{\mathsf{\Phi }}$ is sublinear and has the property $\tilde{\mathsf{\Phi }}\left(x\right)=\tilde{\mathsf{\Phi }}\left(-x\right)\in Y_{+} \forall x\in X.$ Next, we prove the Lipschitz property of $\tilde{\mathsf{\Phi }}.$ To do this, one uses the subadditivity property of $\tilde{\mathsf{\Phi }},$ the fact that the norm of $Y$ is monotone on$Y_{+}$, and relation (6). Namely, the following implications hold:

$$x_1, x_2\in X, |\tilde{\mathsf{\Phi }}\left(x_1\right)-\tilde{\mathsf{\Phi }}\left(x_2\right)|\le \tilde{\mathsf{\Phi }}\left(x_1-x_2\right)\Rightarrow$$

$$||\tilde{\mathsf{\Phi }}\left(x_1\right)-\tilde{\mathsf{\Phi }}{\left(x_2\right)||}_Y\le ||\tilde{\mathsf{\Phi }}{\left(x_1-x_2\right)||}_Y\le \left|\right|\frac{||x_1-x_2||{}_X}{r}{u}_0\left|\right|{}_Y=\frac{||x_1-x_2||{}_X}{r}.$$

Hence, $\tilde{\mathsf{\Phi }}$ is a Lipschitz mapping from $X$ to $Y_{+}.$ This concludes the proof. □

Remark 3.

Under the hypothesis of Theorem 2, each element of$\Phi \in \mathcal{S}$is a Lipschitz operator, with the same Lipschitz constant$1/r.$

Remark 4.

It seems that topological completeness of$Y$is not necessary for the above results. However, the usual concrete spaces verifying the hypothesis of Theorem 2 are Banach spaces.

Remark 5.

The set$\mathcal{C}$of all continuous sublinear operators $\mathsf{\Phi }$from $X$ to $Y_{+}$, such that $\mathsf{\Phi }\left(x\right)=\mathsf{\Phi }\left(-x\right) \forall x\in X, \forall \mathsf{\Phi }\in \mathcal{C},$ $\underset{\phi \in \mathcal{C}}{\mathit{sup}}||\phi {\left(x\right)||}_Y<\infty \forall x\in X,$ is a convex cone. With the notations and under the assumptions of Theorem 2, the subset of all $\mathsf{\Phi }\in \mathcal{C}$formed by all elements of $\mathcal{C}$ with the property $\phi \left(B_{1,X}\right)\subset B_{1,Y}$ is convex, and its elements are the non-expansive operators from $\mathcal{C}.$ If $r$ of the proof of Theorem 2 is strictly greater than $1,$ then the elements of $\mathcal{S}$ (as well as the operator $\tilde{\mathsf{\Phi }})$ are contractions.

Remark 6.

An arbitrary sublinear operator $\phi :X\to Y_{+}$ is a Lipschitz operator if and only if $\mathsf{\Phi }$ is continuous at ${\mathbf{0}}_X.$

Corollary 3.

Let X and Y be as in Theorem 2,$\mathcal{S}=\left\{{\mathsf{\Phi }}_n;n\in {\rm N}\right\}$a countable set of sublinear continuous operators from$X$to$Y$, such that${\mathsf{\Phi }}_n\left(x\right)={\mathsf{\Phi }}_n\left(-x\right) \forall x\in X, \forall n\in \mathbb{N},$and$\underset{n\in \mathbb{N}}{\mathit{sup}}||{\mathsf{\Phi }}_n{\left(x\right)||}_Y<\infty \forall x\in X.$Then, the relation

$$\tilde{\mathsf{\Phi }}\left(x\right)=\underset{n\in \mathbb{N}}{\mathit{sup} }{\mathsf{\Phi }}_n\left(x\right) \forall x\in X$$

defines a sublinear Lipschitz operator$\tilde{\mathsf{\Phi }}:X\to Y_{+},$such that$\tilde{\mathsf{\Phi }}\left(x\right)=\tilde{\mathsf{\Phi }}\left(-x\right) \forall x\in X.$

Corollary 4.

Let$X,Y$be as in Theorem 2,$\mathcal{T}=\left\{{\mathsf{\Phi }}_n;n\in {\rm N}, n\ge 1\right\}$a countable set of sublinear continuous operators from$X$to$Y_{+}$, such that${\mathsf{\Phi }}_n\left(x\right)={\mathsf{\Phi }}_n\left(-x\right) \forall x\in X, \forall n\in \left\{1,2,\dots \right\},$and

$$\underset{\begin{array}{c}n\in N,\\ n\ge 1\end{array}}{\mathit{sup}}{\displaystyle ||}{\displaystyle \sum }_{k=1}^n{\mathsf{\Phi }}_k{\left(x\right){\displaystyle ||}}_Y<\infty \forall x\in X.$$

Then, the relation

$$\tilde{\mathsf{\Phi }}\left(x\right)=\underset{\begin{array}{c}n\in N,\\ n\ge 1\end{array}}{\mathit{sup}}\left({\displaystyle \sum }_{k=1}^n{\mathsf{\Phi }}_k\left(x\right)\right) \forall x\in X,$$

defines a sublinear Lipschitz operator$\tilde{\mathsf{\Phi }}:X\to Y_{+},$such that$\tilde{\mathsf{\Phi }}\left(x\right)=\tilde{\mathsf{\Phi }}\left(-x\right) \forall x\in X.$

Example 2.

Let$K$be a Hausdorff compact topological space, endowed with a regular Borel probability measure$\mu ,$ $X≔C\left(K\right)$the Banach lattice of all real valued, continuous functions on$K,$ $Y≔l_{\infty }$the space of all bounded sequences of real numbers. The norm${\left|\right|·\left|\right|}_{sup}$on the space$X$is the sup-norm and the norm on${\left|\right|·\left|\right|}_Y$is the usual norm${\left|\right|·\left|\right|}_Y={\left|\right|·\left|\right|}_{\infty },$ $||{\left(x_n\right)}_{n\ge 1}||{}_{\infty }=\underset{n\ge 1}{\mathit{sup}}|x_n|.$The space$Y=l_{\infty }$verifies the hypothesis of Theorem 2, since it is an order complete normed vector lattice, the appropriate strong order unit being the sequence$u_0$, which has all the terms equal to$1$. Define the scalar valued norms on$X$

$$N_k\left(f\right)≔{\left(\underset{K}{\overset{}{{\displaystyle \int }}}{|f|}^{k}d\mu \right)}^{1/k}, f\in X, k\in \mathbb{N}, k\ge 1,$$

and the finite dimensional vector-valued norms on$X$

$$S_n\left(f\right)≔||f_n||:X\to Y,$$

$$||f_n||≔\left(N_1\left(f\right), 2^{1/2}{N}_2\left(f\right),\dots ,n^{1/n}{N}_n\left(f\right), 0,\dots ,0,\dots \right), n\in \mathbb{N}, n\ge 1, f\in X,$$

$$N_k\left(f\right)\le {||f||}_{sup}{\left(\mu \left(K\right)\right)}^{1/k}={||f||}_{sup}, N_k(\P )=1\Rightarrow \underset{{||f||}_{\mathit{sup}}=1}{\mathit{sup}}{N}_k\left(f\right)=1,$$

$$k\in \left\{1,2,\dots \right\},f\in X.$$

Consider the elementary function$t\to g\left(t\right)≔ln\left(t\right)/t, t\in \left[1,\infty \right)$, which is increasing on$\left[1,e\right]$and decreasing on the interval$\left[e,\infty \right).$This function has a global maximum point at$t_0=e\in \left(2,3\right).$It results that the function

$$h:\left(1,\infty \right)\to \left(0,\infty \right), h\left(t\right)≔t^{1/t}=e^{ln\left(t\right)/t}$$

has the same monotonicity properties; hence,

$$\underset{1\le k\le n}{\max }{k}^{1/k}\le \max \left\{2^{1/2},3^{1/3}\right\}=3^{1/3} \forall n\in \left\{1,2,\dots \right\}$$

Thus, we obtain

$$f\in X\Rightarrow S_n\left(f\right)={||f||}_n\le \underset{1\le k\le n}{\max }{k}^{1/k}{||f||}_{sup}{u}_0\le 3^{1/3} {||f||}_{sup}{u}_0 \forall n\in \left\{1,2,\dots \right\}\Rightarrow$$

$$\tilde{\mathsf{\Phi }}\left(f\right)=\underset{n\in \mathbb{N}}{\mathit{sup}} S_n\left(f\right)={\left(n^{1/n}{N}_n\left(f\right)\right)}_{n\ge 1}\le 3^{1/3}≔{||f||}_{sup}{u}_0,$$

$$u_0=\left(1,\dots ,1,\dots \right), f\in X,$$

where$\tilde{\mathsf{\Phi }}$is the sublinear operator from Corollary 4. Observe that$\tilde{\mathsf{\Phi }}$has as Lipschitz constant 3^1/3 > 1. Next, we apply the same method, replacing$n^{1/n}$by

$$n^{-1/n}=exp\left(-ln\left(n\right)/n\right)\le 1 \forall n\in \left\{1,2,\dots \right\}.$$

In this case, the above estimations turn into the following ones:

$$\tilde{\mathsf{\Phi }}\left(f\right)=\underset{n\in \mathbb{N}}{\mathit{sup}} S_n\left(f\right)={\left(n^{-1/n}{N}_n\left(f\right)\right)}_{n\ge 1}\le {||f||}_{sup}{u}_0 \forall f\in X\Rightarrow$$

$$|\tilde{\mathsf{\Phi }}\left(f\right)-\tilde{\mathsf{\Phi }}\left(g\right)|\le \tilde{\mathsf{\Phi }}\left(f-g\right)\le {||f-g||}_{sup}{u}_0\Rightarrow$$

$$||\tilde{\mathsf{\Phi }}\left(f\right)-\tilde{\mathsf{\Phi }}{\left(g\right)||}_Y\le ||\tilde{\mathsf{\Phi }}\left(f-g\right)||\le {||f-g||}_{sup} \forall f,g\in X.$$

To conclude, in this case,$\tilde{\mathsf{\Phi }}$ is a nonexpansive vector valued norm from$X$to$Y.$To obtain contractions $\tilde{\mathsf{\Phi }},$ consider

$${\left(c_n\right)}_{n\ge 1}\in Y=l_{\infty }, 0\le c_n\le q<1 \forall n\ge 1,$$

$$S_n\left(f\right)=\left(c_1{N}_1\left(f\right),\dots ,c_n{N}_n\left(f\right),0,\dots ,0,\dots \right),$$

$$\tilde{\mathsf{\Phi }}\left(f\right)=\underset{n\ge 1}{\mathit{sup}} S_n\left(f\right)={\left(c_n{N}_n\left(f\right)\right)}_{n\ge 1}\le q{||f||}_{sup}{u}_0 \forall f\in X\Rightarrow$$

$${|\tilde{\mathsf{\Phi }}\left(f\right)-\tilde{\mathsf{\Phi }}\left(g\right)|}_Y\le \tilde{\mathsf{\Phi }}\left(f-g\right)\le q{||f-g||}_{sup}{u}_0\Rightarrow$$

$$||\tilde{\mathsf{\Phi }}\left(f\right)-\tilde{\mathsf{\Phi }}{\left(g\right)||}_Y\le q{||f-g||}_{sup}||u_0||{}_Y=q{||f-g||}_{sup} \forall f.g\in X.$$

Thus $\tilde{\mathsf{\Phi }}:X\to Y_{+}$ is a contraction vector-valued norm, of contraction constant$q,$and the best value for$q$is$q=\underset{n\ge 1}{\mathit{sup}} c_n.$In particular, if$0\le \underset{n\ge 1}{\mathit{inf}} c_n\le \underset{n\ge 1}{\mathit{sup}} c_n=1/2$ $\forall n\ge 1,$then $\tilde{\mathsf{\Phi }}$ is a contraction operator, of contraction constant$q=1/2$. In this example, the operators ${\mathsf{\Phi }}_n$ mentioned in Corollary 4 stand for$\left(0,\dots ,0,c_n{N}_n\left(f\right),0,0,\dots \right),$and$c_n{N}_n\left(f\right)$is the$n-th$coordinate of the vector$S_n\left(f\right)\in Y_{+}$.

3.2. A Constrained Minimization Problem Related to a Markov Moment Problem

The present subsection has as a motivation proving similar results to some of those of [28]. One proves a result in a general setting, obtained by means of Theorem 3 stated below. A constrained related optimization problem in infinite dimensional spaces is solved too. The results presented in the sequel were published in [34]. In particular, using the latter theorem, one obtains a necessary and sufficient condition for the existence of a feasible solution (see theorem 4 from below). Under such a condition, the existence of an optimal feasible solution follows too. On the other hand, the uniqueness and the construction of the optimal solution does not seem to be obtained easily by such general methods. Therefore, we focus mainly on the existence problem. For other aspects of such problems on an optimal solution (uniqueness or non-uniqueness, construction of a unique solution, etc.), see [28]. In the latter work, one considers the following primal problem (P): study the constrained minimization problem:

$$\nu =inf\left\{{||\phi ||}_{\infty };\phi \in L_{\mu }^{\infty }\left(Z\right), {{\displaystyle \int }}_X^{}\phi f_{j}d\mu =b_j, j=1,\dots ,n, \mathbf{0}\le \alpha \le \phi \le \beta \right\},$$

where $\alpha ,\beta$ are in $L_{\mu }^{\infty }\left(Z\right)$, ${\left(f_j\right)}_{j=1}^n$ is a subset of $L_{\mu }^1\left(Z\right),$and $b={\left(b_1,\dots ,b_n\right)}^t\in {ℝ}^n.$ The function $\phi$ is unknown, and in general, it is not determined by a finite number of moments. The next theorem discusses some of the above existence type results for a feasible solution. Here, $\left(Z,ℳ\right)$is a measure space endowed with a $\sigma -$finite positive measure $\mu ,$ and $ℳ$ is the $\sigma -$algebra of all measurable subsets of $Z.$

Theorem 3.

See [22]. Let $X$ be an ordered vector space, $Y$ an order complete vector lattice,${\left\{{\phi }_j\right\}}_{j\in J}\subset X, {\left\{y_j\right\}}_{j\in J}\subset Y$ given arbitrary families, $T_1,T_2\in L\left(X,Y\right)$ two linear operators. The following statements are equivalent:

(a) 

there is a linear operator$T\in L\left(X,Y\right)$, such that

$$T_1\left(x\right)\le T\left(x\right)\le T_2\left(x\right) \forall x\in X_{+}, T\left({\phi }_j\right)=y_j \forall j\in J;$$

(b) 

for any finite subset$J_0\subset J$and any$\left\{{\lambda }_j;j\in J_0\right\}\subset ℝ,$ the following implication holds true:

$$\left({\displaystyle \sum }_{j\in J_0}{\lambda }_j{\phi }_j={\psi }_2-{\psi }_1,{\psi }_1,{\psi }_2\in X_{+}\right)≔{\displaystyle \sum }_{j\in J_0}{\lambda }_j{y}_j\le T_2\left({\psi }_2\right)-T_1\left({\psi }_1\right);$$

If $X$ is a vector lattice, then assertions (a) and (b) are equivalent to (c), where (c) is formulated as follows:

(c) 

$T_1\left(w\right)\le T_2\left(w\right)$ for all$w\in X_{+}$and for any finite subset$J_0\subset J$and$\forall \left\{{\lambda }_j;j\in J_0\right\}\subset ℝ,$we have

$${\displaystyle \sum }_{j\in J_0}{\lambda }_j{y}_j\le T_2\left({\left({\displaystyle \sum }_{j\in J_0}{\lambda }_j{\phi }_j\right)}^{+}\right)-T_1\left({\left({\displaystyle \sum }_{j\in J_0}{\lambda }_j{\phi }_j\right)}^{-}\right).$$

The next result is an application of Theorem 3 stated above, also using a constrained minimization argument.

Theorem 4.

Let$p\in \left(1,\infty \right)$and let q be the conjugate of p. Let${\left(f_j\right)}_{j\in J}$be an arbitrary family of functions in$L_{\mu }^p\left(Z\right),$where the measure$\mu$is$\sigma$–finite, and${\left(b_j\right)}_{j\in J}$a family of real numbers. Assume that$\alpha ,\beta \in$ $L_{\mu }^q\left(Z\right)$are such that$\mathbf{0}\le \alpha \le \beta$. The following statements are equivalent:

(a) 

there exists $\phi \in L_{\mu }^q\left(Z\right)$such that${{\displaystyle \int }}_Z^{}\phi f_{j}d\mu =b_j,\forall j\in J, \mathbf{0}\le \alpha \le \phi \le \beta ;$

(b) 

for any finitesubset$J_0\subseteq J$and any${\left\{{\lambda }_j\right\}}_{j\in J_0}\subset ℝ$, the following implication holds:

$${\displaystyle \sum }_{j\in J_0}{\lambda }_j{f}_j={\psi }_2-{\psi }_1, {\psi }_1, {\psi }_2\in {\left(L_{\mu }^p\left(Z\right)\right)}_{+}⟹ {\displaystyle \sum }_{j\in J_0}{\lambda }_j{b}_j\le {{\displaystyle \int }}_Z^{}\beta {\psi }_2 d\mu -{{\displaystyle \int }}_Z^{}\alpha {\psi }_1 d\mu ;$$

Moreover, the set of all feasible solutions$\phi$(satisfying the conditions (a)) is weakly compact with respect the dual pair$\left(L^p,L^q\right)$and the inferior

$$\nu =inf\left\{{||\phi ||}_q;\phi \in L_{\mu }^q\left(Z\right), {{\displaystyle \int }}_Z^{}\phi f_{j}d\mu =b_j, j\in J, \mathbf{0}\le \alpha \le \phi \le \beta \right\}\ge {||\alpha ||}_q,$$

is attained for at least one optimal feasible solution${\phi }_0.$

Proof. 

Since the implication $\left(\mathrm{a}\right)⟹\left(\mathrm{b}\right)$ is obvious, the next step consists in proving that $\left(\mathrm{b}\right)⟹\left(\mathrm{a}\right)$. We define the linear positive (continuous) forms $T_1, T_2$ on $X=L_{\mu }^p\left(Z\right)$, by

$$T_1\left(f\right)={{\displaystyle \int }}_Z^{}\alpha fd\mu , T_2\left(\phi \right)={{\displaystyle \int }}_Z^{}\beta fd\mu , f\in X.$$

Then, condition (b) of the present theorem coincides with condition (b) of Theorem 3. A straightforward application of the latter theorem leads to the existence of a linear form $T$on $X$, such that the interpolation conditions $T\left({\phi }_j\right)=b_j, j\in J$ are verified and

$${{\displaystyle \int }}_Z^{}\alpha \psi d\mu \le T\left(\psi \right)\le {{\displaystyle \int }}_Z^{}\beta \psi d\mu , \psi \in X_{+}.$$

In particular, the linear form $T$ is positive on$X=L_{\mu }^p\left(Z\right)$, and this space is a Banach lattice. It is known that on such spaces, any linear positive functional is continuous (see [5], or [8], or [23]). The conclusion is that $T$ can be represented by means of a nonnegative function $\phi \in L_{\mu }^q\left(Z\right)$. From the previous relations, we infer that

$${{\displaystyle \int }}_Z^{}\alpha \psi d\mu \le {{\displaystyle \int }}_Z^{}\phi \psi d\mu \le {{\displaystyle \int }}_Z^{}\beta \psi d\mu , \psi \in X_{+}.$$

Writing these relations for $\psi ={\chi }_B$, where $B$ is an arbitrary measurable set of positive measure $\mu \left(B\right)$, one deduces

$${{\displaystyle \int }}_B^{}\left(\phi -\alpha \right)d\mu \ge 0, {{\displaystyle \int }}_B^{}\left(\beta -\phi \right)d\mu \ge 0, B\in ℳ, \mu \left(B\right)>0.$$

Now, a standard measure theory argument shows that $\alpha \le \phi \le \beta$ almost everywhere in $Z.$ This finishes the proof of $\left(\mathrm{b}\right)⟹\left(\mathrm{a}\right).$ To prove the last assertion of the theorem, observe that the set of all feasible solutions is weakly compact in $L_{\mu }^q\left(Z\right),$ by Alaoglu’s theorem; it is a weakly closed subset of the closed ball centered at the origin, of radius ${||\beta ||}_q$, and $L_{\mu }^q\left(Z\right)$ is reflexive. On the other hand, the norm of any normed linear space is lower weakly semi-continuous, as the supremum of continuous linear forms, which are also weak continuous with respect to the dual pair $\left(L_{\mu }^q\left(Z\right), L_{\mu }^p\left(Z\right)\right), 1<p<\infty , 1/p+1/q=1$. Since $L_{\mu }^q\left(Z\right)$ is reflexive for $1<q<\infty ,$ we conclude that the norm ${||·||}_q$ is weakly lower semi-continuous on the weakly (convex) and compact set described at point (a), so that it attains its minimum at a function ${\phi }_0$ of this set. Hence, there exists at least one optimal feasible solution. This concludes the proof. □

Remark 7.

If the set${\left\{f_j\right\}}_{j\in J}$is total in the space$L_{\mu }^p\left(Z\right),$then the set of all feasible solutions is a singleton, so that there exists a unique solution.

Remark 8.

In the proof of Theorem 4, we claimed that any positive linear function on$L_{\mu }^p\left(Z\right), 1<p<\infty$is continuous. Actually, there is a much more general result on this subject. Namely, any positive linear operator acting between two ordered Banach spaces is continuous (see [8] and/or [23]). In particular, this result holds for positive linear operators acting between Banach lattices.

4. Discussion

In the first part of Section 3, this paper brings a few new elements and completions with respect to the basic results previously published on this subject. The main completions are formulated as Corollaries, Remarks, and two examples. The second subsection of Section 3 reviews the main Theorem 3 and gives one of its applications, stated as Theorem 4. The latter theorem can be applied to the existence of at least one feasible solution for the constrained minimization problem formulated in the same theorem. The problem under attention is solved on a concrete function space. The index set $J$ appearing in Theorems 3 and 4 is arbitrary, finite, countable, or uncountable. In the case of the full moment problem on a closed subset of ${ℝ}^n,$ we have $J={\mathbb{N}}^n, n\in \mathbb{N}, n\ge 1,$ so in this case, $J$ is a countable infinite set of indexes. Theorem 4 provides a necessary and sufficient condition for the feasible set of a minimization problem with many countable constraints being non-empty. The common point of the two subsections of Section 3 is the notion of convexity, applied to real-valued functions and to operators. The connection of convex functions (respectively, convex operators) with the linear functionals (respectively, linear operators) is emphasized in both subsections. As a direction for future work, we recall the importance of Markov linear operators. Many such operators arise as solutions of Markov moment problems. They are dominated by a given continuous sublinear operator and apply the strong order unit of the domain space to the strong order unit of the codomain space (assuming that both the domain and the codomain are endowed with a strong order unit).