![Comparison of Order Continuous and Norm Continuous Linear Mappings (Limited Distribution) Between Two Ordered Banach Spaces Was Studied in [2]](http://data.docslib.org/img/88c21f69b359c045be102d9bf9508a88-1.webp)
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IC/87/3U8 INTERNAL REPOPT Comparison of order continuous and norm continuous linear mappings (Limited distribution) between two ordered Banach spaces was studied in [2]. In this paper the study is carried further to relatively uniformly continuous and order bounded linear mappings. It la observed that relatively Uniformly J ..^International Atomic Energy Agency / and continuous mappings provide a natural link between norm continuous and s Educational Scientific and Cultural Organization order bounded linear mappings. In the process some results of [9] have also been improved. The lattice structure has been avoided to consider
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS linear mappings between arbitrary ordered Banach spaces. In the first section we study conditions on the domain nn
cones X+ and Y+ respectively. X+ is said to be generating Iff
X = X - X+ and X+ is normal Iff a < P 4 b in X implies that M. Nasir Chaudhary •• Up) < mw{[a|,|b||}. Daviea [5] defines X to have property (f^) iff International Centre for Theoretical Physics, Trieste, Italy. -a * x 4 a implies that |x| < Ra|[ . Obviously, a space with nortsal cone has prop(R ) but the converse need not be true. In the real sequence space V if a = (-1,0,0, ...); b * (0,1,0, ...); p = (-1,1,0, ...)
ABSTRACT then a < p < b, |a|| a |b|| « 1 but |p| = 2. Thus the usual cone in J.'
Is not normal. On the other hand; -a « x < a Implies that -a.i < x^ ^ ^ Comparison of order continuous and norm continuous linear mappings i.e. \x±\ 4 |e.jj and ||4 < M- between tvo ordered Banach spaces was studied by Chaudhary and finleemi. A sequence tx } in X is said to be relatively uniformly In thiB paper the study 1B carried further to relatively uniformly convergent to x£ X iff there exists a decreasing sequence l*n) of continuous and order bounded linear mappings. positive real numbers with Inf(X ) • 0 and an element u in X+ such that -Xu and -e < xn - x < e . The sequence (xn) %n order atar converges to x if every subsequence of (x } has a subsequence which order converges to x. * To be submitted for publication. •• Permanent address: Deportment of Mathematics, University of Engineering -2- and TechnoloRy, Lahore, Pakistan. 0» We write x —+ x. Similarly we define relative uniform star convergence 1. RELATIVELY UNIFORMLY CONTINUOUS MAPPINGS and write " x —P•* x, n It is known that p"-convergence and norm convergence of sequences p-convergence implies O-convergence but the converse may not hold. are identical If X+ is generating and X has proptl^) [3;8]. Thus The sequence {x ) in m (the space of bounded real sequences) defined by p"-convergence has a natural closeness to norm convergence. The same 0 i 4 n closeness is present in the case of p*-cont and norm-cont linear mappings: (i) x - n Proposition 1.1 1 i > n Ca) Let X+ be generating and Y have prop(R-). Then a p'-cont linear order converges to the zero element but it does not converge relatively mapping form X into Y Is norm continuous. uniformly to zero [9l. (b) Let Y+ be generating and X have prop(lO. Then a norm cont linear mapping from X into Y Is p"-eont. X is said to have property (A) iff every sequence (x ) in X —™* —^— n Proof (a) Let {x ) be a sequence in X such that x + 0. By with x +0 is norm convergent to zero. X has order continuous norm •— n n n .^—^——__ [1. Lemma 1] x +0. If T : X -» Y is a p*-cont linear mapping. Then 0 iff x -» x implies that x •+ x. Clearly an order continuous normed space Tx - 0. Y has prop(R-) and so by [3] Tx • 0. has prop(A). The converse is true if the cone in X is normal and n X n Thus T is nor generating. An example of an ordered Banach space which is not a lattice continuous. but has prop(A) Is given by May and McArthur in [8]: t with usual (b) The proof is similar to that of part (a). topology ordered by the Lorentzian cone Combining now (a) and (b) ve obtain: Proposition 1.2 Let X and Y be ordered Banach spaces with closed 0 4 x, and E generating cones. If both X and Y have prop(B ) then a linear mi n=2 from X into Y is p^-eont Iff it Is norm-cont. A linear napping T: X * Y is said to be relatively uniformly A vector lattice E has the diagonal property [9] if whenever P ^ { m) continuous (p-cont) Iff whenever x + x in X then Tx •• T in Y. x "* : m, n - 1, 2, ... 1 C E and • n n x Similarly we define p'-cont, order continuous (0-cont) and 0*-cont (a) O-converges to x(n)(n) £ linear mappings. T is said to be order bounded (0-bounded) iff T mans (b) >) x £ order bounded subsets of X into order bounded subsets of Y. 0-converges to x' A counterexample of a positive linear functional on • (the space then there is a strictly Increasing sequence {B 1, 2, ...I of of finite sequences) which is not norm continuous is presented in [7]. x r 0-converges to x.(0v"') . An order A postlve linear functional is both order bounded and p-continuous. Thus we note that there are p-cont/O-bounded linear mappings which are complete Banoch lattice with prop(A) has the diagonal property [9- IV 2.6], not continuous. Conversely [7:Ex.5.l'f] is an example of a continuous linear Order convergence and p-convergence are known to be equivalent In a o-order 2 2 mapping from Jt into Jt which is not 0-bounded and not p-cont. complete vector lattice with the diagonal property. It would be natural to expect similar relationshio in an ordered Banach Bpace with Proo(A) and For other definitions and notations, the reader is referred to [It] which may not be a lattice. In fact we have: and [9]. -It- -3- then {y ) Is a subsequence Of {x ) and so there exists a subsequence Proposition 1.3 Let X+ be normal and generating. If further X k P n P has prop(A) then order convergence is equivalent to relative uniform ly, ) of {y } such that y * 0. Consequently, TyK , •+ 0, which lmplieB convergence for sequences In X. Tx -5-* Q. By [3] Tx, + 0 and T is norm continuous, n n Proof Let tx } be a sequence In X such that x •+ 0. By definition i (c) Let A be an order bounded subset of X and (y ] be an there exists a positive sequence {& } such that a 4 0 and arbitrary sequence In T(A). There are x € A such that 'i'Xj, * y . x i a . Since X has propU), a + 0. X is generating and n n Since A is 0-bounded there are p, q in X with p < x< q. X is so by [1. Lemma ll a 0. Consequently, there is a subsequence {a } + generating and we can find an element u £ X+ such that -u 4 x < u. of {a J, a seq (\) of positive numbers decreasing to zero and an If {X } is a sequence of positive real numbers decreasing to zero then n element y in \ that - v < a« X y. Let us put P fe n -X u^X x 4X u and this means that X x + 0. But T Is p-cont n nnnp pnn for ^ s n < n^j, k 1, 2, ... Then a + 0 and a^ .j a y = ce y « X n ' and hence T(X xft) * 0 i.e. Xn yfl * 0. This implies further that that for n. i.e. given e > 0 we can find M such * X y •+ 0. Since y has the boundedness property we conclude that T(A) p a * a y (Vn > M). n n n n Thus a -» 0 [ U ], and from is 0-bounded. Thus T Is Q-bounded. -o y * -a x s a -< o y {n » n,) we see that x + 0. Thus 0-convergenee n n n n n i- n Implies p-convergence in X. The converse being always true, the equivalence is established. 2. VARIOUS C0MPAEIS033 From the proof one can also note that a decreasing sequence which is p*-convergent is in fact p-convergent. In this section we look at the converse of the results of Section 1. We also compare norm cont and 0-bounded linear mappings. We shall use the above proposition in our next result where we turn to p-cont linear mappings. X is said to have boundednesa property 19] Proposition 2.1 Let T: X * Y be an 0-bounded linear mapping. Then If a subset B of X is 0-bounded whenever {X x ) O-converges to zero (a) If K+ Is generating, T i3 p-cont; n n (b) If X+, Y+ are generating and Y has propt^), T is norm continuous. for each sequence {x } in B and each sequence {X } of positive numbers Proof (a) Let {x } be a sequence In X which is p-convergent to zero. decreasing to zero. Any order unit nonned space has this property. By 16: Lemma 3.2] there are positive numbers a^ tending to infinity such Proposition l.U Let T be a p-cont linear napping. that {a x } le 0-bounded. Since T is 0-bounded linear mapping (a) If X+ is normal, generating and X has prop(A) then T is 0-cont. n n {T(a x )} is 0-bounded in Y. Again applying [6: Lemma 3-2] we get (b) If X+ is generating and Y has propff^) then T is norm continuous. n n (c) If X+ is generating and Y has the boundedness property then T is (Tx } is p-converfeent to zero in Y. Hence T Is p-cont. 0-bounded. (b) Follows from part (a) and Proposition l.U (b). Proof (a) If { is a sequence In X such that x •• 0 then Part (a) of the above proposition may be combined with part (c) of " p n 0 Proposition l.U to obtain: Proposition 1.3 Implies that x •* 0. T 0 n being p-cont, we have Tx •+ 0 I.e. Tx -••0. Hence T is 0-cont. Proposition 2.2 Let X+, Y+ be generating and Y have the boundedness n property. A linear mapping from X into Y la p-cont iff it is (b) Let {x } be a sequence in which is norm convergent to 0-bounded. ia B «ro. By [1. Leiana l] x -^-» 0. If {T subsequence of (Tjt } -6- -5- Next we consider norm continuous linear mappings. We have: Proposition £.5 Let Ti X + I he an 0-cont linear mapping. If Y Proposition 2.3 has prop(A) and Y+ Is. normal, generating then T is p-cont. (a) If X has propfF^) and Y+ Is order unit normed then T is p-cont; Proof If a sequence (x } Is p-convergent to the zero element then it (b) If X+ is normal, generating and Y Is order unit normed then T is is also 0-convergent. T being 0-cont (Tx } is 0-convergent to zero 0-bounded. in Y, Applying Proposition 1.3 we see that TXR IS p-convergent to Proof (a) If x + o then x and so x^ + 0. Since T is zero and hence T Is p-cont. n n continuous, Tx •*• 0. Put y = Tx -» 0. By [l. Lemma, l] y -^* 0. Combining Proposition l.J» (a) and Proposition 2.5 we see that if the Let e be the order unit In Y. For all 1, 2, positive cones X+, Y+ are normal, generating and X, Y both have prop(A) £ [-e,ej is a nbhd of zero. Given k, we can find a positive number then a linear mapping from X into Y is 0-cont iff it is p-cont. p a. such that y £ — [-e,e]. ThiB means that yn •• 0 i.e. T Is p-cont. « •n* K.-* ii (b) Since X+ is normal, X has propt^) and so by part (a) T Is p-cont. An order unit normed space has the boundedness property and so 3. ORDfiR IDEALS part (c) of Proposition l.k implies that Y Is 0-bounded. P Example Consider the identity mapping I from (m,K) to (n,P) where Let L (X,Y), L°(X,Y), L^fX.Y) respectively denote the spaces of m is the space of bounded real sequences vith sup norm and P is the p-cont, 0-cont and 0-bounded linear mappings from X into Y. One can usual positive cone, K Is the cone of P -ordering (all the partial sums easily prove that all of them are subspaces In the space L(X,Y) of all of seq are non-negative). Define x • linear mappings from X into Y. In fact they are order convex in <.£•>. L(X.Y)j e.g. if 0 t T « S, S € LP(X,Y) and T £ MX.Y) then T being P 1 < 2n positive is p-cont and so T G L (X,Y). Thus the three are order Ideals in L(X,Y). (-1)1 2n This sequence is p-eonvergent to zero In the ordering of cone K (choose regulator u - (l, 1. ...) then - u - x fe K) but it is not n n p-convergent to zero in the ordering of cone P. Thus I is not p-cont though It is continuous. In fact (m,K) does not have prop(lL), Consider a = (i, —, ,..); ACKNOWLEDGEMENTS x • {0, 0, +1, -1, +1, -l, ...). Then -a < x -s a with regard to ordering The author would like to thank Professor Abdus Salam, the of cone K but |x| - 1 and ||a| = i International Atomic Energy Agency ana UMESCO for hospitality at the From Proposition 2.2 and Proposition 2.3 we note that: International Centre for Theoretical Physics, Trieste. Proposition 2.U Let X+ be normal, generating and Y be an order unit normed space. A linear mapping from X into Y is p-cont iff it is norm cont iff It Is 0-bounded. Finally as a converse of Proposition l.k (a) we have: -7- -6- REFERENCES [l] Chaudharv H.Ti. and Atkinson 11.R., "Ordered Banach spaces with compact order intervals", J. Lond. Math. Soc. (2) rT (1978) 10T-109. 121 Chaudhary M.tl. and SaJ.w* A.li., "Comparison of continuous and order continuous linear mappings between ordered nonned linear Bpaces", Riaii J. Kar. Math. Aasoc. Vol.8 (1986) 1-5. [3] Chaudhary M.H., "Convergence in regular ordered Banaeh spaces", P.U. J. Math. Vol.X-XI (1977/78) 55-58. [It] Cristescu R. , Ordered Vector Spaces and Linear Operators. (Abacus Press, Turnbridge Veils, Kent, United Kingdom, [5] Davies E.B. , "The structure and ideal theory of tho predual of a Banach lattice", Trans. Amer. Math. Soe. 131 U9fM) 5UU-55. [6] CJordon H. , "Relative uniform convergence", Math. Annalen Ul8-l(27. [7] Khaleelulla G.M. , Counterexamples In topological vector spaces, Lecture Notes in MathB. 936 {Springer-Verlag, 1982). [8] Mav P. and He Arthur If. , "Comparison of two types of order convergence with topological convergence in an ordered tonolORical vector snace", Proc. Amer. Math. Soc. Vol.63, Ho.l (1977) ^9-55. [o] PeretJ"3ini 4.L. t Ordered Topological Vector Spaces. (Harper and Row, Hew York, London, 196T). -9-