Using Linear Functionals to Study Nonlinear Functionals and Their Support Sets

Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets

Jessica E. Stovall

University of North Alabama

May 4, 2018

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets

1 Preliminaries

2 Associating a Linear Operator with T

3 Theorems Regarding Support Sets

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Background

Topology Measure Theory Banach Lattices Operator Theory

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Topology

Any union of open sets is open Any ﬁnite intersection of open sets is open Any intersection of closed sets is closed Any ﬁnite union of closed sets is closed

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Clopen Sets

Deﬁnition A set is clopen if it is both open and closed.

Any ﬁnite union of clopen sets is clopen Any ﬁnite intersection of clopen sets is clopen

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Extremally Disconnected

Deﬁnition A topological space X is extremally disconnected if and only if the closure of every open set in X is open.

For any space X , the following are equivalent: 1 X is extremally disconnected 2 Every two disjoint open sets in X have disjoint closures

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Separation Axioms

The following are called the separation axioms.

T1: Given two distinct points x and y, there is an open set that contains y but not x.

T2: Given two distinct points x and y, there are disjoint open sets O1and O2 such that x ∈ O1 and y ∈ O2.

T3: In addition to T1, given a closed set F and a point x not in F, there are disjoint open sets O1 and O2 such that x ∈ O1 and F ⊂ O2.

T4: In addition to T1, given two disjoint closed sets F1 and F2, there are disjoint open sets O1 and O2 such that F1 ⊂ O1 and F2 ⊂ O2.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Spaces Satisfying the Separation Axioms

A topological space satisfying T1 is called a Tychonoﬀ space.

A topological space which satisﬁes T2 is called a Hausdorﬀ space.

A topological space which satisﬁes T3 is called a regular space

A topological space which satisﬁes T4 is called a normal space.

For topological spaces, T4 ⇒ T3 ⇒ T2 ⇒ T1.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Measure Theory

Deﬁnition An outer measure µ∗ is a nonnegative extended real-valued set function deﬁned on all subsets of a space X and having the following properties: 1 µ∗(∅) = 0 2 A ⊂ B ⇒ µ∗(A) ≤ µ∗(B)

3 S∞ ∗ P∞ ∗ E ⊂ i=1 Ei ⇒ µ (E) ≤ i=1 µ (Ei )

The second property above is called monotonicity and the third property is called countable subadditivity.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Measure Theory

Theorem (Monotone Convergence Theorem)

Let {fn} be a sequence of nonnegative measurable functions which converge almost everywhere to a function f and suppose that R R fn ≤ f for all n. Then, f = lim fn.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Measure Theory

Theorem (Riesz Representation Theorem) Let X be a compact Hausdorﬀ space and C(X ) the space of continuous real-valued functions on X . Then to each bounded linear functional F on C(X ) there corresponds a unique ﬁnite signed Baire measure µ on X such that F (f ) = R fdµ for each f in C(X ).

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Banach Lattices

Deﬁnition A real vector space E is an ordered vector space if for x, y, and z in E and for all real α > 0, the following properties hold: 1 If x ≤ y, then x + z ≤ y + z 2 If x ≤ y, then αx ≤ αy

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Banach Lattices

Deﬁnition An ordered set (M, ≤) is called a lattice if any two elements in M have a least upper bound and a greatest lower bound.

Deﬁnition A Riesz space or vector lattice is an ordered vector space with the property that for each pair of elements in E, both the inﬁmum and the supremum are also in E.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Banach Lattices

Deﬁnition A vector lattice E with norm k · k with the property that |x| ≤ |y| implies kxk ≤ kyk for x and y in E and |x| = x ∨ (−x) is called a normed lattice.

Deﬁnition If E is a normed lattice which is complete with respect to the norm, that is every Cauchy sequence converges, then E is called a Banach lattice.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Examples of Banach Lattices

Some examples of Banach lattices include:

1 The Lp-spaces, Lp(X , B, µ), where (X , B, µ) is a measure space and where the ordering is deﬁned by f ≤ g if and only if f (x) ≤ g(x) almost everywhere. 2 The collection of continuous real-valued functions on a compact Hausdorﬀ space X , denoted by C(X ), with the supremum norm kf k∞ = sup{|f (x)| : x ∈ X } and the same ordering as in example 1.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Properties of Banach Lattices

Deﬁnition The positive cone of E, denoted E +, is all of the x in E such that x ≥ 0.

Deﬁnition A subspace A of E is called order bounded if and only if A is contained in some order interval [x, y] = {z ∈ E : x ≤ z ≤ y}.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Dedekind Complete

Deﬁnition E is called Dedekind complete if every non-empty order bounded set has a supremum and an inﬁmum in E.

Some examples of Dedekind complete Banach lattices include: 1 The Lp-spaces 2 C(X ) where X is extremally disconnected

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Ideals

A subset A of E is called solid if |x| ≤ |y| for some y in A implies that x is in A. Additionally, every solid subspace I of E is called an ideal in E. The ideal generated by a non-empty subset A of E is the smallest, with respect to inclusion, ideal containing A. It coincides with the intersection of all ideals that contain A and is denoted by I (A). Furthermore, the ideal generated by a single element x is called the principal ideal generated by x and is denoted by I (x). Additionally, I (x) = {y ∈ E : there exists λ ≥ 0 S such that |y| ≤ λ|x|} = n[−nx, nx].

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of a Principal Ideal

Consider the Banach lattice C(X ) where X is compact and Hausdorﬀ and the function e : X → R deﬁned by e(x) = 1 for all x in X . Then the principal ideal generated by e is given by I (e) = {f ∈ C(X ) : there exists λ > 0 with |f | ≤ λe}. That is, I (e) = {f ∈ C(X ) : there exists λ > 0 with |f (x)| ≤ λ for all x ∈ X }. Since we are dealing with continuous, real-valued functions on a compact space, we have for every f in C(X ), there exists a λ > 0 such that kf k∞ ≤ λ. Thus, I (e) = C(X ).

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Order Units

Deﬁnition If I (e) = E, then e ∈ E + is called an order unit or strong unit. That is, e is an order unit if for each x in X there exists some λ ≥ 0 such that |x| ≤ λe.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of an Order Unit

Let C([0, 1]) represent the continuous functions on the interval [0, 1]. Then, C([0, 1]) has multiple order units. Since any function that is continuous on [0, 1] must be bounded above by some multiple of 1 and bounded below by some multiple of 1, it follows that e = 1 is an order unit for C([0, 1]). In fact, any function bounded away from zero would serve as an order unit for C([0, 1]).

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of Space with No Order Units

The space C(R) of continuous functions on the real line has no order units. Suppose the u is an order unit for C(R). Then all functions from C(R) would have to be bounded by a multiple of u. x But ue cannot be bounded by a multiple of u. Thus, C(R) has no order units.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Quasi-Interior Points

Deﬁnition Assume that E is a normed Riesz space. If I (e) is dense in E, then e ∈ E + is called a quasi-interior point.

All order units are quasi-interioir points, but the reverse implication is not always true.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of a Quasi-Interior Point that is Not an Order Unit

The space L2([0, 1]) with the usual norm has a quasi-interior point e = 1. However, e = 1 is not an order unit for the space.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Order Continuous Norm

Deﬁnition A Banach lattice E is said to have an order continuous norm if xα ↓ 0 implies kxαk ↓ 0. Equivalently, a norm is order continuous if and only if xα ↑ x implies kx − xαk ↓ 0.

It follows from the above deﬁnition that if E has an order continuous norm and {xα} are directed upwards with ∨xα = x, then xα → x in norm. This will be denoted by xα % x. Likewise, if E has an order continuous norm and {xα} are directed downwards with ∧xα = x, then xα → x in norm. This case will be denoted by xα & x.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of Order Continuous Norm

p Consider the space (L ([0, 1]), k · kp). Now consider ∨fn = f . p Then, fn % f . Thus, |fn − f | → 0. It follows that |fn − f | → 0 R 1 p and 0 |fn − f | → 0. Therefore, kfn − f kp → 0. Hence, k · kp is an order continuous norm.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Proposition

Any Banach lattice with an order continuous norm is Dedekind complete.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Representation Theory

Any Dedekind complete Banach lattice, E, with a quasi-interior point, e, is lattice isomorphic to a space of continuous, extended real-valued functions deﬁned on a compact Hausdorﬀ space X . This representation is a generalization of Kakutani’s Representation Theorem and was due to H.H. Schaefer and further explored by John L. Kelley and Isaac Namioka.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Operator Theory

Definition A mapping T : E → F between two vector spaces over the same field is called a linear operator or simply an operator if T (αx + βy) = αT (x) + βT (y) holds for all x and y in X and all scalars α and β. An operator norm is defined by kT k = supkx| ≤1kT (x)k = supkxk=1kT (x)k. If kT k < ∞, then T is called a bounded operator, and when kT k = ∞ the operator T is called an unbounded operator.

A linear functional on vector space E is a linear operator that maps E into the set of real numbers.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Conditions on the Operator

In addition to T : E → R being nonlinear, T must also be: Orthogonally Additive Continuous Monotonic Subhomogeneous

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Deﬁnitions

Deﬁnition The operator T is called orthogonally additive if T (f + g) = T (f ) + T (g) for f ≥ 0, g ≥ 0, and f ∧ g = 0.

Deﬁnition An operator T : E → F between two Banach lattices is called monotonic if T (f ) ≤ T (g) whenever f ≤ g.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Deﬁnitions

Deﬁnition The operator T is subhomogeneous if for f ≥ 0 and α > 0, there exist positive constants m(α) and M(α) with m(α) a monotone function of α and unbounded so that m(α)T (f ) ≤ T (αf ) ≤ M(α)T (f ) and M(α) goes to zero as α goes to zero.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of the Types of Operators Being Studied

2 The nonlinear operator T : C(X ) → R deﬁned by T (f ) = L(f ) for f ∈ C(X ), where L is a linear functional, is an example of an operator that satisﬁes all of the above conditions. More generally, for appropriate functions φ and λ, the map T (f ) = φ(L(λf )), would also satisfy these conditions.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Observations

T (0) = 0 T is orthogonally additive and 0 ∧ 0 = 0 T (0) + T (0) = T (0) T is positive T is monotonic and T (0) = 0 f ≥ 0 implies T (f ) ≥ T (0)

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Process

Find a premeasure based on T Use Carathéodory’s Extension Theorem to find a measure µ induced by µ Define a linear operator based on µ

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Setting

Let E be a Banach lattice with an order continuous norm and a quasi-interior point and let T : E → R be an orthogonally additive, continuous, monotonic, and subhomogeneous operator. Any Banach lattice with an order continuous norm and a quasi-interior point is Dedekind complete. The elements of E are identiﬁed with their representation in C ∞(X ), where X is an extremally disconnected compact Hausdorﬀ space.

In this setting, if K is a clopen set, then the function χK is continuous.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Premeasure

µ is an premeasure if it satisﬁes the following three conditions: 1 µ(∅) = 0 2 µ is ﬁnitely additive 3 µ is countably monotone

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets A Premeasure Based on T

Deﬁnition Let K be the set of all clopen subsets of X and deﬁne a set function µ : K → [0, ∞] by µ(K) = T (χK ) for any K ∈ K.

Theorem (J.S. and W. Feldman, 2017) µ is a premeasure.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets µ(∅) = 0

Note that ∅ is clopen. Therefore, µ(∅) = T (χ∅) = T (0) = 0.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets µ is Finitely Additive

n Let {Ek }k=1 be a ﬁnite collection of disjoint clopen sets. Then n n S S Sn k=1 Ek is clopen. So, µ( k=1 Ek ) = T (χ Ek ). But, because n k=1 Sn P the Ek are disjoint, T (χ Ek ) = T ( k=1 χEK ). Also, T is k=1Pn Pn orthogonally additive, so T ( k=1 χEK ) = k=1 T (χEk ). Pn Pn But k=1 T (χEk ) = k=1 µ(Ek ). Sn Pn Thus, µ( k=1 Ek ) = k=1 µ(Ek ). Hence, µ is ﬁnitely additive.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets µ is Countably Monotone - Highlights

Sk−1 Deﬁne B1 = E1 and Bk = Ek \ i=1 Ei for k > 1. B1 = E1

B2 = E2 \ E1

B3 = E3 \ (E1 ∪ E2) and so on... S∞ S∞ Notice that Bk ⊂ Ek for all k, k=1 Bk = k=1 Ek , and all of the Bk are disjoint clopen sets.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Measure

Let (X , M, µ) denote the Carath´eodory measure space induced by µ. Since µ is a premeasure and K is closed with respect to relative complements, µ : M → [0, ∞] is the Carath´eodory extension of µ. So every clopen subset K of X is measurable with respect to the outer measure µ∗ induced by µ and µ(K) = µ∗(K) = µ(K).

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Baire Measure

Then for every f ∈ C(X ), S∞ 1 {x|f (x) > α} = n=1 {x|f (x) > α + n }, which is a countable union of clopen sets. Since M is a σ-algebra that contains all of the clopen sets, it follows that {x|f (x) > α} is in M. Hence, f is measurable with respect to M, and thus µ is a Baire measure.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Deﬁning L

Using the Baire measure µ we just found, deﬁne L(f ) = R fdµ for any f ≥ 0 in E. Then L is a positive operator from E + to [0, ∞]. Restricting to I (e)+, the bounded nonnegative functions on X , then L is a positive linear functional.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Support of an Operator

The support of T , denoted KT , is deﬁned as S KT = X \ {h∈I (e)+:T (h)=0}{x ∈ X : h(x) > 0} I (e)+ denotes the positive elements in the ideal generated by e. That is, I (e)+ is the bounded non-negative functions on X .

Notice that if T is linear, then KT is equivalent to the support of the corresponding measure.

T ≡ 0 if and only if KT = ∅.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Relationships between T and L

L and T agree on the characteristic functions. The support of L is equal to the support of T

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets L and T agree on the characteristic functions

Proof. Let K be a clopen set. It follows that χ is continuous. By R K construction, L(χK ) = χK dµ = µ(K) = µ(K). But from the

deﬁnition of µ, µ(K) = T (χK ). So T (χK ) = L(χK ). Thus, L and T agree on the characteristic functions.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Support Sets are Equal

Let z be in KfT . This means S z ∈ {h∈I (e)+:T (h)=0}{x ∈ X : h(x) > 0}. So, there exists a continuous function h > 0 such that h ∈ I (e)+, h(z) > 0, and T (h) = 0. Since h is continuous, bounded, and h(z) > 0, there exists a clopen set H and a constant α > 0 such that z is in H,

αχH (z) 6= 0, and αχH ≤ h. Since T is monotonic, and thus

positive, 0 ≤ T (αχH ) ≤ T (h) = 0. Therefore, T (αχH ) = 0.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Proof -continued

Additionally, T is subhomogeneous. So, there exists a positive

constant m(α) such that 0 ≤ m(α)T (χH ) ≤ T (αχH ) = 0. It

follows that m(α)T (χH ) = 0 and thus T (χH ) = 0. Since L and T

agree on characteristic functions, L(χH ) = 0. Thus,

z ∈ {x : χH (x) > 0} ⊂ KfL. So, KfT ⊂ KfL and therefore KL ⊂ KT .

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Proof - continued

Now, let z be in KfL. This means S z ∈ {h∈I (e)+:L(h)=0}{x ∈ X : h(x) > 0}. So, there exists a continuous function h > 0 such that h ∈ I (e)+, h(z) > 0, and L(h) = 0. Since h is continuous, bounded, and h(z) > 0, there exists a clopen set H and a constant α > 0 such that z is in H,

αχH (z) 6= 0, and αχH ≤ h. Since L is positive and monotonic,

0 ≤ L(αχH ) ≤ L(h) = 0. Hence, L(αχH ) = 0.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Proof - continued

Additionally, L is linear, and thus, 0 = L(αχH ) = αL(χH ). It

follows that L(χH ) = 0. Since L and T agree on characteristic

functions, T (χH ) = 0. Thus, z ∈ {x : χH (x) > 0} ⊂ KfT . So, KfL ⊂ KfT and therefore, KT ⊂ KL.

Hence KL = KT , that is, the support of L is equal to the support of T .

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Discussion

Notice in the discussion of L, the quasi-interior point e where e mapped to 1 and the decomposition e = χ + χ were used. K Ke But if a Banach lattice has a quasi-interior point, then it is unique in the sense that if u and v are two quasi-interior points, then the principal ideals I (u) and I (v) are lattice isometric. The relationship between a linear operator J and a nonlinear operator T related to the choice of quasi-interior point is now described.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets A Linear Functional Associated with T

Deﬁnition An order continuous map, J : E + → [0, ∞], which is a linear functional restricted to the positive elements in a dense principal ideal, is associated with T if there exists a quasi-interior point, u of E such that for every decomposition u = u1 + u2 of u where u1 ∧ u2 = 0 and u1 and u2 are greater than or equal to zero, then J(u1) = T (u1) and J(u2) = T (u2).

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Relationships between T and J

Since u = u + 0 meets the conditions for the above decomposition, it follows that if J is associated with T , then J(u) = T (u) for the quasi-interior point u as described above. Since this is true for every decomposition of u as described above, J and T still agree on the characteristic functions with respect to the corresponding spectrum from u. If J is associated with T , then the support of J is equal to the support of T .

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Using Linear Functionals to Study Nonlinear Functionals

Thus, the conditions observed for L are also satisﬁed for other operators associated with T . The associated linear operators can now be utilized in studying this class of nonlinear operators.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Theorem 1

Let E be a Banach lattice with an order continuous norm and a quasi-interior point, e. Furthermore, let T : E → R be an orthogonally additive, continuous, monotonic, and subhomogeneous nonlinear operator. Deﬁne Fp = {fα : fα(p) = 1}. If Fp is directed downward, then the following hold true:

1 µ(p) = 0 if and only if ∧T (Fp) = 0 2 µ(p) > 0 if and only if for every f there exists λ > 0 such that T (f ) ≥ λf (p)

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Theorem 2

Let E be a Banach lattice with an order continuous norm and a quasi-interior point, e, and let T : E → R be an orthogonally additive, continuous, monotonic, and subhomogeneous nonlinear operator. Then KT = int(KT ).

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Corollary

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Theorem 3

Let E be a Dedekind complete Banach lattice with a quasi-interior point, e, and let T : E → R be an orthogonally additive, continuous, monotonic, and subhomogeneous nonlinear operator. Then, int(KT ) = ∅ is equivalent to for every f , there exists g, where 0 < g ≤ f and T (g) = 0.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets References

1 H. Schaefer. Banach Lattices and Positive Operators. Springer-Verlag, new York, 1974. 2 J. L. Kelly and I. Namioka. Linear Topological Spaces. Springer-Verlag, New York, 1976. 3 J. E. Stovall and W. A. Feldman. Associating Linear and Nonlinear Operators. Problems and Recent Methods in Operator Theory, Contemporary Mathematics, 687:225-230, 2017. 4 W. Feldman and P. Singh. A characterization of positively decomposable non-linear maps between banach lattices. Positivity, 12(3):495-502, 2007. 5 J. E. Stovall. Nonlinear Functionals on Banach Lattices and Their Support Sets. PhD thesis, The University of Arkansas, 2011.

University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Questions

Thank you!

Jessica E. Stovall Department of Mathematics University of North Alabama

University of North Alabama Jessica E. Stovall