VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS

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1 Dedicated to Professor Philippe G. Ciarlet o his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst sectio of this paper deals with the properties of some type of vector semiorms. I the secod sectio, vector spaces with vector orm are cosidered while i the third sectio some theorems about the regular operators are give. Operators deed o a vector space with vector orm are cosidered i the fourth sectio. The termiology is that of [3]. AMS 2000 Subject Classicatio: Key words: vector semiorm, regular operators. 1. VECTOR SEMINORMS If X is a vector space over R ad Y a ordered vector space, a mappig P of X ito Y is said to be a vector semiorm if P (x 1 + x 2 ) P (x 1 ) + P (x 2 ), x 1, x 2 X, P (αx) = α P (x), α R, x X. If X ad Y are ordered vector spaces, a vector semiorm P : X Y is said to be mootoe if 0 a b X P (a) P (b) while it is said to be absolute mootoe if ± x c X P (x) P (c). A ordered vector space is said to be (o)-complete if every majorized subset has a supremum. The followig theorem geeralize some results give i [7] for real semiorms. Theorem 1.1. Let X be a directed vector space, G a majorized vector subspace of X, ad Y a (o)-complete ordered vector space. If P 0 : G Y is REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56,

2 408 Romulus Cristescu 2 a vector semiorm, the by puttig (1.1) P (x) = if {P 0 (a) + P 0 (z) a, z G; ± (x z) a}, x X, the followig statemets hold. (i) The operator P : X Y give by (1.1) is a absolute mootoe vector semiorm ad P (z) P 0 (z), z G. (ii) If Q : X Y is a absolute mootoe vector semiorm ad if Q(z) P 0 (z), z G, the Q(x) P (x), x X. (iii) We have P G = P 0 if ad oly if the vector semiorm P 0 is absolute mootoe. Proof. (i) For ay x X there exists the elemet P (x) give by (1.1). Ideed, sice G is a majorizig vector subspace of the directed vector space X, for ay elemets x X ad z G there exists a elemet a G such that ± (x z) a. O the other had, we have P 0 (c) 0, c G, ad the space Y is (o)-complete. As i the case of real semiorms, it is easily veried that the operator P give by (1.1) is a vector semiorm. We also have (1.2) P (x) = if {P 0 (b) x b G}, x X +. Ideed, deotig by P (x) the right-had side i (1.2), we obviously have P (x) P (x). Let ow x X + ad ± (x z) a with a, z G. The 0 x a + z G, whece P (x) P 0 (a + z) P 0 (a) + P 0 (z). So, P (x) P (x). Therefore (1.2) holds. The vector semiorm P is absolute mootoe. Ideed, if ± x v i X the, takig a G such that v a, it follows from ± x a that P (x) P 0 (a) by (1.1). The, by (1.2), we get P (x) P (v). Obviously P (z) P 0 (z), z G. (ii) Let Q : X Y be a absolute mootoe vector semiorm such that Q(z) P 0 (z), z G. If x X ad ± (x z) a with a, z G, the Q(x) Q(x z) + Q(z) Q(a) + Q(z) P 0 (a) + P 0 (z) whece Q(x) P (x). (iii) Suppose that P 0 is absolute mootoe ad let x G. Let also, z, a G such that ± (x z) a. We have P 0 (x z) P 0 (a) whece, as i the precedig case regardig the vector semiorm Q, we obtai P 0 (x) P (x). We also have P (x) P 0 (x) by statemet (i). Coversely, if P G = P 0 the P 0 is obviously absolute mootoe.

3 3 Vector semiorms, spaces with vector orm, ad regular operators 409 Deitio 1.1. If X is a directed vector space ad Y a (o)-complete ordered vector space, a vector semiorm P : X Y is said to be solid if P (x) = if {P (a) ± x a X}, x X. Remark 1.1. Let X be a directed vector space ad Y a (o)-complete ordered vector space. If G is a majorizig vector subspace of X ad P 0 : G Y a solid vector semiorm, the the formula P (x) = if{p 0 (a) ± x a G}, x X, yields a solid vector semiorm P : X Y such that P G = P 0. Remark 1.2. Let X be a vector lattice ad Y a (o)-complete ordered vector space. A vector semiorm P : X Y is solid if ad oly if x 1 x 2 i X implies P (x 1 ) P (x 2 ). This assertio is veried as i the case Y = R. Remark 1.3. Let X be a vector lattice, Y a (o)-complete ordered vector space ad P : X Y a solid vector semiorm. The P is additive o the positive coe X +, if ad oly if P is of the form (1.3) P (x) = U( x ), x X, where U : X Y is a positive liear operator. Ideed, if P is additive o X + the, by puttig, U(x) = P (x + ) P (x ), x X, we obtai a positive liear operator U : X Y for which (1.3) holds. Coversely, if U : X Y is a positive liear operator, the by (1.3) we obviously obtai a solid vector semiorm. We recall ow that if X ad Y are vector lattices, a operator U : X Y is said [3] to be disjuctive if x 1 x 2 i X implies U(x 1 ) U(x 2 ). Theorem 1.2. If X ad Y are vector lattices ad P : X Y is a mootoe ad disjuctive vector semiorm, the (1.4) P (x 1 x 2 ) = P (x 1 ) P (x 2 ), x 1, x 2 X +, ad if x 1 x 2 = 0 i X, the (1.5) P (x 1 x 2 ) = P (x 1 ) P (x 2 ). whece Proof. Let x 1, x 2 X +. We have (x 1 (x 1 x 2 )) (x 2 (x 1 x 2 )) = 0, (1.6) P (x 1 (x 1 x 2 )) P (x 2 (x 1 x 2 )) = 0.

4 410 Romulus Cristescu 4 O the other had ad, by (1.6), 0 P (x i ) P (x 1 x 2 ) P (x i (x 1 x 2 )), i = 1, 2, (P (x 1 ) P (x 1 x 2 )) (P (x 2 ) P (x 1 x 2 )) = 0. Therefore, (1.4) holds. If x 1 x 2 = 0 the x 1 x 2 = x 1 + x 2, hece P (x 1 x 2 ) = P (x 1 + x 2 ) P (x 1 ) + P (x 2 ) = P (x 1 ) P (x 2 ) sice P (x 1 ) P (x 2 ). O the other had, we have so that (1.5) holds. P (x 1 ) P (x 2 ) P (x 1 x 2 ), Remark 1.4. Let X ad Y be vector lattices ad let P : X Y be a solid vector semiorm. The P is disjuctive if ad oly if x 1 x 2 = 0 i X implies P (x 1 ) P (x 2 ) = 0. Theorem 1.3. Let X be a vector lattice ad Y a (o)-complete ordered vector space. If P : X Y is a (o-zero) solid vector semiorm, the there exists a (o-zero) positive liear operator U : X Y such that (1.7) U(x) P (x), x X. Proof. The proof is similar to that give i the case of fuctioals [4]. By puttig Q(x) = P (x + ), x X, we obtai a subliear operator Q : X Y. Let us cosider a elemet a > 0 of X such that P (a) > 0. By a corollary of the Hah-Baach-Katorovich theorem [5] there exists a liear operator U : X Y such that U(a) = Q(a) ad U(x) Q(x), x X. It is easy to see that U is positive ad that iequality (1.7) holds. Theorem 1.4. Let X be a vector lattice, Y a (o)-complete ordered vector space ad P (X, Y ) the set of all solid vector semiorms mappig X ito Y. If P i P(X, Y ), i = 1, 2, the, with respect to the poitwise order i the set P(X, Y ), there exists P 1 P 2 i this set, ad we have (P 1 P 2 )(x) = if {P 1 (a) + P 2 (b) a, b X + ; a + b = x }, x X. The proof is similar to that give i the case Y = R (see [2]).

5 5 Vector semiorms, spaces with vector orm, ad regular operators SPACES WITH VECTOR NORM Cauchy sequece if there exists a sequece (a ) N of elemets of X such that 0 ad a N (2.1) f i f j X a, i, j. We say that a sequece (f ) N (v)-coverges to a elemet f F X if the sequece ( f f X ) N (o)-coverges (i.e. coverges with respect to the order) to 0 i the space X. The elemet f is called the (v)-limit of the sequece (f ) N, ad we write f = (v)-lim f. Let F be a real vector space ad X a vector lattice. A vector semiorm P : F X is said to be a vector orm if P (x) = 0 implies x = 0. If the vector space F is edowed with a vector orm P : F X, the F is called a v-ormed vector space ad is deoted by F X. We shall also write f X = P (f). Let F X be a v-ormed vector space. A subset A of F X is said to be (v)-bouded if there exists x 0 X such that f X x 0, f A. A sequece (f ) N of elemets of the space F X is said to be a (v)- The space F X is said to be (v)-complete if ay (v)-cauchy sequece of elemets of F X (v)-coverges to a elemet of the space. The space F X is called a strictly v-ormed vector space if wheever f X = a 1 + a 2 with a i X +, i = 1, 2, there exists f i F X, i = 1, 2, such that f i X = a i, i = 1, 2, ad f = f 1 + f 2. A strictly v-ormed vector space which is also (v)-complete is said to be a space of type (B K ). The otio of a (v)-coverget series i a v-ormed vector space ca be itroduced i a atural maer: (v)- f = (v)- lim f i =1 if the right-had member (v)-limit exists. If F X is a v-ormed vector space with respect to a Archimedea vector lattice X, a sequece (f ) N of elemets of F X is said to (vρ)-coverge to a elemet f F X if the sequece ( f f X ) N (ρ)-coverges (i.e. coverges with regulator [3]) to 0 i the space X. Remark 2.1. If a (v)-cauchy sequece (f ) N of elemets of a v-ormed vector space has a (v)-coverget subsequece, the the sequece (f ) N is (v)-coverget. I the ext theorem, by regular space we mea (as i [4]) a Archimedea vector lattice i which ay (o)-coverget sequece is also (ρ)-coverget. i=1

6 412 Romulus Cristescu 6 Theorem 2.1. Let F X be a v-ormed vector space ad cosider the followig two coditios: (i) the space F X is (v)-complete; the series (ii) if a series of the form =1 =1 f is (v)-coverget. f X, f F X, is (o)-coverget, the Coditio (i) implies (ii), ad if X is a σ-complete vector lattice which is also a regular space, the coditios (i) ad (ii) are equivalet. Proof. Suppose that coditio (i) is satised ad let (f ) N be a sequece of elemets of F X such that the series f X be (o)-coverget. Deotig s = f i, N, there exists a sequece (w ) N of elemets of X such that w i=1 s s m X w m, m, N; m. 0 ad N Therefore, (s ) N is a (v)-cauchy sequece ad sice F X is (v)-complete, the sequece (s ) N is (v)-coverget. Cosequetly, (i) implies (ii). Suppose ow that X is a σ-complete vector lattice which also is a regular space ad suppose that coditio (ii) is satised. Let (f ) N be a (v)-cauchy sequece of elemets of F X. Let (a ) N be a sequece of elemets of X such that a N 0 ad (2.1) f i f j X a, i, j. Sice X is a regular space, there exists w X + ad a sequece (ε ) N of real umbers, such that ε 0 ad a ε w, N. Accordig to (2.1) N there exists a strictly icreasig sequece (j ) N of atural umbers such that (2.2) f j+1 f X j 1 w, N. 2 Sice the vector lattice X is σ-complete, it follows from (2.2) that the series correspodig to the sequece ( f j+1 f j X ) N is (o)-coverget. By hypothesis, the series (f j+1 f j ) is (v)-coverget. It follows from the equatio f j+1 = f j1 + ( ) fjk+1 f jk, N, k=1

7 7 Vector semiorms, spaces with vector orm, ad regular operators 413 that the sequece (f j ) N is (v)-coverget while, by Remark 2.1, the sequece (f ) N is (v)-coverget. Cosequetly, if X is a σ-complete vector lattice which also is a regular space, the coditios (i) ad (ii) are equivalet. Theorem 2.2. Let X ad Y be complete vector lattices ad G a vector sublattice of X edowed with a solid vector orm with values i Y. If X is a Dedekid extesio of G, the the vector orm give o G ca be exteded to a solid vector orm o X by the formula (2.3) x Y = if { a Y x a G}, x X. we have Proof. If x 1, x 2 X the, deotig A = { a Y a = b + c, x 1 b G, x 2 c G}, B = { b Y x 1 b G}, C = { c Y x 2 c G}, x 1 + x 2 Y if A if (B + C) = if B + if C = x 1 Y + x 2 Y. O the other had, oe ca easily verify that αx Y = α x Y, α R, x X. If 0 x X the there exists b G such that 0 < b x. If x a G the 0 < b a, hece 0 < b Y a Y. It follows is immediately that x Y 0. Therefore, by (2.3), we obtai a vector orm o X. It is easily veried that this vector orm is solid ad that it is a extesio of the vector orm give o G. 3. REGULAR OPERATORS Let X be a vector lattice ad Y a complete vector lattice. We shall deote by R(X, Y ) the set of all regular operators which map X ito Y. It is kow that R(X, Y ) is a complete vector lattice with respect to the usual operatios ad the order give by the coe of positive operators. We shall also deote X r = R (X, R). As i the rst sectio (see Theorem 1.4) we deote by P(X, Y ) the set of all solid vector semiorms o X ito Y ad shall cosider the poitwise order i this set. Theorem 3.1. Let U i R(X, Y ) ad P i P(X, Y ), i = 1, 2, such that (3.1) U i (x) P i (x), x X, i = 1, 2.

8 414 Romulus Cristescu 8 The the iequality (3.2) ( U 1 U 2 ) (x) (P 1 P 2 ) (x), x X. holds. Proof. It follows from (3.1) that ad by Theorem 1.4 we have U i (x) P i (x), x X, i = 1, 2, ( U 1 U 2 ) (x) ( U 1 U 2 ) ( x ) = = if { U 1 (a) + U 2 (b) a, b X + ; a + b = x } if {P 1 (a) + P 2 (b) a, b X + ; a + b = x } = (P 1 P 2 ) (x). Therefore, (3.2) holds. Deitio 3.1. Let p : X R be a semiorm. A operator U : X Y is said to be p-bouded if there exists y 0 Y such that U(x) p(x)y 0, x X. Remark 3.1. Theorem 3.1 implies the followig propositio give i [2]. If U 1, U 2 R (X, Y ), p 1, p 2 are solid real semiorms o X ad U i is p i -bouded, i = 1, 2, the U 1 U 2 is p 1 p 2 -bouded. Theorem 3.2. Let P : X Y be a solid vector semiorm, G a ormal subspace of X ad V 0 : G Y a positive liear operator such that (3.3) V 0 (a) P (a), a G. The there exists a positive liear operator V : X Y such that V G = V 0 ad (3.4) V (x) P (x), x X. Proof. By the Hah-Baach-Katorovich theorem [5], there exists a liear operator U : X Y such that U G = V 0 ad U(x) P (x), x X. It follows that U is a (o)-bouded liear operator, hece U R(X, Y ). Let us ow cosider the positive part U + of U. If x X + the it follows from 0 z x that U(z) P (z) P (x), hece U + (x) = sup {U(z) 0 z x} P (x). For ay x X we ow have U + (x) U + ( x ) P ( x ) = P (x).

9 9 Vector semiorms, spaces with vector orm, ad regular operators 415 Therefore, puttig V = U +, iequality (3.4) holds. O the other had, sice G is a ormal subspace of X, if 0 a G the [0, a] G. It follows that V (a) = sup U([0, a]) = V 0 (a) ad, G beig a vector sublattice of X, we have V (x) = V 0 (x), x G. Remark 3.2. It follows by Theorems 3.2 ad 3.1 that if G is a ormal subspace of the space X, if 0 V i R (G, Y ), P i P (X, Y ), i = 1, 2, ad V i (a) P i (a), a G, i = 1, 2, the there exists a positive V R(X, Y ) such that V G = V 1 V 2 ad V (x) (P 1 P 2 ) (x), x X. 4. LINEAR OPERATORS ON v-normed VECTOR SPACES Let X, Y be vector lattices ad F X, G Y v-ormed vector spaces. We recall that a operator U : F X G Y is said to be a (v)-regular operator [3] if it is additive ad if there exists a positive liear operator V : X Y such that (4.1) U(f) Y V ( f X ), f F X. We shall deote by R v (F X, G Y ) the set of all (v)-regular operators which map F X ito G Y. If the space Y is Archimedea, the a (v)-regular operator which maps F X ito G Y is a liear operator. Remark 4.1. If Y is a complete vector lattice ad i the spaces X, Y we cosider the modulus as vector orm, the R v (X, Y ) = R(X, Y ). Remark 4.2. By a propositio give i [3], if F X is a strictly v-ormed vector space, Y is a complete vector lattice ad U R v (F X, G Y ), the there exists the smallest positive liear operator V which satises iequality (4.1). I this case we shall deote such a operator by U R. The map P : R v (F X, G Y ) R(X, Y ) give by the formula P (U) = U R is a vector orm o the vector space R v (F X, G Y ). We recall ow that a subset A of a Archimedea vector lattice is said to be (o)-aihilatig [3] if for ay sequece (x ) N of elemets of A ad ay sequece (α ) N of real umbers which coverges to zero, oe has (o)- lim α x = 0. A Archimedea vector lattice Z is said to be of (o)-boudedess type if ay (o)-aihilatig subset of Z is (o)-bouded. The followig theorem geeralizes a result give i [8].

10 416 Romulus Cristescu 10 Theorem 4.1. Let X, Y be Archimedea vector lattices ad F X, G Y v- ormed vector spaces. Let (U ) N be a sequece of liear operators which map F X ito G Y. Cosider the followig statemets. (i) For ay (v)-bouded subset A of F X, the set U (A) is (v)-bouded. (ii) If (vρ)-lim f = 0 i F X, the (vρ)-lim U j (f ) = 0 for ay sequece (j ) N of atural umbers. Statemet (i) implies (ii) ad if Y is of (o)-boudedess type, the (i) ad (ii) are equivalet. Proof. Suppose (i) holds ad let (vρ)-lim f = 0 i F X. The [5] there exists a sequece (λ ) N of atural umbers such that λ ad (ρ)- N N lim λ f X = 0. It follows that the sequece (λ f ) N is (v)-bouded. By hypothesis, there exists y 0 Y such that (4.2) U k (λ f ) Y y 0, k, N. If (j ) N is a arbitrary sequece of atural umbers, the by (4.2) we have U j (f ) Y 1 λ y 0, N, hece (vρ)-lim U j f = 0. Cosequetly, (i) (ii). Let us ow assume that Y is of (o)-boudedess type ad suppose (ii) holds. Let A be a (v)-bouded subset of the space F X. Deote B = k N U k (A) ad let us cosider a sequece (g ) N of elemets of B. Let j N ad f A with g = U j (f ). Cosider ow a sequece (α ) N of real umbers coverget to zero. Sice the sequece ( f X ) N is (o)-bouded, the sequece (α f X ) N is ρ-coverget to 0, hece (vρ)-lim α f = 0. By hypothesis, (vρ) -lim U j (α f ) = 0 hece (ρ)-lim α g Y = 0. Cosequetly, the set { g Y g B} is (o)- aihilatig, therefore it is (o)-bouded. I other words, the set B is (v)- bouded. Remark 4.3. Let F X be a strictly v-ormed vector space, G Y a v-ormed vector space with Y complete vector lattice ad U R v (F X, G Y ), N. If the sequece ( U R ) N is (o)-bouded i R (X, Y ) the statemet (i) of Theorem 4.1 holds.

11 11 Vector semiorms, spaces with vector orm, ad regular operators 417 I the ext theorem we cosider the modulus as vector orm i a complete vector lattice Y. Theorem 4.2. Let F be a vector lattice ad E a vector sublattice of F. Cosider a solid vector orm o F with values i a vector lattice X such that F X ad E X are strictly v-ormed vector spaces. If Y is a complete vector lattice ad U 0 : E X Y is a positive (v)-regular operator, the there exists a positive (v)-regular operator U : F X Y such that U E = U 0 ad U R = U 0 R. Proof. Deotig by F + the positive coe of the space F ad puttig Q(f) = if { U 0 R ( h X ) f h F +}, f F, we obtai a subliear operator Q : F Y. It is easily veried that (4.3) Q(f) U 0 R ( f X ), f F. The iequality (4.4) U 0 (f) Q(f), f E, also holds. Ideed, if f E ad f h F +, the f f + h with f + E ad we have U 0 (f) U 0 (f + ) U 0 R ( f + X ) U 0 R ( h X ) whece (4.4) follows. By the Hah-Baach-Katorovich theorem [5], there exists a liear operator U : F X such that U E = U 0 ad (4.5) U(f) Q(f), f F, whece, by (4.3), U(f) U 0 R ( f X ), f F. Therefore, U R v (F X, Y ) ad U R = U 0 R. O the other had, if f F ad f 0, the Q(f) = 0 ad it follows by (4.5) that U(f) 0, that is, U is a positive operator. REFERENCES [1] C.D. Alipratis, O order properties of order bouded trasformatios. Caad. J. Math. 27 (1975), [2] I. C tueau, P-bouded operators ad F-bouded operators. A. Uiv. Bucure³ti Mat. 36 (1987), 912. [3] R. Cristescu, Ordered Vector Spaces ad Liear Operators. Abacus Press, Tubridge, [4] R. Cristescu, Topological Vector Spaces. Editura Academiei Româe, Bucharest & Noordho I.P., Leyde, [5] R. Cristescu, Notios of Liear Fuctioal Aalysis. Editura Academiei Româe, Bucure³ti, (Romaia)

12 418 Romulus Cristescu 12 [6] R. Cristescu, O some liear operators ad o some vector itegrals. I: Order Structures i Fuctioal Aalysis. Vol. 4, pp Editura Academiei Româe, Bucure³ti, [7] O. va Gaas, Extedig mootoe semiorms o partially ordered vector spaces. Idag. Math. (N.S.) 9 (1998), 3, [8] C. Swartz, The uiform boudedess priciple for order bouded operators. Iterat. J. Math. Math. Sci. 12 (1989), Received 16 July 2008 Uiversity of Bucharest Faculty of Mathematics ad Computer Sciece Str. Academiei Bucharest, Romaia