Polynomials on Riesz Spaces. John Loane. Supervisor: Dr. Ray Ryan

Transcription

1 Polynomials on Riesz Spaces John Loane Supervisor: Dr. Ray Ryan Department of Mathematics National University of Ireland, Galway December 2007

2 Declaration I declare that this thesis is entirely my own work. It has not been submitted for a degree or any other award at any other institution. i

3 To my parents ii

4 Abstract Mathematicians have been exploring the concept of polynomial and holomorphic mappings in infinite dimensions since the late 1800 s. From the beginning the importance of representing these functions locally by monomial expansions was noted. Recently Matos studied the classes of homogeneous polynomials on a Banach space with unconditional basis that have pointwise unconditionally convergent monomial expansions relative to this basis. More recently still Grecu and Ryan noted that these polynomials coincide with the polynomials that are regular with respect to the Banach lattice structure of the domain. In this thesis we investigate polynomial mappings on Riesz spaces. This is a natural first step towards building up an understanding of polynomials on Banach lattices and thus eventually gaining an insight into holomorphic functions. We begin in Chapter 1 with some definitions. A polynomial is defined to be positive if the corresponding symmetric multilinear mappings are positive. We discuss monotonicity for positive homogeneous polynomials and then give a characterization of positivity of homogeneous polynomials in terms of forward differences. In Chapter 2 we show that, as in the linear case positive multilinear and positive homogeneous polynomial mappings are completely determined by their action on the positive cone of the domain and furthermore additive mappings on the positive cone extend to the whole space. We conclude by proving formulas for the positive part, the negative part and the absolute value of a polynomial mapping. iii

5 In Chapter 3 we prove extension theorems for positive and regular polynomial mappings. We consider the Aron-Berner extension for homogeneous polynomials on Riesz spaces. In Chapter 4 we first review the Fremlin tensor product for Riesz spaces and then consider a symmetric Fremlin tensor product. We discuss symmetric k- morphisms and define the concept of polymorphism. We give several characterizations of k-morphisms in terms of these polymorphisms. Finally we consider orthosymmetric multilinear mappings. iv

6 Acknowledgements There are lots of people I would like to thank for helping me get to this stage of my thesis. Firstly, I would like to thank my supervisor, Dr. Ray Ryan. I always seem to land on my feet, but in having Ray as a supervisor I have been particularly lucky. Thank you for your constant patience and support. Thank you for being so generous with your time and for sharing your ideas and knowledge with me. I have always felt fortunate to be able to work with you because of your teaching ability and because of your concern for people. When I decided to follow Meave to New Zealand for three months you understood the reasons why and I will always be grateful for that. I would like to thank everyone in the Mathematics Department at NUI, Galway for their help and support. It was a pleasure working with Bogdan Grecu and Padraig Kirwan during the time they spent here. Much of my knowledge of Banach Lattice theory was learnt in front of Ray s blackboard with Ray and Bogdan. I have to thank James Ward for the reference on Stirling numbers. I have always enjoyed the friendly and relaxed atmosphere that exists in the Mathematics Department. Many people have come and gone during my time in the mathematics postgraduate room but there has always been a nice friendly atmosphere there. I would like to thank the wider community of mathematicians who work in this area. During my time working on this thesis I have been fortunate to attend quite a few conferences and I have always been pleasantly surprised by how approachable and personable the big shots in this area are. It has been v

7 inspiring to meet people like Sean Dineen, Richard Aron, Chris Boyd, Manolo Maestre and Domingo Garcia. Anthony Wickstead, Gerard Buskes and Anatoli Kusraev have also been extremely helpful with speedy replies to requests for papers. It is nice to feel part of a caring community. I also acknowledge with thanks the financial support I have received, by way of a Basic Research Grant from Enterprise Ireland. My parents and my family have always been there for me when I needed them. It has always been a pleasure to go home and enjoy the warm loving atmosphere at Hillcrest. Thanks Mam and Dad for always encouraging me to follow my dreams. My final words of thanks are for my girlfriend, Meave. Thanks for the wonderful years we spent together in Galway. Thanks for your love and understanding over the last few months. John Loane Galway 2007 vi

8 Contents Abstract iii Acknowledgements v Introduction Preliminaries Positive Polynomials Definitions and Basic Facts Positivity and Monotonicity for Polynomials Forward Differences and Positivity Symmetry and Additivity of Forward Differences Forward Differences for Homogenous Polynomials Kantorovič Theorems and Fremlin Formulae Multilinear and Homogeneous Polynomial Kantorovič Theorems Regular Multilinear and Regular Polynomial Mappings Extension Theorems Hahn-Banach Extensions Extension Theorems on Majorizing Spaces The Aron-Berner Extension Tensor Products of Riesz Spaces Review of the Fremlin Tensor Product Associativity of the Fremlin Tensor Product The Symmetric Fremlin Tensor Product Symmetrization of the Fremlin Tensor Product vii

9 4.5. Properties of the Symmetric Fremlin Tensor Product S and Operators Symmetric k-morphisms Polymorphisms Orthosymmetric Mappings Bibliography viii

10 Introduction Around the turn of the twentieth century Von Koch and Hilbert outlined a theory of holomorphic functions using monomial expansions converging on polydiscs. Hilbert s [25] results were published in The works of Fréchet [12, 13, 14, 15, 16] and Gâteaux [21, 22, 23] provided a new, more wide ranging approach to holomorphic functions. They represented holomorphic functions by power series of homogeneous polynomials. Michal, a student of Fréchet and his own students Clifford, Martin, Highbert and Taylor developed the theory of holomorphic mappings along this line. In 1978 Boland and Dineen [2] brought back the monomial expansion approach and they studied holomorphic functions on nuclear spaces. In the 1980 s and 1990 s Matos and Nachbin [31, 32] explored holomorphic functions and homogeneous polynomials on Banach spaces with unconditional basis, which have unconditionally convergent monomial expansions. They defined natural norms on these classes and considered some of their basic properties. Previously in 1961 Gelbaum and de Lamadrid [19] provided an example to show that even in Hilbert spaces, unconditional convergence may fail. More recently Defant and Kalton [10] have shown the space of k-homogeneous polynomials on an infinite dimensional Banach space with an unconditional basis does not have an unconditional basis, thus proving a conjecture of Dineen [11]. Grecu and Ryan [24] showed that homogeneous polynomials with unconditionally convergent monomial expansions coincide with the homogeneous polynomials that are regular with respect to the Banach lattice structure of the domain. They defined a homogeneous polynomial to be regular if it can be written as the difference of two 1

11 positive homogeneous polynomials. In turn positivity of homogeneous polynomials is defined in terms of their associated symmetric multilinear mappings. This was where we started to get interested in this topic. There is very little in the literature about polynomial mappings on Banach lattices. A natural place to start is to simplify the question by leaving out the norm and to consider polynomials on Riesz spaces. In order to consider this question we first need to understand the existing work in this area. Linear operators on Riesz spaces are now well understood and a number of excellent books have been published on the subject [1, 35, 39]. However it is only in more recent years that bilinear mappings have been investigated. The starting point for this theory is Fremlin s [17] fundamental construction of the tensor product of two Riesz spaces. This tensor product approach means that known results on positive linear maps can be transferred to positive bilinear mappings. However this tensor product is not well understood and this is why the theory of multilinear and polynomial mappings in Riesz spaces has been slow to develop. Schaefer [40], Wittstock [43, 44], Grobler and Labuschagne [20] and Buskes and Van Rooij [7] have all made significant contributions towards simplifying and extending Fremlin s results. The notion of orthosymmetric bilinear mappings naturally arose out of work on almost f-algebras by Buskes and van Rooij [8] in Their importance derives from the fundamental fact that they are symmetric. Important contributions to the theory of orthosymmetric mappings have been made in a sequence of papers by Kusraev [28], Kusraev and Shotaev [29], Kusraev and Tabuev [30], a paper by Boulabiar [4], a paper by Boulabiar and Toumi [6] as well as a paper by Buskes and Boulabiar [5]. In 1991 Sundaresan [42] introduced the class of orthogonally additive homogeneous polynomials on Banach lattices. 2

12 In this thesis we consider multilinear and polynomial mappings on Riesz spaces and extend some of the results for linear operators. This is a natural first step towards building up an understanding of polynomials on Banach lattices and thus eventually gaining an insight into holomorphic functions. In Chapter 1 we introduce the basic theory of positive multilinear mappings and positive polynomials on Riesz spaces. We give the definitions of positive multilinear mappings and positive polynomials as first introduced by Grecu and Ryan [24]. A k-linear mapping A : E 1 E k F is positive if A(x 1,..., x k ) 0 whenever x 1,..., x k lie in the positive cones of E 1,..., E k respectively. A k-homogeneous polynomial P on a Riesz space E is defined to be positive if the (unique) symmetric k-linear mapping that generates P is positive. A polynomial is defined to be positive if each of its homogeneous components are positive. For P a homogeneous polynomial, we say P is monotone on the positive cone if x y 0 implies P (x) P (y). We show that if a homogeneous polynomial is positive then it is monotone on the positive cone. We provide an example to show that the converse is false when the degree of the polynomial is greater than two. It proves very useful to be able to define positivity of a homogeneous polynomial without reference to its associated symmetric multilinear form. Forward differences give us an intrinsic characterization of the positivity of homogeneous polynomials. We recall the Mazur-Orlicz polarisation formula that relates a homogeneous polynomial to the unique symmetric multilinear mapping that generates it. This polarisation formula is much more useful than the usual polarisation formula when working with positive mappings as it keeps all the arguments positive. In Chapter 2 we first formulate and prove multilinear and homogeneous polynomial Kantorovič theorems. These theorems tell us that, as in the linear 3

13 case, positive multilinear and positive homogeneous polynomial mappings are completely determined by their action on the positive cone of the domain and furthermore additive mappings on the positive cone extend to the whole space. Fremlin gave a formula, without proof, for the absolute value of a regular bilinear mapping. Here we prove this formula and give similar formulae for multilinear and polynomial mappings. In Chapter 3 we continue in the vein of the first part of Chapter 2 and extend known results for positive linear operators to results for positive multilinear and positive polynomial mappings. First we recall the most general form of the Hahn-Banach theorem for operators into Riesz spaces and we use this to prove an extension theorem for positive multilinear mappings. We then prove a similar result for positive symmetric multilinear mappings and this leads directly to a result for positive homogeneous polynomials. We continue generalising the result as much as possible, first to positive polynomials (componentwise) and then to regular multilinear and regular polynomial mappings. We then prove an extension theorem for positive multilinear mappings on majorizing vector subspaces. Again we prove this result for positive symmetric multilinear mappings, positive homogeneous polynomial mappings and positive polynomial mappings. Finally we consider the Aron-Berner extension for homogeneous polynomials on Riesz spaces. We show that it works exactly as for the Banach space case. In Chapter 4 we first provide a review of the Fremlin tensor product of two Riesz spaces. We wish to consider the k-fold tensor product so we show that the Fremlin tensor product is associative. This result can be found, without proof, in the literature [41]. We then investigate the symmetric Fremlin tensor product. We consider properties of the symmetrization operator. 4

14 We define the symmetric Fremlin tensor product, E s E to be the Riesz subspace of the Fremlin tensor product, E E generated by the symmetric tensors E s E. The symmetrization operator S : E E E s E is also defined and we show that S is a positive projection of E E onto E s E. We demonstrate the equivalence of applying an operator to an element of k se with that of applying the symmetrized operator to an element of k E. Next we discuss symmetric k-morphisms and show that if A is symmetric then A is a k-morphism if and only if A(x 1,..., x k ) = A(x 1,..., x k 1, x k ) for all x 1,..., x k 1 0 and for all x k. Then we give a characterization of symmetric k-morphisms in terms of their associated homogeneous polynomial mappings. We define a polymorphism to be a homogeneous polynomial mapping P : E F that satisfies P (x) = P ( x ) for all x E. We characterize the polymorphisms on R 2 and use this characterization to give an example to show that if P is a polymorphism, then A its associated symmetric multilinear mapping may not be a k-morphism. We then show that if A is a symmetric k-linear mapping with associated homogeneous polynomial P and A is a k-morphism, then all derivatives of P are polymorphisms. We give an example to show that the converse is false. Finally we show that if A is a symmetric multilinear mapping with associated homogeneous polynomial mapping P then A is a k-morphism if and only if each homogeneous derivative of P is a polymorphism. We conclude with a study of orthosymmetric multilinear mappings. Buskes and van Rooij [8] defined a bilinear mapping B on E E to be orthosymmetric if x y = 0 implies B(x, y) = 0. They established a very surprising fact: every orthosymmetric bilinear mapping is symmetric. We show that this result can be viewed as the dual of a result about tensor products. Sundaresan [42] introduced the class of orthogonally additive homogeneous polynomials. A real-valued function f on a Riesz space E is said to be orthogonally additive 5

15 if f(x + y) = f(x) + f(y) whenever x y. We show that a homogeneous polynomial mapping is orthogonally additive if and only if its associated symmetric multilinear mapping is orthosymmetric. 6

16 Preliminaries Each chapter is divided into sections numbered by two digits, the first one being the number of the chapter. Propositions, theorems, lemmas, corollaries, examples and definitions in each chapter are labeled by two digits, the first indicating the chapter. We will use the symbol to mark the end of a proof. We refer to [1, 35] for details of the results on Riesz spaces that we use. Definition. A real vector space E is said to be an ordered vector space whenever it is equipped with an order relation (i.e. is reflexive, antisymmetric and transitive) that is compatible with the algebraic structure of E in the sense that is satisfies the following two axioms: 1) If x y, then x + z y + z holds for all z E. 2) If x y, then αx αy holds for all α 0. Definition. An element x in an ordered vector space E is called positive whenever x 0 holds. The set of all positive elements of E will be denoted by E + (i.e. E + = {x E : x 0}) and will be referred to as the positive cone of E. By the term operator T : E F between two vector spaces, we shall mean a linear operator, i.e. that T (αx + βy) = αt (x) + βt (y) holds for all x, y E and α, β R. Definition. An operator T : E F between two ordered vector spaces is said to be positive (in symbols T 0) whenever T (x) 0 holds for all x 0. An operator T : E F between two ordered vector spaces is, of course, positive if and only if T (E + ) F + (and equivalently if x y implies T x T y). 7

17 Definition. A Riesz space (or a vector lattice) is an ordered vector space E with the additional property that for each pair of elements x, y E the supremum and infimum of the set {x, y} both exist in E. We shall write x y = sup{x, y} x y = inf{x, y}. Typical examples of Riesz spaces are provided by the function spaces. A function space is a vector space of real-valued functions on a set Ω such that for each pair f, g E the functions and f g(w) = max{f(w), g(w)} f g(w) = min{f(w), g(w)} both belong to E. Every function space with pointwise ordering (i.e. f g holds in E if and only if f(w) g(w) for all w Ω) is a Riesz space. Here are some important examples of function spaces: a) R Ω, all real valued functions on a set Ω. b) C(Ω), all continuous real-valued functions on a topological space Ω. c) C b (Ω), all bounded real-valued continuous functions on a topological space Ω. d) l (Ω), all bounded real-valued functions on a set Ω. e) l p (1 p < ), all real sequences (x 1, x 2,... ) with n=1 x n p <. Definition. A net {x α } in a Riesz space is said to be decreasing (in symbols, x α ) whenever α β implies x α x β. The notation x α x means that x α and inf{x α } = x both hold. Definition. A Riesz space E is called Archimedean whenever n 1 x 0 holds in E for each x E +, n N. 8

18 All classical spaces of functional analysis (notably the function spaces and the L p -spaces) are Archimedean. For this reason we shall assume that all the Riesz spaces we consider are Archimedean. Throughout the thesis we shall denote an Archimedean Riesz space over the reals by a capital letter, usually E, F, G or H. Theorem. If A is a subset of a Riesz space for which sup A exists, then a) inf( A) exists and inf( A) = sup(a). b) The supremum of the set x + A = {x + a : a A} exists and sup(x + A) = x + sup A. c) For each α 0 the supremum of the set αa = {αa : a A} exists and sup(αa) = α sup A. For any vector x in a Riesz space we define x + = x 0, x = ( x) 0, x = x ( x). The element x + is called the positive part, x the negative part and x the absolute value of x. The vectors x +, x and x satisfy the following important properties: Theorem. If x is an element of a Riesz space, then we have 1) x = x + x. 2) x = x + + x. 3) x + x = 0. Moreover the decomposition in 1) is unique in the sense that if x = y z holds with y z = 0, then y = x + and z = x. Definition. For an operator T : E F between two Riesz spaces we say that its modulus T exists whenever T = T ( T ) 9

19 exists (in the sense that T is the supremum of the set {T, T } in the ordered vector space L(E; F )). We refer to Dineen [11] for results on polynomials. Definition. Let k be a positive integer and let E, F be two vector spaces. We say that P : E F is a k-homogeneous polynomial if there exists a k-linear mapping A : E E F such that P is given by the restriction of A to the diagonal: P (x) = A(x,..., x). We denote by P( k E; F ) the space of k-homogeneous polynomials from E to F. We say that the k-linear mapping A generates P and we write P = Â. Notation. We denote the set of k-linear mappings from E 1 E k to F by L(E 1 E k ; F ). If F denotes the scalars then we denote this space by L(E 1,..., E k ). In addition if E j = E for j = 1,..., k then we denote this space by L( k E). In the following by operator we mean a linear map between vector spaces. Definition. An operator T : E F is said to be a regular operator if it can be written as the difference of two positive operators. L r (E; F ) denotes the space of all regular operators from E to F. Definition. A set A is called order bounded if it is bounded both from above and from below. Definition. An operator T : E F that maps order bounded subsets of E onto order bounded subsets of F is called order bounded. We denote the set of order bounded operators from E into F by L b (E; F ). Definition. A Riesz space is called Dedekind complete whenever every nonempty subset that is bounded above has a supremum. 10

20 When F is Dedekind complete, the ordered vector space L r (E; F ) has the structure of a Riesz space. This important result was established first by Riesz [36] for the case F = R, and later Kantorovič [26, 27] extended it to the general setting. Theorem (Riesz-Kantorovič). If F is Dedekind complete, then L r (E; F ) is a Dedekind complete Riesz space. Moreover T + (x) = sup{t y : 0 y x} T (x) = inf{t y : 0 y x} T (x) = sup{t (2y x) : 0 y x} = sup{ T y : y x} for all T L r (E; F ) and all x E +. Definition. A subset D of a Riesz space is said to be upwards directed (in symbols D ) whenever for each pair x, y D there exists some z D with x z and y z. The symbol D x means that D is upwards directed and x = sup D holds. Corollary. Assume that F is Dedekind complete 1) If T, S L r (E; F ) and x E +, then (T S)(x) = sup{t (x y) + S(y) : y [0, x]} = sup { r r } T x i Sx i : r N, x i 0 : x = x i. i=1 i=1 (T S)(x) = inf{t (x y) + S(y) : y [0, x]} = inf { r r } T x i Sx i : r N, x i 0 : x = x i. i=1 i=1 Moreover T z T z for every z E. 2) If A L r (E; F ) is upwards directed then sup(a)(x) = sup{t x : T A} for every x E +. 11

21 The vector space E of all order bounded linear functionals on a Riesz space E is called the order dual of E, i.e, E = L b (E; R). Since R is a Dedekind complete Riesz space, it follows from the Riesz-Kantorovič theorem that E is precisely the space generated by the positive linear functionals. Moreover, E is a Dedekind complete Riesz space. From the above corollary we get formulas for T +, T and T that use partitions. For example for x E + : T + (x) = (T 0)(x) = sup { r (T x i ) + : r N, x i 0 : x = i=1 r } x i. i=1 The sup is taken over all positive partitions of x where a partition of x is a finite sequence of elements of E + whose sum equals x. The partitions of x form a set x. We often denote a partition (a 1,..., a n ) of x by just the letter a. If a = (a 1,..., a N ) and b = (b 1,..., b M ) are partitions of x we call a a refinement of b if the set {1,..., N} can be written as a disjoint union of sets I 1,..., I M in such a way that b m = n I m a n (m = 1,..., M). Any two partitions of x have a common refinement. Thus, in a natural way x is a directed set. If T is a linear map of E into a Riesz space F, if a, b x and a is a refinement of b, then (T an ) + (T b m ) +. Hence for every linear T : E F ( (T an ) + ) a x is an increasing net in F. In particular the supremum is actually a limit and we can rewrite the formula for T + : T + (x) = (T 0)(x) = lim a x 12 { r (T a i ) +}. i=1

22 Writing a supremum in this way as the limit of a net is useful as it allows us to manipulate expressions involving sums more easily. If A is a k-linear form on E k, the associated linear mapping L A : E L( k 1 E) is defined by L A (x)(x 1,..., x k 1 ) = A(x, x 1,..., x k 1 ). Let {E i : i I} be a family of Riesz spaces. The Cartesian product E i is a Riesz space, under the ordering (x i ) (y i ) whenever x i y i holds for all i I. If x = (x i ) and y = (y i ) are elements of E i then x y = {x i y i } and x y = {x i y i }. The direct sum E i is the vector space of all elements {x i } of E i for which x i = 0 holds for all but a finite number of i. With the pointwise algebraic and lattice operations, E i is a Riesz subspace of E i (and hence a Riesz space in its own right). Note that, if, in addition each E i is Dedekind complete, then Ei and E i are likewise Dedekind complete Riesz spaces. Definition. A vector subspace G of a Riesz space E is called a Riesz subspace whenever G is closed under the lattice operations of E (i.e. whenever for each pair x, y G the element x y, taken in E, belongs to G). Definition. A subset A of a Riesz space is called solid whenever x y and y A imply x A. A solid vector subspace of a Riesz space is referred to as an ideal. From the identity x y = 1 (x + y + x y ), it follows immediately that every 2 ideal is a Riesz subspace. A few words about linear operators that preserve the order structure of a Riesz space are called for. 13

23 Definition. A linear mapping u : E F between two Riesz spaces is called a lattice homomorphism if for any x 1, x 2 E u(x 1 x 2 ) = u(x 1 ) u(x 2 ). u(x 1 x 2 ) = u(x 1 ) u(x 2 ). Proposition. Let u : E F be a linear operator between the Riesz spaces E and F. Then the following are equivalent: 1) u is a lattice homomorphism. 2) u(x) = u( x ) for each x E. 3) u(x + ) u(x ) = 0 for each x E. A norm. on a Riesz space is said to be a lattice norm whenever x y implies x y. A Riesz space equipped with a lattice norm is known as a normed Riesz space. If a normed Riesz space is also norm complete, then it is referred to as a Banach lattice. A Banach lattice is called an AM-space if its norm satisfies x y = x y for all x, y E +. This condition is, in particular satisfied if the closed unit ball U of E contains a greatest element e (so that U = [ e, e]); these Banach lattices are called AM-spaces with unit. While c 0 and C 0 (X) (continuous functions on a locally compact space X vanishing at infinity) are AM-spaces without unit, examples of AM-spaces with unit are furnished by c, L (µ) and C(K). The following representation theorem shows that every AM-space with unit is (Riesz and norm) isomorphic to C(K) for a unique compact space K. Notation. Let E be a normed Riesz space. We denote by E dual of E. 14 the topological

24 Theorem (Kakutani-Krein). Every AM-space E with unit e is order and norm isomorphic to some C(K) space. More precisely: If K denotes the weak* compact extreme boundary of {f E + : f(e) = 1}, then evaluation at the points of K defines a Riesz and norm isomorphism of E onto C(K). Definition. Let A be a convex and absorbing subset of a vector space E. Then the Minkowski functional (or the supporting functional or the gauge) p A of A defined by p A (x) = inf{λ > 0 : x λa}, x E. One reason for the importance of the spaces C(K) is their occurrence as principal ideals of arbitrary Banach lattices. In fact, if E is a Banach lattice and if u E +, the ideal of E generated by {u} consists of all x E satisfying x cu for suitable c R + ; that is E u = n=1 n[ u, u]. Since [ u, u] is a convex, circled absorbing subset of E u containing no vector subspace other than {0}, its gauge function p is a norm on E u. Since [ u, u] is complete in E, the space (E u, p) is a Banach space and hence an AM-space with unit u. We summarise Proposition. If E is any Banach lattice and u E +, then under the norm whose closed unit ball is [ u, u], E u is an AM-space with unit u. 15

25 CHAPTER 1 Positive Polynomials In this chapter we introduce the basic theory of positive multilinear mappings and positive polynomials on Riesz spaces. We give the definitions of positive multilinear mappings and positive polynomials as first introduced by Grecu and Ryan [24]. Let E 1,..., E k, F be Riesz spaces. A k-linear mapping A : E 1 E k F is positive if A(x 1,..., x k ) 0 whenever x 1,..., x k lie in the positive cones of E 1,..., E k respectively. P is a positive polynomial if the corresponding symmetric multilinear mappings are positive. We investigate some order theoretic properties of positive homogeneous polynomials. We show that if a homogeneous polynomial is positive then it is monotone on the positive cone. We provide an example to show that the converse is false when the degree of the polynomial is greater than two. It proves very useful to be able to define positivity of a homogeneous polynomial without reference to its associated symmetric multilinear form. Forward differences give us an intrinsic characterization of the positivity of homogeneous polynomials. We recall the Mazur-Orlicz polarisation formula that relates a homogeneous polynomial to the unique symmetric multilinear mapping that generates it. This polarisation formula is much more useful than the usual polarisation formula when working with positive mappings as it keeps all the arguments positive. 16

26 1.1. Definitions and Basic Facts Definition 1.1. Let E 1,..., E k, F be Riesz spaces. A k-linear mapping A : E 1 E k F is positive if A(x 1,..., x k ) 0 whenever x 1,..., x k lie in the positive cones of E 1,..., E k respectively. A partial order is defined on the space of k-linear mappings by A 1 A 2 if A 1 A 2 is positive. L(E 1 E k ; F ) is an ordered vector space. In general it is not a Riesz space. We need the extra condition of regularity to ensure that it is a Riesz space. Definition 1.2. A k-linear mapping is regular if it can be written as the difference of two positive k-linear mappings. Definition 1.3. Let E, F be Riesz spaces. A k-homogeneous polynomial P P( k E; F ) is positive if the associated symmetric k-linear mapping is positive. This is more than saying that the k-homogeneous polynomial P is positive on the positive cone of E. To see this we will briefly consider positive multilinear mappings and positive homogeneous polynomials on R k with the standard ordering. Every k-linear mapping A : R k F can be expanded relative to the standard basis as follows: (1) A(x 1,..., x k ) = A(e j1,..., e jk )x 1j1... x kjk 1 j 1,...,j k k where A(e j1,..., e jk ) F are the coefficients of the expansion. These coefficients determine the positivity of the k-linear mapping. Lemma 1.4. A k-linear mapping A : R k coefficients in its expansion are positive. F is positive if and only if all Proof: Each A L( k R k ; F ) has an expansion as above in (1). Since the basis vectors lie in the positive cone of R k, it follows that A is positive if and only if all coefficients in its expansion are positive. 17

27 Proceeding in the same way, we see that every k-homogeneous polynomial P : R k F has a monomial expansion relative to the standard basis. This expansion is: P (x) = A(e j1,..., e jk )x j1... x jk. j 1,...,j k R k Now looking at this formula and Lemma 1.4 we get the following: Lemma 1.5. If P : R k F is a k-homogeneous polynomial map then P is positive if and only if all coefficients in its expansion are positive. We note that every positive k-homogeneous polynomial is positive on the positive cone: if P 0 then P (x) 0 for every x E +. Now we give an example to show that, for a k-homogeneous polynomial, P, on a Riesz space E, positivity is more than saying that P is positive on the positive cone of E. Example 1.6. On E = R 2 with the standard ordering consider the 2-homogeneous polynomial P (x) = (x 1 x 2 ) 2. Clearly P is positive everywhere. However from Lemma 1.5 we see that P is not a positive 2-homogeneous polynomial. A mapping P : E F is said to be a polynomial if there exists k N and j-homogeneous polynomials P j, 0 j k such that P = P 0 + P P k. If P k 0 then the degree of P is defined to be k. Definition 1.7. A polynomial P = P P k of degree k is positive if each of its homogeneous components P j is positive, 0 j k. 18

28 1.2. Positivity and Monotonicity for Polynomials Let E, F be Riesz spaces. Recall that for a linear operator T : E F, T is positive if T maps positive elements of E to positive elements of F. T is said to be monotone if x y implies T x T y. Positivity and monotonicity are easily seen to be equivalent for linear mappings. Monotonicity for homogeneous polynomials only makes sense on the positive cone. Consider, for example the 2-homogeneous polynomial P (x) = x 2 on R. Outside the positive cone we have points where P (x) P (y) even though y x. Thus we say that a polynomial P is monotone on the positive cone if x y 0 implies P (x) P (y). For homogeneous polynomials positivity and monotonicity are not equivalent. We will show this below but first we need some notation. Notation. Let E be a Riesz space and let P : E R be a k-homogeneous polynomial. We denote by P (x) the directional derivative of P at x in the v direction v. Thus for x, v R k we get P v (x) = k j=1 P x j (x)v j. In general we are working on Riesz spaces so we are taking the Gâteaux derivative. Thus the directional derivative of a k-homogeneous polynomial P at the point x in the direction v is P P (x + tv) P (x) (x) = lim. v t 0 + t 19

29 To see that this limit exists we expand P (x + tv): k P (x + tv) = k Ax j (tv) k j j=0 j k 1 = P (x) + k Ax j (tv) k j. j j=0 Hence P (x) = lim v t 0 + = kax k 1 v. P (x) + k 1 j=0 k Ax j (tv) k j P (x) j t For k-homogeneous polynomials we can characterize monotonicity on the positive cone in terms of these directional derivatives. Proposition 1.8. Let P be a k-homogeneous polynomial on a Riesz space E with associated symmetric k-linear form A. Then the following are equivalent: a) P is monotone on the positive cone. b) Each of the directional derivatives of P at every positive point and in every positive direction is positive. c) Ax k 1 y 0 for all x, y E +. Proof: a) implies b): Suppose P is monotone on the positive cone. Thus x y 0 implies P (x) P (y). Now consider the directional derivative at any positive point x in any positive direction v: P P (x + tv) P (x) (x) = lim. v t 0 + t Now P is monotone on the positive cone and x, v, t 0. Thus P (x + tv) P (x) 0. Hence P (x) 0 for all v 0, x 0. v 20

30 b) is equivalent to c): This follows immediately from b) implies a): P v (x) = kaxk 1 v. Now suppose each directional derivative at every positive point in every positive direction is non-negative. P P (x + tv) P (x) (x) = lim v t 0 + t 0 for all x, v, t 0. Now we want to show that if 0 x z then P (x) P (z). Consider the directional derivative of P at x in the positive direction y = z x. P P (x + t(z x)) P (x) (x) = lim y t 0 + t Now if we can show that the function 0. g(t) : t P (x + t(z x)) t 0 is increasing for all t 0 it will follow that P (x) P (z). We have Thus we get g P (x + (t + h)(z x)) P (x + t(z x)) (t) = lim. h 0 + h g P (x + (t + h)y) P (x + ty) (t) = lim h 0 + h P ((x + ty) + hy) P (x + ty) = lim. h 0 + h Letting x = x + ty 0, we have g (t) = lim h 0 + P (x + hy) P (x ) h = P y (x ) 0. Therefore g (t) 0 for all t > 0. Similarly g +(0) 0 where g +(0) is the right derivative of g at t = 0. We wish to stay in the positive cone of E so we have 21

31 to be careful to make this distinction at t = 0. Hence g (t) 0 for all t 0. Thus g is an increasing function for t 0. In particular g(0) = P (x) g(1) = P (z). Therefore P is monotone on the positive cone. Corollary 1.9. Let P be a homogeneous polynomial on R k. Then P is monotone on the positive cone if and only if all of its partial derivatives are nonnegative at every positive point. Proof: Note that P v (x) = k i=1 P x i (x)v i 0 for all x, v 0 P x i (x) 0 for all x 0. The following Proposition shows that every positive homogeneous polynomial is monotone on the positive cone. Proposition Let P be a k-homogeneous polynomial on a Riesz space E. If P is positive then P is monotone on the positive cone of E. Proof: Suppose P is positive. Then its associated positive symmetric k-linear mapping A is also positive. Hence Ax k 1 y 0 for all x, y 0. Now from Proposition 1.8 it follows that P is monotone on the positive cone. In general the converse of Proposition 1.10 is not true. However it is valid for homogeneous polynomials of degree 2. Proposition For 2-homogeneous polynomials positivity and monotonicity on the positive cone are equivalent. 22

32 Proof: From Proposition 1.10 we know that positivity implies monotonicity on the positive cone. So now suppose that we have a 2-homogeneous polynomial, P, which is monotone on the positive cone. Let A be the 2-linear symmetric generator of P. From Proposition 1.8 it follows that A(x, y) 0 for all x, y 0. Hence P is positive. Now we would like to find an example of a polynomial which is monotone on the positive cone but not positive. 2-homogeneous polynomials are ruled out by Proposition So the first place to look is 3-homogeneous polynomials on R 2. Lemma For 3-homogeneous polynomials on R 2 positivity and monotonicity on the positive cone are equivalent. Proof: The most general 3-homogeneous polynomial on R 2 is of the form P (x) = ax bx cx 2 1x 2 + dx 1 x 2 2. Suppose that P is monotone on the positive cone. This means, by Corollary 1.9 that each of its partial derivatives is non-negative. Hence: P x 1 (x) = 3ax cx 1 x 2 + dx for all x 1, x 2 0, and P x 2 (x) = 3bx cx dx 1 x 2 0 for all x 1, x 2 0. Taking x 1 = 0, x 2 > 0 we get d, b 0. Similarly taking x 2 = 0, x 1 > 0 we get a, c 0. Hence all the coefficients in the polynomial are positive and by Lemma 1.5 P is positive. Hence P monotone on the positive cone implies that P is positive. Now we have two options in our search for a homogeneous polynomial which is monotone on the positive cone but not positive. We can increase the degree of the polynomial and look at 4-homogeneous polynomials on R 2 or we can increase the dimension of the space and look at 3-homogeneous polynomials on 23

33 R 3. Both of these approaches give us examples of homogeneous polynomials which are monotone on the positive cone but not positive. Example Consider the 4-homogeneous polynomial on R 2 given by P (x) = x x x 3 1x 2 + x 3 2x 1 x 2 1x 2 2. By Lemma 1.5 P is not positive. Now we will show that P is monotone on the positive cone. If each of the partial derivatives of P at every positive point is positive then P is monotone by Corollary 1.9. Thus P (x) = 4x x 2 x 1x 2 + x 3 2 2x 1 x should be positive for all x 1, x 2 0. P (x) = 4x x 2 x 2x 1 + x 3 1 2x 2 x should be positive for all x 1, x 2 0. To see that this is true for P (x) consider the options. If x 1 = 0, x 2 = 0 then x 1 P (x) = 0. If one of x 1, x 2 is 0 then P (x) 0. Now if x 1 = x 2 > 0 x 1 x 1 P x 1 (x) = 4x x x 3 1 2x 3 1 = 6x If x 2 > x 1 > 0 then let h = x 2 x 1 > 0. Then x 2 = x 1 + h. Expanding we get 4x x 2 1(x 1 + h) + (x 1 + h) 3 2x 1 (x 1 + h) 2 = 6x x 2 1h + x 1 h 2 + h 3 0. Similarly if x 1 > x 2 > 0 then P (x) 0. The same approach gives P (x) x 1 x 2 0. So P is monotone on the positive cone but not positive. We now present the second example of a homogeneous polynomial which is monotone on the positive cone but not positive. In this case we increase the dimension of the space and consider 3-homogeneous polynomials on R 3. 24

34 Example Consider the 3-homogeneous polynomial on R 3 defined by P (x) = x x 2 1x 2 + 3x 2 1x 3 + 3x 2 2x 1 + 3x 2 2x 3 + 3x 2 3x 1 + 3x 2 3x 2 6x 1 x 2 x 3 + x x 3 3. By Lemma 1.5 P is not positive. If each of the partial derivatives of P at every positive point is positive then P is monotone by Corollary 1.9. Note P x 1 (x) = 3x x 1 x 2 + 6x 1 x 3 + 3x x 2 3 6x 2 x 3 = 3x x 1 x 2 + 6x 1 x 3 + 3(x 2 x 3 ) 2 0 for all x 1, x 2, x 3 0. Similarly we find that P (x) 0 and P (x) 0. Hence P is monotone on x 2 x 3 the positive cone but not positive Forward Differences and Positivity The definition of positivity of a homogeneous polynomial is given in terms of its associated symmetric multilinear form. This makes it inconvenient to work with. We would like an intrinsic characterization of positivity. Finite difference calculus leads us to such a characterization. We begin by recalling the basic definitions as given originally by Boole [3]. Definition Let f be a real function defined on a vector space E. For each positive integer k and h 1,..., h k E the k-th forward difference, k f(x; h 1,..., h k ) is defined recursively as follows: 1 f(x; h 1 ) = f(x + h 1 ) f(x), and k f(x; h 1,..., h k ) = 1 ( k 1 f( ; h 1,..., h k 1 ))(x; h k ). We also denote the first forward difference operator in the direction h by 1 h. Similarly the kth forward difference in directions h 1,..., h k is denoted k h 1,...,h k. In other words k h 1,...,h k f(x) = k f(x; h 1,..., h k ). 25

35 For example the second forward difference is given by: 2 f(x; h 1, h 2 ) = 1 ( 1 f( ; h 1 ))(x; h 2 ) = 1 (f(x + h 1 ) f(x))(x; h 2 ) = f(x + h 1 + h 2 ) f(x + h 2 ) f(x + h 1 ) + f(x) and the third forward difference is: 3 f(x; h 1, h 2, h 3 ) = f(x + h 1 + h 2 + h 3 ) f(x + h 1 + h 2 ) f(x + h 1 + h 3 ) f(x + h 2 + h 3 ) + f(x + h 1 ) + f(x + h 2 ) + f(x + h 3 ) f(x). A clear pattern is emerging. forward difference: k f(x; h 1,..., h k ) = In fact there is a general formula for the kth k ( 1) k δ i f(x + δ 1 h δ k h k ). δ i =0,1 i= Symmetry and Additivity of Forward Differences Let f be any real function on a vector space E. Fix x E and let h 1,..., h j E. Consider the shift operators S j defined by S j f(x) = f(x + h j ). These operators clearly commute: S i S j = S j S i. Now 1 f(x; h j ) = f(x + h j ) f(x) = (S j I)f(x). Thus 1 h j = (S j I). Similarly k k h 1,...,h k = (S j I). j=1 26

36 Since the operators S 1,..., S k commute, it follows that k f(x; h 1,..., h k ) is a symmetric function of h 1,..., h k. Forward differences also have a useful additivity property. We will demonstrate this below. First note that if f is a function on a Riesz space E such that 1 f(x; h) = 0 for all x, h E +, then f(x + h) f(x) = 0 for all x, h E +. Thus f(h) = f(0) for all h E +. Hence f is constant on the positive cone. We will use this observation in proving the following: Lemma Let E be a Riesz space and let f be a real function on E such that 2 f(x; h 1, h 2 ) = 0 for every x, h 1, h 2 E +. Then 1 f(x; h) is additive in h E + for every x E +. Proof: Suppose 2 f(x; h 1, h 2 ) = 0 for all x, h 1, h 2 E +. Thus 1 ( 1 f(x; h 1 ))(x; h 2 ) = 0. This implies 1 f(x; h 1 ) is a constant function of x for every h 1 E +. Thus f(x + h 1 ) f(x) = C(h 1 ). On choosing x = 0 we see that f(h 1 ) f(0) = C(h 1 ). So f(x + h 1 ) = f(x) + f(h 1 ) f(0). Now for h 1, k 1 E + we have 1 f(x; h 1 + k 1 ) = f(x + h 1 + k 1 ) f(x) = f(h 1 + k 1 ) f(0) = f(h 1 ) + f(k 1 ) f(0) f(0) = 1 f(x; h 1 ) + 1 f(x; k 1 ). Therefore 1 f(x; h) is additive in h. 27

37 We now prove a general version of the above Lemma. We say k f(x; h 1,..., h k ) is additive in h 1,..., h k for every x E + if it is additive in each of the variables h 1,..., h k for every fixed x E +. Proposition Let E be a Riesz space and let f be a real function on E such that k+1 f(x; h 1,..., h k+1 ) = 0 for all x, h 1,..., h k+1 E +. Then k f(x; h 1,..., h k ) is additive in h 1,..., h k for every x E +. Proof: First notice the following: k+1 f(x; h 1,..., h k+1 ) = 1 ( k f(x; h 1,..., h k ))(x; h k+1 ) = 2 ( k 1 f(x; h 1,..., h k 1 ))(x; h k, h k+1 ). Thus from Lemma ( k 1 f(x; h 1,..., h k 1 ))(x; h k ) is additive in h k. Hence k f(x; h 1,..., h k ) is additive in h k. By symmetry of k it follows that k f(x; h 1,..., h k ) is additive in each variable Forward Differences for Homogenous Polynomials When P is a k-homogeneous polynomial with symmetric generator A, we have a general formula for forward differences: Proposition Let P be a k-homogeneous polynomial on a vector space E whose associated symmetric k-linear form is A. Then for every m N and every x, h 1,..., h m E, m P (x; h 1,..., h m ) = k 1 j 1 1 j 1 =0 j 2 =0 j m 1 1 j m=0 j m 1... j 1 k Ax jm h k j h j m 1 j m j 1 j m Proof: The proof is by induction on m. For m = 1, we have k 1 1 P (x; h 1 ) = k Ax j h k j 1. j j=0 28 j 2 m.

38 Assume our formula is true for forward differences of order m. We must show that it is also true for forward differences of order m + 1. Note m+1 P (x; h 1,..., h m+1 ) = 1 [ m P (x; h 1,..., h m )](x; h m+1 ) k 1 j 1 1 j m 1 1 = 1[ j m 1... k Ax jm h k j h j m 1 j m ] (x; hm+1 ) j 1 = k 1 j 1 =0 j 1 =0 j 2 =0 j m 1 j m+1 =0 j m=0 j m+1 j m j m j m 1... k Ax j m+1 h k j h jm j m+1 j 1 j m We are interested in particular instances of the above proposition. m m+1. Corollary a) When we take m = k 1 in the Proposition we get k 1 P (x; h 1,..., h k 1 ) = k! 2 [Ah2 1h 2... h k Ah 1 h 2... h 2 k 1] + k!a(x, h 1,..., h k 1 ). b) When we take m = k in the Proposition we get k P (x; h 1,..., h k ) = k!a(h 1,..., h k ). c) m P (x; h 1,..., h m ) = 0 for every m > k. Proof: a) When m = k 1 k 1 P (x; h 1,..., h k 1 ) = k 1 j 1 1 j 1 =0 j 2 =0 j k 2 1 j k 1 =0 j k 2... j 1 j k 1 j 2 k j 1 Ax j k 1 h k j h j k 2 j k 1 k 1. This is non-zero only when k 1 j 1 j 2 j k 1 0. Since each j i ranges between 0 and j i 1 1 and we have k 1 terms the inner inequalities are strict. Note k 1 j 1 > j 2 > > j k

39 So one option is j 1 = k 1, j 2 = k 2, j i = k i, j k 2 = 2, j k 1 = 1 giving k!a(x, h 1,..., h k 1 ). All other terms are of the form j 1 = k 1, j 2 = k 2,..., j i 1 = k (i 1), j i = k (i + 1), j i+1 = k (i + 2),..., j k 1 = 0 where 1 i k 1 giving k! 2 Ah 1h 2... h 2 i... h k 1 where 1 i k 1. Hence the result. b) When m = k, k P (x; h 1,..., h k ) = k 1 j 1 1 j 1 =0 j 2 =0 j k 1 1 j k =0 The right hand side is non-zero only if j k 1... j 1 k Ax j k h k j h j k 1 j k j 1 j k j 2 k 1 j 1 > j 2 > > j k 0. k. Thus our only option is to take j 1 = k 1, j 2 = k 2,..., j k = 0. Thus the only term is k!a(h 1,..., h k ). c) When m > k, k P (x; h 1,..., h k ) = k!a(h 1,..., h k ). This is independent of x. Hence m P (x; h 1,..., h m ) = 0 for every m > k. We can also prove Corollary 1.19 a) directly by a counting argument. We will now outline this approach as it uses a different technique. Note k 1 P (x; h 1,..., h k 1 ) =P (x + h h k 1 ) P (x + h h k 1 )... + P (x + h h k 1 ) +... = ( 1) k 1 δ i P (x + δ 1 h δ k 1 h k 1 ). δ i =0,1 30

40 Here we are grouping terms according to the number of h i s dropped, for example 1 in the first group. We expand each of these terms using the multinomial formula. Note P (x + h h k 1 ) = j 1 + +j k =k ( ) k! j 1!... j k! A(x)j 1 (h 1 ) j 2... (h k 1 ) j k. We will now use a counting argument to prove Corollary 1.19 a). We need to expand out each of the polynomials and count the number of times each term occurs. Everything cancels out except the claimed terms. To see how this works consider homogeneous terms first Ax k, Ah k 1,..., Ah k k 1. We get one copy of Ax k from the first group of terms, k 1 copies from the second group and so on. Thus we get 1 (k 1) + k 1 k k 1 k 3 k 4 0 = (1 1) k 1 = 0. Thus the coefficient of Ax k is 0. Similarly all homogeneous terms have coefficient 0. In fact we can find a general formula for the coefficients of the expanded terms. Ax k p h p 1 l 1 h p 2 l 2... h p j l j occurs with coefficient [ k! 1 (k (j + 1)) + (k p)!p 1!... p j! ] k (j + 1)... k (j + 3) where 1 l 1 < l 2 < < l j k 1, p p j = p = (1 1) k (j+1) = 0 unless k = j + 1. So the constraints are j = k 1, k-homogeneous polynomial, p p j = p. The only non-zero terms are Axh 1... h k 1, Ah h k 1,..., Ah 1... h 2 k 1. Hence our formula. 31

41 We now derive the Mazur-Orlicz polarisation formula. This polarisation formula is much more useful than the regular polarisation formula when working with positive mappings as it keeps all the arguments positive. Proposition 1.20 (Mazur-Orlicz). Let P be a k-homogeneous polynomial on a vector space E and let A be the associated symmetric k-linear mapping. Then for x, h 1,..., h k E we have the following polarisation formula: A(h 1,..., h k ) = 1 ( 1) k δ i P (x + δ 1 h δ k h k ). k! δ i =0,1 Proof: Using Corollary 1.19 b) we see that A(h 1,..., h k ) = 1 k! k P (x; h 1,..., h k ). Now using the general formula for k A(h 1,..., h k ) = 1 ( 1) k δ i P (x + δ 1 h δ k h k ). k! δ i =0,1 Compare the Mazur-Orlicz polarisation formula with the usual polarisation formula: A(h 1,..., h k ) = 1 2 k k! ε i =±1 ε 1... ε k P (ε 1 x ε k x k ). The Mazur-Orlicz polarisation formula leads to a polarisation inequality on normed spaces that does not give sharp bounds: A 2k k k P k! A judicious choice of x gives the modern version, with sharp bounds: and this is attained on l 1. A kk k! P When working with positive polynomials on a Riesz space the Mazur-Orlicz polarisation formula is much more useful. keeps the arguments of P positive, keeping us in the positive cone where we 32 It

42 know P is positive. The usual polarisation formula is inconvenient for working with positive mappings on Riesz spaces as when x 1,..., x k 0, the terms ε 1 x ε k x k will often lie outside the positive cone. Taking x = 0 the Mazur-Orlicz formula is much more useful. Now we give a characterization of positivity of homogeneous polynomials in terms of forward differences. The following theorem gives the full picture of how we can characterize positivity of homogeneous polynomials in terms of forward differences. Theorem Let E be a Riesz space and let P be a k-homogeneous polynomial on E. Then following are equivalent: a) P is positive. b) k P (x; h 1,..., h k ) 0 for all x, h j E +. c) k 1 P (x; h 1,..., h k 1 ) 0 for all x, h j E +. d) m P (x; h 1,..., h m ) 0 for all m and for all x, h j E +. Proof: a) is equivalent to b): k P (x; h 1,..., h k ) = k!a(h 1,..., h k ) from Corollary 1.19 b). This implies k P (x; h 1,..., h k ) is positive if and only if A(h 1,..., h k ) is positive for all h i 0. This is equivalent to P being positive. c) implies a): From Corollary 1.19 a) we see that k 1 P (x; h 1,..., h k 1 ) = k! 2 [Ah2 1h 2... h k Ah 1 h 2... h 2 k 1] + k!a(x, h 1,..., h k 1 ). So k 1 P (x; h 1,..., h k 1 ) is positive if and only if for every x, h i 0 k! 2 [Ah2 1h 2... h k Ah 1 h 2... h 2 k 1] + k!a(x, h 1,..., h k 1 ) 0. 33