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Banach Lattices in Applications

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Banach Lattices in Applications J. Banasiak Department of Mathematics and Applied Mathematics University of Pretoria, Pretoria, South Africa BANACH LATTICES IN APPLICATIONS Contents 1 Introduction ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5 2 Basics ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 7 1 Defining Order . .7 2 Order and Norm . 13 3 Sublattices, Ideals and Bands . 17 3.1 Complexification . 19 4 Some other stuff . 21 3 Operators on Banach Lattices ::::::::::::::::::::::::::::::::::::::::::::::::::::::: 23 1 Types of operators . 23 1.1 Positive functionals . 27 1.2 Positive operators on complex Banach lattices. 27 2 Perron{Frobenius theorems . 29 2.1 Basics of spectral theory . 29 References ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 33 Index :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 35 1 Introduction The common feature of many models coming from natural and engineering sciences is that the solutions represent the distribution of the particles of various sizes and thus they should be coordinate-wise nonnegative (provided the initial distribution is physically realistic; that is, also non-negative). Though mostly we work with spaces of continuous or integrable functions, or sequences, where the nonnegativity of an element f is understood as f(x) ≥ 0 pointwise or µ-almost everywhere in the former case and fn ≥ 0 for any n in the latter case, ertain results are easier to formulate in a more general setting. Thus we shall briefly present basic concepts of the theory of Banach lattices; that is, Banach spaces with order compatible with the norm. The presentation is based on classical textbooks on Banach lattices, see e.g. [1, 2, 3, 4, 5, 6]. 2 Basics 1 Defining Order In a given vector space X an order can be introduced either geometrically, by defining the so-called positive cone (in other words, by saying what it means to be a positive element of X), or through the axiomatic definition: Definition 2.1. Let X be an arbitrary set. A partial order (or simply, an order) on X is a binary relation, denoted here by ` ≥', which is reflexive, transitive, and antisymmetric, that is, (1) x ≥ x for each x 2 X; (2) x ≥ y and y ≥ x imply x = y for any x; y 2 X; (3) x ≥ y and y ≥ z imply x ≥ z for any x; y; z 2 X. If x; y 2 X, we write x > y if x ≥ y and x 6= y and x ≤ y if y ≥ x: Remark 2.2. In some books, e.g. [5], partial order is introduced through the relation `>' defined as above. More precisely, the relation `>' is defined by: for any x; y; z 2 X (1') x > y excludes x = y; (3') x > y and y > z imply x > z: It is easy to see that both definitions are equivalent. If Y ⊂ X, then x 2 X is called an upper bound (respectively, lower bound) of Y if x ≥ y (respectively, y ≥ x) for any y 2 Y . An element x 2 Y is said to be maximal if there is no Y 3 y 6= x for which y ≥ x.A minimal element is defined analogously. A greatest element (respectively, a least element) of Y is an x 2 Y satisfying x ≥ y (respectively, x ≤ y) for all y 2 Y . We note here that in an ordered space there are generally elements that cannot be compared and hence the distinction between maximal and greatest elements is important. A maximal element is the `largest' amongst all comparable elements in Y , whereas a greatest element is the `largest' amongst all elements in Y . If a greatest (or least) element exists, it must be unique by axiom (2). By the order interval [x; y] we understand the set [x; y] := fz 2 X; x ≤ z ≤ yg: The supremum of a set is its least upper bound and the infimum is the greatest lower bound. For a two-point set fx; yg we write x ^ y or inffx; yg to denote its infimum and x _ y or supfx; yg to denote supremum. In 8 2 Basics the same way we define sup A and inf A for an arbitrary set A. The supremum and infimum of a set need not exist. We observe the following associative law. S Proposition 2.3. Let a subset E of X be represented as E = ξ2Ξ Eξ. Then 1. If yξ = sup Eξ exists for any ξ 2 Ξ and y = supξ2Ξ yξ exists, then y = sup E; 2. If zξ = inf Eξ exists for any ξ 2 Ξ and z = infξ2Ξ yξ exists, then z = inf E. Proof. 1. If x 2 E, then x 2 Eξ for some ξ and thus x ≤ yξ ≤ y. Thus, y is an upper bound for E. If v is an arbitrary upper bound for E, then v is an upper bound for Eξ for each ξ. Hence yξ ≤ v for any ξ 2 Ξ and consequently y ≤ v. Thus v is the least upper bound and hence y = sup E. 2. The proof is analogous. ut Definition 2.4. We say that an ordered space X is a lattice if for any x; y 2 X both x ^ y and x _ y exist. Example 2.5. The set of all subsets of a given set X with partial order defined as inclusion is a lattice with X being its greatest element. From now on, unless stated otherwise, any vector space X is real. Definition 2.6. An ordered vector space is a vector space X equipped with partial order which is compatible with its vector structure in the sense that (4) x ≥ y implies x + z ≥ y + z for all x; y; z 2 X; (5) x ≥ y implies αx ≥ αy for any x; y 2 X and α ≥ 0. If the ordered vector space X is also a lattice, it is called a vector lattice, or a Riesz space. The set X+ = fx 2 X; x ≥ 0g is referred to as the positive cone of X. Example 2.7. A convex cone in a vector space X is a set C characterised by the properties: (i) C + C ⊂ C; (ii) αC ⊂ C for any α ≥ 0; (iii) C \ (−C) = f0g. We show that X+ is a convex cone in X. In fact, from axiom (4) we see that if x; y ≥ 0, then x + y ≥ 0 + y = y ≥ 0, so (i) is satisfied. From (5) we immediately have (ii) and, again using (2), we see that if x ≥ 0 and −x ≥ 0, then 0 = x: Convexity then follows as if x; y 2 C, then αx; βy 2 C for any α; β ≥ 0 (in particular, for α + β = 1) and thus αx + βy 2 C. On the other hand, let C be a convex cone in a vector space X. If we define the relation ` ≥' in X by the formula y ≥ x if and only if y − x 2 C, then X becomes an ordered vector space such that X+ = C. In fact, because x − x = 0 2 C, we have x ≥ x for any x 2 X which gives (1). Next, let x − y 2 C and y − x 2 C. Then by (iii) we obtain axiom (2). Furthermore, if x − y 2 C and y − z 2 C, then we have x − z = (x − y) + (y − z) 2 C by (i). Hence ≥ is a partial order on X. To prove that X is an ordered vector space, we consider x − y 2 C and z 2 X; then (x + z) − (y + z) = x − y 2 C which establishes (4). Finally, if x − y 2 C and α ≥ 0, then αx − αy = α(x − y) 2 C by (ii) so that (5) is satisfied. Moreover, X+ = fx 2 X; x ≥ 0g = fx 2 X; x − 0 2 Cg = C. The cone C of X is called generating if X = C − C; that is, if every vector can be written as a difference of two positive vectors or, equivalently, if for any x 2 X there is y 2 X+ satisfying y ≥ x. 1 Defining Order 9 The Archimedean property of real numbers is that there are no infinitely large or small numbers. In other −1 words, for any r 2 R+, limn!1 nr = 1 or, equivalently, limn!1 n r = 0. Following this, we say that a −1 Riesz space X is Archimedean if infn2Nfn xg = 0 holds for any x 2 X+. In this book we only deal with Archimedean Riesz spaces. The operations of taking supremum or infimum have several useful properties which make them similar to the numerical case. Proposition 2.8. For arbitrary elements x; y; z of a Riesz space, the following identities hold. 1. x + y = supfx; yg + inffx; yg; 2. x + supfy; zg = supfx + y; x + zg and x + inffy; zg = inffx + y; x + zg; 3. supfx; yg = − inf{−x; −yg and inffx; yg = − sup{−x; −yg; 4. α supfx; yg = supfαx; αyg and α inffx; yg = inffαx; αyg for α ≥ 0. 5. For any x; y; z 2 X+ we have inffx + y; zg ≤ inffx; zg + inffy; zg: Proof. 1. From inffx; yg ≤ y we obtain x + inffx; yg ≤ x + y so that x ≤ x + y − inffx; yg and similarly y ≤ x + y − inffx; yg. Hence, supfx; yg ≤ x + y − inffx; yg; that is, x + y ≥ supfx; yg + inffx; yg: On the other hand, because y ≤ supfx; yg, in a similar way we obtain x + y − supfx; yg ≤ x and also x + y − supfx; yg ≤ y so that x + y ≤ supfx; yg + inffx; yg and the identity in property 1 follows. 2. Clearly, x+y ≤ x+supfy; zg and x+z ≤ x+supfy; zg and thus supfx+y; x+zg ≤ x+supfy; zg.
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