State on a partially ordered generalized 2-normed spaces

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1 South Asian Journal of Mathematics 2011, Vol. 1 ( 2): ISSN RESEARCH ARTICLE State on a partially ordered generalized 2-normed spaces P.V. REEJA 1, K.T. RAVINDRAN 1 1 Department of Mathematics, Payyanur College, Payyanur, India pvreeja@gmail.com; drravindran@gmail.com Received: ; Accepted: *Corresponding author The research is supported by the Teacher fellowship awarded to the first author by the Faculty Improvement Programme of University Grants Commission of India during XI plan period. Abstract In this paper, we study a bilinear mapping on a sub cone of a partially ordered generalized 2-normed space, its extension to the positive cone of the space, then decomposition of a bilinear functional on the space using the same. Key Words generalized 2-normed space, bounded bilinear mapping, ordered vector space, cone MSC A15, 46A40 1 Introduction Ordered vector spaces were developed at the beginning of twentieth century in parallel with functional analysis and operator theory. L.V. Kantorovich, Krein brothers, H. Nakanao, H.H. Schafer and Zannen were among them who started its systematic study. It is an indispensable tool for studying a variety of problems in engineering and economics. The concept of generalized 2-normed space was introduced by Zofia Lewandowska as a generalization of the 2-normed space given by Gahler. The properties of 2-normed space were suggested by the area function for a triangle determined by a triple in Euclidean space. In this paper we consider a generalized 2-normed space with a partial order, that is a partially ordered generalized 2-normed space. 2 Preliminaries First we recall some basic definitions and facts. Definition 2.1([3]). Let X be a real linear space of dimension greater than 1. Suppose, is a real valued function on X X satisfying the following conditions: 1. x, y = 0 if and only if x and y are linearly dependent; 2. x, y = y, x ; Citation: P.V. Reeja and K.T. Ravindran, State on a partially ordered generalized 2-Normed spaces, South Asian J Math, 2011, 1(2),

2 South Asian J. Math. Vol. 1 No αx, y = α x, y ; 4. x + y, z x, z + y, z. Then, is called a 2-norm on X and the pair (X,, ) is called a 2-normed space. Some of the basic properties of 2-norms, are that they are non-negative and x, y + αx = x, y, x, y X and α R. Definition 2.2([3]). Let X and Y be a real linear spaces of dimension greater than 1. Let X and Y be real linear spaces. Denote by D, a non-empty subset of X Y such that for every x X, y Y, the sets D x = {y Y : (x, y) D} and D y = {x X : (x, y) D} are linear subspaces of the space Y and X respectively. A function, will be called a generalized 2-norm on D if it satisfies the following conditions: 1. αx, y = α x, y for any real number α and all (x, y) D; 2. x, y + z x, y + x, z for x X, y, z Y such that (x, y), (x, z) D; 3. x + y, z x, z + y, z for x, y X, z Y such that (x, z), (y, z) D. The set D is called a 2-normed set. In particular, if D = X Y the function, will be called a generalized 2-norm on X Y and the pair (X Y,, ) generalized 2-normed space. Moreover, if X = Y then the generalized 2-normed space will be denoted by (X,, ). Example([4]). Let X be a real linear space having two semi norms 1 and 2 and then (X,, ) is a generalized 2-normed Space with the 2-norm defined by x, y = x y. Definition 2.3([4]). Let D X X be a 2-normed set, Y a normed space. A map f is called a bilinear if it satisfies the conditions 1. f(a + c, b + d) = f(a, b) + f(a, d) + f(c, b) + f(c, d) for a, b, c, d X such that a, c D b D d ; 2. f(αa, βb) = αβf(a, b) for α, β R and (a, b) D. Definition 2.4([4]). A bilinear map f : D Y is said to be bounded if there exists a non negative real number M such that f(a, b) M ab for all a, b D and the norm of a bilinear map f is defined by f =inf {M > 0 : f(a, b) M a, b, (a, b) D}. Theorem 2.5([4]). Let D X X be a 2-normed set, Y a Banach space and L 2 (D, Y ) be the set of all bounded bilinear operators from D into Y. Then L 2 (D, Y ) is a Banach space with respect to the norm f =inf {k > 0; f(a, b) k a, b, (a, b) D}. Theorem 2.6([5]). Let (X,, ) be a generalized 2-normed space and M be a linear subspace of (X,, ). If F 0 is a real bounded bilinear functional on M, then there exists a real bounded bilinear extension F on X such that F(a, b) = F 0 (a, b) for all a, b M, and F = F 0. Definition 2.7([1]). Let X be a vector space. A cone of X is a non empty subset of X for which αa A, whenever α R +. Definition 2.8([1]). The cone A is called sharp if A A = 0. Definition 2.9([1]). The cone A is convex if and only if A + A A. 45

3 P.V. Reeja, et al: State on a partially ordered generalized 2-normed spaces Definition 2.10([2]). Lexicographic Order: Given two posets (A 1, R 1 ) and (A 2, R 2 ) we construct an induced partial order R on A 1 A 2 as x 1, y 1 R x 2, y 2 if and only if x 1 R 1 x 2 or if x 1 = x 2 and y 1 R 2 y 2. Definition 2.11([1]). An ordered vector space is a vector space X with an order relation such that given x, y, z X and α R +, x+z y+z and αx αy whenever x y. Defining X + = {x X/x 0}, we call the elements of X + positive and elements of X negative. Theorem 2.12([1]). If X is an ordered vector space then X + is a sharp convex cone in X. Definition 2.13([2]). Let X is an ordered vector space and K a cone in X. Then u K is called an order unit for K if for each x X there is some λ > 0 such that x λu. An ordered vector space with an order unit u is denoted by (X, u). Theorem 2.14([1]). Let X be a vector space and C a sharp convex cone in X. Then there is a unique order relation on X, with respect to which X is an ordered vector space with E + = C. Theorem 2.15([2]). If X is an ordered vector space then X X is an ordered vector space with Lexicographic order (or with product order). 3 Main results Definition 3.1. Let (X, u) be a partially ordered vector space and M a sub cone of the positive cone of X. Then a mapping f on M is called a pre-state if 1. f is real bilinear; 2. f is positive, that is f(x, y) 0; 3. f(u, u) = 1. Definition 3.2. Let (X, u) be a partially ordered vector space. Let M be a sub cone of the positive cone of X containing u. Then a mapping s on M is called a state if 1. s is real bilinear; 2. s is monotone. That is (x 1, y 1 ) (x 2, y 2 ) s(x 1, y 1 ) s(x 2, y 2 ); 3. s(u, u) = 1. Remark 3.3. Every State is a Pre-state. Remark 3.4. s(x, y) x, y. Proof. There exists λ 1, λ 2 such that 0 x λ 1 u and 0 y λ 2 u. Then (0, 0) (x, y) (λ 1 u, λ 2 u) 0 s(x, y) λ 1 λ 2 s(u, u) 0 s(x, y) λ 1 λ 2. Hence s(x, y) Inf{λ 1 λ 2 ; 0 x λ 1 u; 0 y λ 2 u}. So 0 s(x, y) x, y. Remark 3.5. Every State s is bounded and s = Sup{s(x, y); x, y 1}. Definition 3.6(Supcone). A sub cone Q of X + is called Supcone if the order in every super cone of the form Q + R + a 0 is such that q 1 + r 1 a 0 q 2 + r 2 a 0 q 1 q 2 and r 1 r 2. 46

4 South Asian J. Math. Vol. 1 No. 2 Theorem 3.7. Let (X, u) be a partially ordered generalized 2-normed space and Q a sub cone of X + containing u. If Qis a Supcone then any state s 0 on Q can be extended to the positive cone of X. Proof. If Q = X + or s 0 = 1, then s = s 0. Without loss of generality assume s 0 = 1. Let E is the set of all pairs (Q i, s i ), where Q i is a sub cone of X + containing Q and s i is a state on Q i. Define a partial order such that (Q i, s i ) (Q j, s j ) if and only if s j is an extension of s i and Q i Q j. By using Zorn s lemma, there exists a maximal element (Q, s). Suppose Q X +.Then there exists (a 0, b 0 ) X + such that (a 0, b 0 ) / Q. Consider Q = Q + R + (a 0, b 0 ). Here R + (a 0, b 0 ) = (ta 0, t b 0 ), where t, t R +. Define s : Q R by s (a + ta 0, b + t b 0 ) = s(a, b) + tt γ where γ is such that s = 1. Then s(a, b) + tt γ a + ta 0, b + t b 0, a, b Q s(a/t, b/t ) + γ a/t + a 0, b/t + b 0, a/t, b/t Q s(a, b) + γ a + a 0, b + b 0, a, b Q. Such a γ exists by Remark 3.4. Now we will prove that s is a state. Clearly s is real bilinear and s (u, u) = 1. Let (a 1 + ta 0, b 1 + t b 0 ) (a 2 + sa 0, b 2 + s b 0 ) a 1 + ta 0 a 2 + sa 0 ; b 1 + t b 0 b 2 + s b 0 a 1 a 2 ; t s; b 1 b 2 ; t s s(a 1, b 1 ) s(a 2, b 2 ) s (a 1 + ta 0, b 1 + t b 0 ) s (a 2 + sa 0, b 2 + s b 0 ) s is monotone, which is a contradiction. Hence Q = X +. Theorem 3.8. Let X be a partially ordered generalized 2-normed space and Q a sub cone of X +. If Qis a Supcone then, any pre-state s 0 on Q can be extended to positive cone of X. Proof. Proof follows as above. Theorem 3.9. Let X be a partially ordered generalized 2-normed space and Q a sub cone of X +. If Q is a Supcone then, any positive bilinear functional s 0 on Q can be extended to positive cone of X. Proof. Similar to the proof of theorem 3.7. Corollary Let (X, u) be a partially ordered generalized 2-normed space. Y a proper subspace of V containing u, whose positive cone Y + is a Supcone. Then any real bilinear mapping on Y, which is a state on Y + can be extended to the real bilinear mapping on X, which is a state on X +. Corollary Let (X, u) be a partially ordered generalized 2-normed space, Y a proper subspace of X containing u, whose positive cone Y + is a Supcone. Then any real bilinear map on Y, can be extended to a real bilinear mapping on X. Corollary Let (X, u) be a partially ordered generalized 2-normed space. Y a proper subspace of X containing u, whose positive cone Y + is a Supcone. Then any real positive bilinear map on Y, which is a pre-state on Y + can be extended to the real bilinear mapping on X, which is a pre-state on X +. Theorem Let (X, u) be a partially ordered generalized 2-normed space. If (a 0, b 0 ) X X is such that (a 0, b 0 ) (0, 0)and (a 0, b 0 ) = α(u, u). If R + u is a Supcone then there exists a state s on X + such that s(a 0, b 0 ) = (a 0, b 0 ). 47

5 P.V. Reeja, et al: State on a partially ordered generalized 2-normed spaces Proof. Define s : M = R + (u, u) R by s(α 1 u, β 1 u) = α 1 β 1. Then s is a state on M. Since M is a Supcone s can be extended to X +. Theorem Let (X, u) be a partially ordered generalized 2-normed space. If (a 0, b 0 ) X X is such that (a 0, b 0 ) (0, 0). If R + u and R + (u, u) + R + (a 0, b 0 ) are Supcones then there exists a state s on X + such that s(a 0, b 0 ) = (a 0, b 0 ). Proof. Define f : M = R + (u, u) + R + (a 0, b 0 ) R by f(α 1 u + α 2 a 0, β 1 u + β 2 b 0 ) = α 1 β 1 + α 2 β 2 (a 0, b 0 ). Then clearly f is bilinear, f(u, u) = 1 and f(a 0, b 0 ) = a 0, b 0. Let (α 1 u+α 2 a 0, β 1 u+β 2 b 0 ) (γ 1 u + γ 2 a 0, δ 1 u + δ 2 b 0 ) α 1 u + α 2 a 0 γ 1 u + γ 2 a 0 ; β 1 u + β 2 b 0 δ 1 u + δ 2 b 0 α 1 γ 1 ; α 2 γ 1 ; β 1 δ 1 ; β 2 δ 2 f(α 1 u + α 2 a 0, β 1 u + β 2 b 0 ) f(γ 1 u + γ 2 a 0, δ 1 u + δ 2 b 0 ). Hence f is state on M. Since M is a Supcone f can be extended to X +. Theorem Let (X, u) be a partially ordered generalized 2-normed space. If the positive cone of the subspace Z = {z : z = t(u, u) (x, x)} is a Supcone, then any bilinear functional on X can be decomposed into two positive bilinear functionals. Proof. Let Z = {z : z = t(u, u) (x, x)} and z 1, z 2 Z where z 1 = t 1 (u, u) (x, x), z 2 = t 2 (u, u) (y, y). Define F(z 1, z 2 ) = t 1 t 2 f f(x, y), where f is bilinear functional on X. Clearly F is bilinear on Z. Let (z 1, z 2 ) 0. Then t 1 u x t 1 u and t 2 u y t 2 u x, y t 1 t 2. Now f(x, y) f(x, y) t 1 t 2 f f(z 1, z 2 ) 0 f is a pre state on Z. Hence there exists a bilinear extension F on X, which is positive. Define g(x, y) = F((x, 0), (0, y)) and h(x, y) = F((0, x), (y, 0)). Clearly g and h are real bilinear, which are positive and f = g h. Theorem Let (X, u) be a partially ordered generalized 2-normed space, then each bounded bilinear functional on X can be decomposed into two bilinear functionals. Proof. Similar to above, by using Theorem 2.4. References 1 Corneliu Constantinescu, C -Algebras Volume1: Banach Spaces, Elsevier Science B.V, Amsterdam, Charalambos D. Aliprantis, Rabee Tourky. Cones and Duality, American Mathematical Society, Zofia Lewandowska, Linear Operators On Generalized 2-Normed Spaces, Bull.Math.Soc.Sc.Math., 42 (90) (4), Zofia Lewandowska, Bounded 2-Linear Operators On 2-Normed Sets, Glasnik Matematicki, 2001, 39(59), Zofia Lewandowska, Mohammad Sal Moslehain, Assieh Saadatpour Moghaddam, Hahn-Banach Theorem In Generalized 2-Normed Spaces, Communications in Mathematical Analysis, 2006, 1(2),