Special Issue of Symmetry: “Symmetry in Mathematical Analysis and Functional Analysis”

This Special Issue consists of 11 papers recently published in MDPI’s journal Symmetry under the general thematic title “Symmetry in Mathematical Analysis and Functional Analysis” (see [1,2,3,4,5,6,7,8,9,10,11]). The deadline for manuscript submissions was 31 July 2022. This Special Issue belongs to the section of the journal entitled “Mathematics and Symmetry/Asymmetry”.

Among other aspects of the theory underlying this area of research, the content of these 11 published papers (and their references) covers, but is not limited to, the following subjects:

• Common fixed-point results in general metric space settings and applications.
• Constrained optimization.
• Optimal control.
• Solving systems of special differential equations.
• Applications of fractional calculus.
• Inclusion and inequalities in interval-valued pre-invex and convex functions.
• Fuzzy fractional integral inequalities in pre-invex fuzzy interval-valued functions.
• Multi-objective convex optimization in real Banach space.
• Well-posedness for certain classes of equations.
• Families of convex operators and related linear operators.
• Symmetry of sublinear continuous operators and its applications (see [11]).

In the first part of paper [11], the symmetry of sublinear continuous operators $P:X\to Y$ $\left(P\left(x\right)=P\left(-x\right) \forall x\in X\right)$ appears in Theorem 2 and in some of its consequences. Of note, if $X,Y$ are Banach lattices, with $Y$ being an order complete, then the norm of a continuous sublinear operator from $X$ into $Y$ controls the norm of all its subgradients. In the second part of the same paper, elements of the theory of the Markov moment problem are explored. Since this thematic area is closely related to many other fields of mathematics, here, we briefly review some of the notions regarding the classical one dimensional and, in particular, the multidimensional moment problem and its relationship with other areas of research, such as the explicit form of any non-negative polynomial on a closed subset of ${ℝ}^n$ in terms of the sums of the squares of some polynomials; the extension of positive linear functionals and operators; the extension of linear operators dominated by a convex continuous operator and dominating a given continuous concave operator (these constraints might hold only on the positive cone of the domain space); measure theory; the notion of a moment determinate measure and study of determinacy; matrix theory; spaces of commuting self-adjoint operators (in particular, spaces of commuting symmetric matrices with real entries); inequalities; the Banach lattices of functions and self-adjoint operators; existence, uniqueness, and the eventual construction of the linear solution to an interpolation problem with one or two constraints; and examples of continuous sublinear (or only convex) operators, operator theory, and the complex functions of complex variables. In the present editorial, only the analysis and functional analysis of the real field are addressed. As is well-known and pointed out by the authors of [12], symmetric matrices with real entries have special properties, and there exits a natural order relation with respect to the real vector space $Sym\left(n\times n,ℝ\right)$ of all such matrices. With respect to this order relation, for $n\ge 2,$ the ordered vector space $Sym\left(n\times n,ℝ\right)$ is not a lattice. On the other hand, the multiplication operation of such $n\times n$ matrices is not commutative for $n\ge 2.$ Clearly, the corresponding assertions hold true regarding the space $\mathcal{A}\left(H\right)$ of all the self-adjoint operators acting on a real or complex Hilbert space $H$, where $\dim \left(H\right)\ge 2.$ The same article [12] contains simple proof of the fact that any positive linear operator applying an ordered Banach space $X$ to an ordered Banach space $Y$ is continuous. In particular, any positive linear operator mapping an arbitrary Banach lattice onto a Banach lattice is continuous. In order to avoid the two main difficulties mentioned above, regarding the space $\mathcal{A}\left(H\right)$ for any $A\in \mathcal{A}\left(H\right)$, as demonstrated in [13], one must construct a commutative real Banach algebra over the real field, denoted by $Y\left(A\right),$ which is also an order complete Banach lattice (endowed with the operatorial norm on $\mathcal{A}\left(H\right)).$ In this Banach lattice, we have $\left|U\right|≔sup\left\{U,-U\right\}=\sqrt{U^2}$ for all $U\in Y\left(A\right).$ In other words, the modulus of $U$ in this Banach lattice equals the positive square root of the positive self-adjoint operator $U^2$. Moreover, due to the order completeness of the vector lattice $Y\left(A\right),$ Hahn–Banach-type extension theorems for linear operators have $Y\left(A\right)$ as a codomain hold. In the classical one-dimensional moment problem, given a sequence ${\left(y_j\right)}_{j\in \mathbb{N}}$ of real numbers, we should find necessary and sufficient conditions for the existence of a positive regular Borel measure $\nu$ on the closed subset $F\subseteq ℝ,$ which satisfies the interpolation conditions ${{\displaystyle \int }}_F^{}{t}^{j}d\nu \left(t\right)=y_j, j\in \mathbb{N}≔\left\{0,1,2,\dots \right\}$. This is an inverse problem, because the measure $\nu$ is not known. Thus, it must be identified starting with its moments ${{\displaystyle \int }}_F^{}{t}^{j}d\nu \left(t\right), j\in \mathbb{N}.$ If such a measure does exist, its uniqueness and, eventually, its construction can be studied. The multidimensional real moment problem can be formulated in a similar way. In the case of an $n-$ dimensional moment problem, we have $j=\left(j_1,\dots ,j_n\right)\in {\mathbb{N}}^n, t=\left(t_1,\dots ,t_n\right)\in F\subseteq {ℝ}^n, n\ge 2, n$, being a fixed integer. Considering the unique linear form $L_0$ on the space of all the polynomials with real coefficients, satisfying the interpolation condition $L_0\left({\phi }_j\right)=y_j, j\in {\mathbb{N}}^n$, the existence of a solution is reduced to the representation of $L_0$ by a positive measure $d\nu .$ Namely, through linearity, the following equality is true for $d\nu :L_0\left(p\right)={{\displaystyle \int }}_F^{}p\left(t\right)d\nu \left(t\right)$ for all the polynomials $p\in ℝ\left[t_1,\dots ,t_n\right].$ This is a motivation for the terminology representing measure for $L_0$. According to the Haviland theorem [14], the sufficient (and necessary) condition for the existence of the representing positive measure $d\nu$ for $L_0$ is $L_0\left(p\right)\ge 0$ for any polynomial $p\in ℝ\left[t_1,\dots ,t_n\right]$ satisfying $p\left(t\right)\ge 0$ for all $t=\left(t_1,\dots ,t_n\right)\in F.$ In the important case of $n=1, F=ℝ,$ this positivity condition can be expressed in terms of the semi-positiveness of quadratic forms, since each polynomial (with real coefficients) which is non-negative on the entirety of the real axes is the sum of two squares of the polynomials from $ℝ\left[t\right]$ (see [15,16]). With the abovementioned notations, the coefficients of the quadratic forms are $y_{i+j}.$ This is the one-dimensional Hamburger moment problem. It represents a good example of symmetry, given by the symmetric matrices ${\left(y_{i+j}\right)}_{0\le i,j\le m}$ and $m\in \mathbb{N}.$ A similar remark is valid for the one-dimensional moment problem on $\left[0,+\infty \right):$ $p\in ℝ\left[t\right], p\left(t\right)\ge 0$ for all $t\in \left[0,+\infty \right)⟺p\left(t\right)=q^2\left(t\right)+t{r}^2\left(t\right) \forall t\in \left[0,+\infty \right)$ for some polynomials $q,r\in ℝ\left[t\right].$ Unlike the one-dimensional case, there are non-negative polynomials on ${ℝ}^2$, which are not sums of the squares of the polynomials in $ℝ\left[t_1,t_2\right]$ (see [16]). Up to now, the terms of the sequence ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ have been numbers. This is the scalar moment problem. Next, we consider a sequence ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ of elements of an ordered vector space $Y$ and, with the notation forms above, we study the existence of a linear positive operator $T:X_1\to Y$, such that $T\left({\phi }_j\right)=y_j$ for all $j\in {\mathbb{N}}^n$. Here, $X_1$ is an ordered vector space of real functions, containing the polynomials and the space $C_c\left(F\right)$ of all the continuous compactly supported functions on $F,$ such that the subspace of the polynomials is a majorizing subspace in $X_1$. For example, if $X≔L_{\nu }^p\left(F\right), p\in \left[1,+\infty \right),$ the space $X_1$ will be the sublattice of $X$ formed by all the functions $f$ from $X$, possessing the modulus $\left|f\right|$ dominated by a polynomial on the entire subset $F.$ Then, it is easy to observe that the subspace of the polynomials is a majorizing subspace in $X_1$ and, clearly, $X_1$ contains $C_c\left(F\right)$ and the space of the polynomials. Assuming that $Y$ is order complete, we consider the unique linear operator $T_0$ mapping the space of the polynomials to $Y,$ $T_0\left({\displaystyle \sum }_{j\in J_0}{\alpha }_j{\phi }_j\right)≔{\displaystyle \sum }_{j\in J_0}{\alpha }_j{y}_j, J_0\subset {\mathbb{N}}^n$, being an arbitrary finite subset. Additionally, assume that $T_0\left(p\right)\in Y_{+}$ for all the non-negative polynomials $p$ on $F.$ The application of the Kantorovich extension theorem for positive linear operators (see [17]) leads to the existence of a linear positive extension $T_1$ of $T_0$, where $T_1$ is mapped $X_1$ to $Y.$ If we prove the continuity of $T_1$ on $X_1,$ then there exists a unique continuous positive extension $T:X\to Y$ of $T_1.$ This follows from the density of $C_c\left(F\right)$ in $X=L_{\nu }^p\left(F\right), p\in \left[1,+\infty \right)$ (see [18]). When an upper constraint on the solution $T$ is required, we have a Markov moment problem. Usually, the following constraints on the solution $T$ of the interpolation problem are required: $0\le T\le T_2$ on the positive cone $X_{+}$, where $T_2$ is a given linear positive operator mapping the Banach lattice $X$ to the order complete Banach lattice $Y.$ In [19], the explicit form of non-negative polynomials on a strip is highlighted in terms of the sums of squares. In the papers [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36], various results on the full and truncated moment problem are provided. These results refer to connection with fixed-point theory (see [23]), the moment problem on compact subsets with a nonempty interior in ${ℝ}^n,$ (see [25]), and the decomposition of positive polynomials on such compact subsets, the moment problem, and the decomposition of positive polynomials on compact semi-algebraic subsets (see [26,27,28,29]. In [29], a class of moment problems on unbounded semi-algebraic sets are also discussed. The truncated Markov moment problem, including the construction of a solution, is emphasized in the articles [30,31]. For optimization related to the truncated moment problem, see [32,33]. A solution to a full moment problem obtained as a limit of the solutions for the associated truncated moment problem is provided by the authors of [34]. In [35], an operator-valued moment problem is solved, while an L-moment problem is discussed in [36]. The geometric aspects of functional analysis in nonstandard spaces are discussed in the papers [37,38], without any connection with the moment problem. Iterative methods regarding fixed-point and related optimization problems are discussed in [39,40,41]. In the monograph [42], the authors study the sandwich condition $T_1\le T\le T_2$ on the positive cone of the domain space, where $T_1, T_2$ are given linear functionals and $T$ is a solution for a finite number of the interpolation moment conditions. The article [43] provides the necessary and sufficient conditions for the existence of a positive linear operator solution dominated by a convex operator. A result of G. Cassier (see [25]) is applied in order to apply the first theorem in [43] to the classical multidimensional Markov moment problem on a compact with a nonempty interior in ${ℝ}^n.$ A characterization of the existence of a linear operator solution $T$ for an arbitrary infinite number of moment conditions, such that the sandwich constraint $T_1\le T\le T_2$ on $X_{+}$ holds, is also provided. Here, $T_1,T_2$ are the given linear operators. In the article [44], sufficient conditions for the determinacy of probability distributions on $ℝ$ or respectively on $\left[0,+\infty \right)$ are studied. We recall that a measure is a determinate measure on the closed subset $F$ if it is uniquely determined by its moments on $F.$ In the paper [45], the notion of a finite simplicial set is reviewed and applied to a non-standard sandwich theorem on that set. Notably, a finite simplicial set can be unbounded in the case of any locally convex topology on the vector space in which the set is contained. As we have already seen, the non-negative polynomials on ${ℝ}^n$ are not expressible in terms of the sums of squares. This is the motivation for the polynomial approximation results provided in [46,47] and applied to the Markov moment problem with the operator solution in [46,47,48,49]. These results are essentially based on the notion of a moment determinate measure. In [46], it was proved that for a moment determinate measure $\nu ,$ the non-negative polynomials on $F$ are dense in the positive cone of $L_{\nu }^1\left(F\right).$ Consequently, the subspace of the polynomials is dense in $L_{\nu }^1\left(F\right).$ Notably, if $n\ge 2,$ there exist moment determinate measures $\nu$ on ${ℝ}^n,$ such that the polynomials are not dense in $L_{\nu }^2\left({ℝ}^n\right)$ (see [22]). We can assume that all the measures are positive Borel regular measures on $F,$ with finite moments of all the orders. In [47,48,49], the authors prove that for the products $\nu ={\nu }_1\times \cdots \times {\nu }_n$ of $n$ moment determinate measures ${\nu }_i$ on $ℝ,$ any function of the positive cone of $L_{\nu }^1\left(F\right)$ can be approximated by finite sums of special products of polynomials, $p\left(t\right)=p_1\left(t_1\right)\cdots p_n\left(t_n\right)$, where each $p_i$ is non-negative on $ℝ$, which, hence, is a sum of (two) squares, $i=1,\dots ,n.$ For such measures $\nu ,$ this enables us to solve the multidimensional Markov moment problems on ${ℝ}^n$ mentioned above in terms of the quadratic forms. The corresponding result for the products of the $n$ moment determinate measures on ${\left[0,+\infty \right)}^n$ holds. Here, assume that $Y$ is an order complete Banach lattice and $T_1,T_2$ are bounded linear operators applying $L_{\nu }^1\left(F\right)$ to $T.$ In this case, the linear solution $T$ of the problem under investigation is also bounded due to the constraint $T_1\le T\le T_2$ on the positive cone of $L_{\nu }^1\left(F\right).$ The uniqueness of the solution follows according to the density of the polynomials in $L_{\nu }^1\left(F\right).$ To conclude, we can observe that polynomial approximation on bounded subsets solves the existence, as well as the uniqueness, of the solution to a large class of Markov moment problems on ${ℝ}^n$ or on ${\left[0,+\infty \right)}^n, n\ge 2.$