Nonlinear operator extensions of Korovkin's theorems

In this paper we extend Korovkin's theorem to the context of sequences of weakly nonlinear and monotone operators defined on certain Banach function spaces. Several examples illustrating the theory are included.


Introduction
Korovkin's theorem [20], [21] provides a very simple test of convergence to the identity for any sequence (L n ) n of positive linear operators that map C ([0, 1]) into itself: the occurrence of this convergence for the functions 1, x and x 2 .In other words, the fact that lim n→∞ L n (f ) = f uniformly on [0, 1] for every f ∈ C ([0, 1]) reduces to the status of the three aforementioned functions.Due to its simplicity and usefulness, this result has attracted a great deal of attention leading to numerous generalizations.Part of them are included in the authoritative monograph of Altomare and Campiti [8] and the excellent survey of Altomare [4].See [5], [6], [7] and [27] for some very recent contributions.
At the core of Korovkin's theorem (as well as of many of its extensions) is the nice behavior of the involved functions, reminding the property of absolute continuity in real analysis.In what follows we refer to it as the Korovkin absolute continuity.For a real-valued function f defined on a subset K of the Euclidean space R N this property means the following a priori estimate: for every ε > 0 there is δ > 0 such that for all x, y ∈ K, Necessarily, such a function is uniformly continuous.The converse fails (for example, see the case of the function √ x on (0, ∞)), but it occurs for all uniformly continuous and bounded functions f : K → R. Examples of unbounded functions on R N playing the property of Korovkin absolute continuity are ± pr 1 , ..., ± pr N and N k=1 pr 2  k , where pr k denotes the canonical projection on the kth coordinate.More details are available in [23].
Based on the concept of Korovkin absolute continuity, the present authors have extended Korovkin's theorem to the framework of sublinear and monotone operators acting on function spaces defined on appropriate subsets K of R N .See [12], [14], [15] and [16].
The aim of the present paper is to further extend these results to the case of sequences of nonlinear and monotone operators converging pointwise to an operator of the same nature, possibly different from the identity.In the linear case, results of this kind have been obtained (in order) by Wang [29], Guessab and Schmeisser [18] and Popa [27].According to Theorem 1 in [27], if L n (n ∈ N) and L are positive linear operators from C ([0, 1]) into itself such that L(1)L(x 2 ) = (L(x)) 2  and L(1)(t) > 0 for all t ∈ [0, 1], then L n (f ) → L(f ) uniformly for all f ∈ C ([0, 1]) if and only if L n (1) → L(1), L n (x) → L(x) and L n (x 2 ) → L(x 2 ) uniformly on [0,1].This is extended by our Theorem 3 (Section 5) to a large class of nonlinear operators defined on a space C(K), with K a compact subset of the Euclidean space R N .The possibility to replace C(K) with other function spaces (such the Lebesgue spaces L P (K) with p ∈ [1, ∞)) makes the statement of Theorem 4 (Section 5).Due to the linearity of L, the condition L(1)L(x 2 ) = (L(x)) 2 in Popa's Theorem 1 is invariant under any translation x → x + α, a fact that makes its result to work for all spaces C ([a, b]) (not just for C ([0, 1])).In the nonlinear case, this remark doesn't work, and in the case of compact subsets K not included in R N + we have to adjust the hypotheses by using an analogue of L(1)L(( For the convenience of the reader, we summarized in Section 2 some very basic facts on ordered Banach spaces, usually omitted by most of the textbooks on Banach lattices, but essential for understanding our approach.So is Theorem 1, that motivates why in this paper we consider only operators with values in a C(K) space.
Section 3 is a thorough presentation of the class nonlinear operators that makes the subject of this paper.These are sublinear and monotone operators acting on ordered Banach spaces of functions which verify the property of translatability relative to the multiples of unity.They was introduced in [15], motivated by our interest in Choquet's theory of integration, but there are many other examples outside that theory, mentioned in this paper.
Section 4 is devoted to a nonlinear generalization of a result due to Popa [27], which proves to be essential in proving our new Korovkin type results in Section 5. Examples illustrating our main results are exhibited in Section 6.The paper ends with a short list of open problems.

Preliminaries on ordered Banach spaces
Most authors (see Aliprantis and Tourky [3] and Schaefer and Wolff [28]) define the ordered Banach spaces as the real Banach spaces E endowed with an order relation ≤ such that the following three conditions are verified: As usually, x ≤ y will be also denoted also y ≥ x and x < y (equivalently y > x) will mean that y ≥ x and x = y.
For convenience, we will consider in this paper only ordered Banch spaces E whose positive cones are closed (in the norm topology), proper (−E + ∩ E + = {0}) and generating (E = E + − E + ).
An ordered vector space E such that every pair of elements x, y admits a supremum sup{x, y} and an infimum is called a vector lattice.In this case for each x ∈ E we can define x + = sup {x, 0} (the positive part of x), x − = sup {−x, 0} (the negative part of x) and |x| = sup {−x, x} (the modulus of x).We have x = x + − x − and |x| = x + + x − .A Banach lattice is any real Banach space E which is at the same time a vector lattice and verifies the condition |x| ≤ |y| implies x ≤ y .
Most classical Banach spaces are actually Banach lattices.So are the Euclidean space R N and the discrete spaces c 0 , c and ℓ p for 1 ≤ p ≤ ∞ (endowed with the coordinate-wise ordering).The same is true for the function spaces and the pointwise ordering modulo null sets.
It is well known that all norms on the N -dimensional real vector space R N are equivalent.When endowed with the sup norm and the coordinate wise ordering, R N can be identified (algebraically, isometrically and in order) with the Banach lattice C ({1, ..., N }), where {1, ..., N } carries the discrete topology.
A convenient way to emphasize the properties of ordered Banach spaces is that described by Davies in [11].Davies calls an ordered Banach space E regularly ordered if x = inf { y : y ∈ E, − y ≤ x ≤ y} for all x ∈ E. This class of spaces brings together the Banach lattices and some other spaces which are not vector lattices, a notorious example being Sym(n, R), the ordered Banach space of all n × n-dimensional symmetric matrices with real coefficients with the ordering A ≤ B if and only if Ax, x ≤ Bx, x and the norm For details, see [24], Section 2.5, pp.97-103.
Lemma 1.Every ordered Banach space can be renormed by an equivalent norm to become a regularly ordered Banach space.
For details, see Namioka [22].Some other useful properties of ordered Banach spaces are listed below.
Lemma 2. Suppose that E is a regularly ordered Banach space.Then: (a) There exists a constant C > 0 such that every element x ∈ E admits a decomposition of the form x = u − v where u, v ∈ E + and u , v ≤ C x .
(b) The dual space of E, E * , when endowed with the dual cone The assertion (a) follows immediately from Lemma 1.For (b), see Davies [11], Lemma 2.4.The assertion (c) is an easy consequence of the Hahn-Banach separation theorem; see [24], Theorem 2.5.3, p. 100.
The assertion (d) is also a consequence of the Hahn-Banach separation theorem; see [28], Theorem 4.3, p. 223.
The concept of strictly positive function admits a natural extension in the framework of ordered Banach spaces.Precisely, a positive element of an ordered Banach space E is said to be strictly positive provided that x * (x) > 0 for every nonzero functional x * ∈ E * + .Strictly positive elements exists in every separable ordered Banach space E. For example, choose a sequence (x n ) n of elements of E, dense in the closed unit ball and consider their decomposition vanishing at x will vanish on all elements x n .By density, this implies that x * = 0. Actually a stronger results holds, precisely, the set of all strictly positive elements of a separable ordered Banach space is dense into the positive cone.See [28], Theorem 7.6, p. 241.The spaces C(K), C b (X) and UC b (X) (separable or not) admit strictly positive elements with stronger properties, called order units.Recall that an element u > 0 of an ordered Banach space E is called an order unit if for each for each x ∈ E there exists a real number λ > 0 such that −λu ≤ x ≤ λu.Necessarily, an order unit is a strictly positive element.Besides the three aforementioned spaces, the case of the space Sym(n, R) outlines the importance of regularly ordered Banach spaces with an order unit whose norms are associated to the unit via the formula The natural order unit of the vector lattice R N is the vector u, whose all components equal 1.The norm associated to this unit is the sup-norm.
An example of Banach lattice without strictly positive elements is C 0 (X), provided that X is a nonseparable metric space.
The following result explains why in this paper we consider only operators with values in C(K) space.
Theorem 1.Every ordered Banach space E can be represented as a vector subspace of the space C(K) of continuous real-valued functions on a compact Hausdorff space K via an order-preserving linear and continuous map Φ : E → C(K).If E has a strictly positive element e, then one can choose K such that Φ(e) = 1, the unity of C(K).
Proof.According to the Alaoglu theorem, the set K = x * ∈ E * + : x * ≤ 1 is compact relative to the w * topology.Taking into account the assertions (c) and (d) of Lemma 2 one can easily conclude that E embeds into C(K) via the positive linear isometry For the second part of Theorem 1, notice that Φ(e) is a strong order unit for C(K).Then the conclusion follows from a classical result due to Kadison [19], stating that every ordered real vector space with an strong order unit can be represented as a vector subspace of the space of continuous real-valued functions on a compact Hausdorff space via an order-preserving map that carries the order unit to the constant function 1.

Weakly nonlinear operators acting on ordered Banach spaces
Our next goal is to describe a class of nonlinear operators which provides a convenient framework for the extension of Korovkin's theorem.
Given a metric space X, we attach to it the vector lattice F (X) of all real-valued functions defined on X, endowed with the metric d and the pointwise ordering.
Suppose that X and Y are two metric spaces and E and F are respectively ordered vector subspaces (or subcones of the positive cones) of F (X) and F (Y ) and that F (X) contains the unity.An operator T : E → F is said to be a weakly nonlinear if it satisfies the following two conditions: (SL) (Sublinearity) T is subadditive and positively homogeneous, that is, for all f, g in E and α ≥ 0; (TR) (Translatability) T (f + α • 1) = T (f ) + αT (1) for all functions f ∈ E and all numbers α ≥ 0.
In the case when T is unital (that is, T (1) = 1) the condition of translatability takes the form A stronger condition than translatability is for all functions f ∈ E and all numbers α ∈ R.
The last condition occurs naturally in the context of Choquet's integral, being a consequence of what is called there the property of comonotonic additivity, that is, See [14] and [13], as well as the references therein.
In this paper we are especially interested in those weakly nonlinear operators which preserve the ordering, that is, which verify the following condition: Remark 1.If T is a weakly nonlinear and monotone operator, then Indeed, for α ≥ 0 the property follows from positive homogeneity.Suppose now that α < 0. Since T (0) = 0 and −α > 0, by translatability it follows that 0 = T (0 Examples weakly nonlinear and monotone operators can be found in [14], [13] and [15]. Suppose that E and F are two ordered Banach spaces and T : E → F is an operator (not necessarily linear or continuous).
If T is positively homogeneous, then and every positively homogeneous and monotone operator T maps positive elements into positive elements, that is, Therefore, for linear operators the property (3.1) is equivalent to monotonicity.Every sublinear operator is convex and a convex operator is sublinear if and only if it is positively homogeneous.
The norm of a continuous sublinear operator T : E → F can be defined via the formulas A sublinear operator may be discontinuous, but when it is continuous, it is Lipschitz continuous.More precisely, if T : E → F is a continuous sublinear operator, then Remarkably, all sublinear and monotone operators are Lipschitz continuous: Theorem 2. Every sublinear and monotone operator T : and thus it is Lipschitz continuous with Lipschitz constant equals to T , that is, See [16] for details.Theorem 2 is a generalization of a classical result of M. G. Krein concerning the continuity of positive linear functionals.See [1].

An a priori estimate
The proof of our main results depend on the following a priori estimate, previously noticed by Popa [27]  In what follows u = (1, ..., 1) denotes the natural order unit of the vector lattice R N .Every compact subset K of the Euclidean space R N can be moved into the positive cone R N + via a translation of the form T a : x → x + αu, associated to a number α ≥ 0. The smallest α doing this job will be denoted α(K, u) and we will refer to it as the deficit of positivity of K in the direction u.We have Suppose that K is a compact subset of the Euclidean space R N , X is a compact Hausdorff space and V and A are weakly nonlinear and monotone operators from C(K) into C(X).Then for every function f ∈ C(K), and every ε > 0 there exists δ > 0 such that whenever α ≥ α(K, u).
Proof.Let f ∈ C(K) and ε > 0. Since f is Korovkin absolutely continuous, there is δ > 0 such that for all x, y ∈ K we have Viewing y as a parameter, the last inequality can be rewritten as The choice of α makes pr k +α ≥ 0 and this allows us to value the properties of positive homogeneity and translatability of the operator T.
Suppose for a moment that f is nonnegative.Then, according to Theorem 2, (pr k (y) + α) 2 T (1) , which yields, for each t ∈ X, the following inequality in C(X) : Applying the weakly nonlinear and monotone operator A to the both sides of the last inequality and taking into account that −2T (− pr k −α) ≥ 0, we obtain As t ∈ T was arbitrarily chosen, this ends the proof in the case of nonnegative functions.
The case where f is a signed function can be reduced to this one by replacing f by f + f ∞ .Indeed, using the property of weak nonlinearity of the operators T and A we have Corollary 1.Under the hypotheses of Lemma 3, the following inequality holds

The main results
We are now in a position to settle the case of uniform convergence.
Theorem 3. Suppose that K is a compact subset of the Euclidean space R N , X is a compact Hausdorff space and T n (n ∈ N) and A are weakly nonlinear and monotone operators from E = C(K) into C(X) such that (5.1) A( 1) is a strictly positive element and for a suitable α ≥ α(K, u). Then if and only if this property occurs for each of the functions Moreover, if K ⊂ R N + , then Theorem 3 works for α = 0. Proof.The necessity part is clear.For sufficiency, notice first that we may restrict ourselves to the case where K ⊂ R N + by performing, if necessary, a change of variable of the form y = x + αu.This allows us to continue the proof with α = 0.
According to Corollary 1, for all f ∈ C (K) and ε > 0 there exists δ > 0 such that for all n ∈ N we have whence, by using hypothesis (5.2), we infer that Combining this fact with the inequality By our hypotheses the function A( 1) is strictly positive, which yields inf t∈X A(1 and the proof is done. Remark 2. In the particular case where E = F = C(K) and A is an algebra homomorphism preserving the unit (in particular, if A = I), the conditions (5.2) and (5.1) are automatically fulfilled and Theorem 3 is covered by our previous results in [12].
The condition lim n→∞ T n (1) − A(1) = 0 in Theorem 3 implies the equicontinuity of the operators T n since T n = T n (1) for all n.
The result of Theorem 3 can be put in a more generality as follows: Theorem 4. Suppose that K is a compact subset of the Euclidean space R N , E is an ordered Banach space that includes C(K) in such a way that the canonical inclusion i K : C(K) → E is a positive linear operator with dense image and X is a compact Hausdorff space.If T n (n ∈ N) and A are weakly nonlinear and monotone operators from E into C(X) such that A(1) is a strictly positive element, (5.4) then lim n→∞ T n (f ) − A(f ) = 0 for all f ∈ E, if and only if this property of convergence occurs for the following set of functions: (5.6) 1, −(pr 1 +α), ..., −(pr N +α) and In particular, Theorem 4 holds for the Banach lattices E = L p (µ), where µ is a positive Borel measure on K.The fact that L p (µ) includes C(K) as a dense subspace can be covered from many sources, e.g., [9], Corollary 4.2.2, p. 252.
Proof.As in the case of Theorem 3, for the sufficiency part we may assume that α = 0 and Necessarily, g is Korovkin absolutely continuous, so there exists a number δ > 0 such that for all x and y in R N .
According to Corollary 1 applied to T n • i K and A • i K for α = 0, we have whence, by denoting M = sup T n (1 and taking into account the hypothesis (5.5), we obtain the inequality Thus, under the presence of condition (5.6), we have Proceeding as in the proof of Theorem 3, we infer that for all g ∈ C (K).This property can be transferred to f due to the inequalities Therefore lim for all f ∈ E and the proof is done.
Remark 3. When the property of translatability is replaced in Theorem 3 and Theorem 4 by the property of strong translatability, the set of test functions can be simplified as follows: 1, − pr 1 , ..., − pr N and N k=1 pr 2 k +2α pr k .
As was noticed by Korovkin [20], [21], his theorem mentioned in the Introduction also works when the unit interval is replaced by the unit circle Noticing that in this case the cosine function represents the restriction of pr 1 to S 1 and the sine function represents the restriction of pr 2 to S 1 , one can easily deduce the following result from Theorem 3: Theorem 5. Suppose that T n (n ∈ N) and A are weakly nonlinear and monotone operators from E = C 2π (R) into itself such that A(1) = 1 and if and only if this property occurs for each of the following test functions, 1, −1 − cos, −1 − sin and 3 + 2 cos +2 sin .
Here C 2π (R) denotes the Banach lattice of all continuous functions f : R → R, periodic, of period 2π, endowed with the sup-norm.
Remark 4. Notice that the hypothesis (5.8) is automatically fulfilled when A is an algebra homomorphism preserving the unit (in particular, if A = I).When the property of translatability is strengthen to strong translatability, then the set of test functions in Theorem 5 can be replaced by 1, − cos, − sin and 2 cos +2 sin .
The analogues of Theorem 5 for some other cases of interest such as the 2dimensional torus S 1 × S 1 and the 2-dimensional sphere S 2 are left to the reader.
Theorem 5 (and the remarks following it) also works in the context of Cesàro convergence, that is, of the convergence of the form Concrete examples illustrating the above results are indicated in the next section.Last but not the least, one can extend our results (following the model of Theorem 2 in [15]) using instead of the classical families of test functions on R N the separating functions, which allow us to replace the a priori estimates of the form (1.1) by inequalities that work in the general context of metric spaces.For details concerning these functions see [23].

Examples
In what follows we will need the following family of polynomials related to Bernstein's proof of the Weierstrass approximation theorem.As is well known they verify a number of combinatorial identities such as for all x ∈ R. See [10], Theorem 8.8.1, p. 256).
Example 1.Given a continuous function ϕ : [0, 1] → [0, 1] one associates to it the sequence of nonlinear operators T n : C ([0, 1]) → C ([0, 1]) defined by the formulas Clearly, these operators are sublinear, strongly translatable, monotone and unital (the last property being a consequence of formula (6.1)).We will show (using Theorem 3) that the sequence (T n ) n is pointwise convergent to the linear positive and unital operator The condition (5.1) is trivial since A is unital.The fulfillment of condition (5.2) is a consequence of the fact that A is an algebra homomorphism preserving the unit; see Remark 2. Using the identities (6.1)-( 6.3), one can show that T n (f ) → A(f ) uniformly for each of the functions 1, −x and x 2 ; this set of test functions suffices since [0, 1] ⊂ R + .Therefore the operators T n and A fulfill the hypotheses of Theorem 3 and we can conclude that T n (f ) → A(f ), uniformly for all f ∈ C([0, 1]).
Example 2. As above, ϕ : [0, 1] → [0, 1] is a continuous function.Attached to it is the sequence of Kantorovich operators K n : L 1 ([0, 1]) → L 1 ([0, 1]) , defined by the formulas and also the operator Clearly, these operators are linear, positive and unital.Also, and the operator A verifies the technical condition (5.5) for α = 0, that is, The operators are sublinear, monotone and strongly translatable.Simple computations (based on the identities (6.1)-(6.3)),show that which imply that T n (f ) → A(f ) uniformly for each of the functions f ∈ 1, −x, x 2 (and thus in the L 1 -norm).According to Theorem 4, this property of convergence occurs for all functions f ∈ L 1 ([0, 1]) .
Example 3. Let ϕ and A as in the preceding example and consider the sequence of Choquet-Kantorovich operators ]) defined by the formulas where the letter C in front of the integral means that we are dealing with a Choquet's integral, in our case, the integral with respect to the capacity µ representing a distortion of the Lebesgue measure m via the formula µ(A) = g(m(A)), where g : [0, 1] → [0, 1] is strictly increasing, differentiable and concave function such that g(0) = 0 and g(1) = 1.The theory developed by Gal and Trifa [17] assures the applicability of Theorem 4 to conclude that T n (f ) → A(f ) for all functions f ∈ L 1 ([0, 1]) .
Examples involving functions of several variables can be exhibited using tensor products of operators.For instance, in the case of Example 1 by the tensor product method we get p n,k (ϕ(x))p n,j (ϕ(y)) • sup{f (t, s); t ∈ [k/(n + 1), (k + 1)/(n + 1)], v ∈ [j/(n + 1), (j + 1)/(n + 1)]}, which for any We already noticed that any algebra homomorphisms preserving the unit (in particular, A = I) is a solution.Is the converse true?The usual books on functional equations (including that by Aczél and Dhombres [2]) seem silent in this case.

Open problems
Not entirely surprising, one can prove results similar to those in this paper by working with other classes of limit operators.The following one was presented by the second named author in a talk given at the Fourth Romanian Itinerant Seminar on Mathematical Analysis and its Applications, Braşov, May 19-20, 2022: Theorem 6. Suppose that K is a connected and compact subset of the Euclidean space R N , A : C(K) → R is a (possibly nonlinear) monotone functional such that A(1) = 1 and (T n ) n is a sequence of sublinear and monotone operators from C(K) into C(K) such that T n (f )(x) → A(f ) • 1 in the sup-norm for each of the test functions 1, ± pr 1 , ..., ± pr N and N k=1 pr 2 k .Then this property of convergence extends to all nonnegative functions f in C(K).
It occurs for all functions in C(K) provided that the operators T n are weakly nonlinear and monotone.
Moreover, in either case the family of test functions can be reduced to 1, − pr 1 , ..., − pr N and N k=1 pr 2 k provided that K is included in the positive cone of R N .Theorem 6 is a nonlinear extension of the Weyl ergodic theorem.This theorem, whose essence is the unique ergodicity of the irrational rotation  [25] and Parry [26] for details.The Weyl ergodic theorem represents the particular case of Theorem 6 when C(K) = C 2π (R), . The classical proof of the Weyl ergodic theorem consists in verifying it in the case of exponentials e inx and next in applying the trigonometric form of the Weierstrass approximation theorem.Theorem (6) reduces the set of test functions to the triplet 1, cos and sin!Notice that Theorem 5 does not apply in this case because the technical hypothesis (5.8) fails.
Problem 2. Find other substitutes for the technical condition 5.2 that avoids the presence of the operation of multiplication in the codomain.
Another problem left open is the following one: Problem 3. Find an extension of Theorem 4 to the case of operators defined on the Lebesgue spaces L p (µ) associated to a finite measure µ on R N .Of a special interest is the case of the Gaussian measure N e − x 2 /2 dx.
and bounded on the metric space X} , UC b (X) = {f : X → R : f uniformly continuous and bounded on the metric space X} , C 0 (X) = {f : X → R : f continuous and null to infinity on the locally compact Hausdorff space X} , each one endowed with the sup-norm f ∞ = sup x |f (x)| and the pointwise ordering.Other Banach lattices of an utmost interest are the Lebesgue spaces L p (µ) (p ∈ [1, ∞]), endowed with norm in the context of linear and positive operators defined on the Banach lattice C ([a, b]) .

5 : 1 .
We end our paper by mentioning few open problems that might be of interest to our readers.The first one concerns a technical hypothesis made in our Theorems 3-Problem How large is the set of solutions of the functional equation − pr k −α))2 , among all weakly nonlinear and monotone operators A : C(K) → C(X)?