A KRENGEL TYPE THEOREM FOR COMPACT OPERATORS BETWEEN LOCALLY SOLID VECTOR LATTICES Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 3, P. 76-80

YAK 517.98

DOI 10.46698/g6863-7709-2981-j

A KRENGEL TYPE THEOREM FOR COMPACT OPERATORS BETWEEN LOCALLY SOLID VECTOR LATTICES

O. Zabeti1

1 Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Sistan and Baluchestan, Zahedan, P.O. Box 98135-674, Iran E-mail: o.zabeti@gmail.com

Abstract. Suppose X and Y are locally solid vector lattices. A linear operator T : X —v Y is said to be «¿-compact provided that there exists a zero neighborhood U C X, such that T(U) is compact in Y; T is 66-compact if for each bounded set B C X, T(B) is compact. These notions are far from being equivalent, in general. In this paper, we introduce the notion of a locally solid AM-space as an extension for AM-spaces in Banach lattices. With the aid of this concept, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices. This extends [1, Theorem 5.7] (established for compact operators between Banach lattices) to different classes of compact operators between locally solid vector lattices.

Keywords: compact operator, the Krengel theorem, locally solid AM-space. AMS Subject Classification: 46B42, 47B65.

For citation: Zabeti, O. A Krengel Type Theorem for Compact Operators Between Locally Solid Vector Lattices, Vladikavkaz Math. J., 2023, vol. 25, no. 3, pp. 76-80. DOI: 10.46698/g6863-7709-2981-j.

1. Motivation and Introduction

Let us start with some motivation. Let E be a Banach lattice. E is called an AM-space provided that for each x,y € E+, we have ||x V y|| = ||x|| V ||y||. The remarkable Kakutani theorem states that every AM-space is a closed sublattice of some C(K)-space, in which K is the compact Hausdorff topological space. Now, suppose E is a Banach lattice and F is an AM -space. The Krengel theorem states that every compact operator T : E — F has a modulus which is defined by the Riesz-Kantorovich formulae; that is |T|(x) = sup{|Ty| : |y| ^ x} for each x € E+; furthermore, |T| is also compact. So, we conclude that AM-spaces have many interesting properties among the category of all Banach lattices. Therefore, it is fascinating and significant to consider the AM-spaces and numerous applications in the operator theory for locally solid vector lattices and operators between them. In this paper, we consider locally solid vector lattices whose family of pseudonorms which generate the topology of X, preserve the finite suprema; we call them: locally solid AM-spaces. This definition extends AM-spaces to the category of all locally solid vector lattices. Note that a variant of this notion has been defined by the author in [2]; however, that definition has a mild gap so that we consider the new definition using the generating pseudonorms. Moreover, observe that there are several different ways to define bounded and compact operators between locally solid vector lattices.

© 2023 Zabeti, O.

In this paper, we are going to generalize the known Krengel theorem [1, Theorem 5.7] for different types of compact operators between locally solid vector lattices.

Note that a subset A of a vector lattice X is called solid provided that x € X, y € A and |x| ^ |y| implies that x € A. Moreover, a pseudonorm p on a vector lattice X is a real-valued function on X that satisfies the following properties.

(i) p(x) ^ 0 for each x € X;

(ii) p(x + y) ^ p(x) + p(y) for each x, y € X;

(iii) p(Ax) ^ 0 as A ^ 0, for each x € X;

(iv) p(x) ^ p(y) whenever |x| ^ |y| for each x,y € X.

For undefined terminology and related notions (see [1, 3]). All locally solid vector lattices in this note are assumed to be Hausdorff.

2. Main Result

We introduce the notion of the locally solid AM-space; a variant of this notion has been defined as the AM-property in [2]; first, let us recall the AM-property.

Suppose X is a locally solid vector lattice. We say that X has the AM-property provided that for every bounded set B C X, Bv is also bounded with the same scalars; namely, given a zero neighborhood V and any positive scalar a with B C aV, we have Bv C aV. Note that by Bv, we mean the set of all finite suprema of elements of B.

That definition has a gap as follows. Let X be R2 with the sup norm. Take B = {(1,1), (\/2,0)} and V = {(x,y) : x2 + y2 ^ 1}. Note that V is a neighbourhood of 0. Then B is contained in but B is not.

On the other hand, by [3, Theorem 2.28] due to Fremlin, the linear topology of every locally solid vector lattice has been generated by a family of the Riesz pseudonorms. So, we can define a locally solid AM-space as follows. A version of the following definition was originally defined at first in [4].

Definition 1. A locally solid AM-space is a locally solid space X together with a family (pi)i^i of the Riesz pseudonorms which generate the topology of X and, in addition, satisfy the following property:

pi(x V y) = pi(x) V pi(y)

for each i € I and for each x, y € X+.

This definition modifies the gap in definition of the AM-property considered in [2]. Now, we restate the following useful fact which is proved initially in [5, Lemma 3] while we employed the AM-property instead of the locally solid AM-space.

Lemma 1. Suppose X is a locally solid AM-space with a family (pi)igI of the Riesz pseudonorms, which generate the topology of X and U is an arbitrary solid zero neighborhood in X. Then, for each m € N, U V ... V U = U, in which U is appeared m-times.

< Without loss of generality, we may assume that

U = {x € X, pik (x) < ek : 1 < k < n}

in which {ii,..., in} C I and {ei,..., en} C R+. It is obvious that U C U V ... V U. For the other direction, assume that x1,..., xm € U+. For each 1 ^ k ^ n, we have pk(x1, V ... Vxm) = pk(x1) V ... V pk(xm) < ek. This shows that U V ... V U C U, as claimed. >

Moreover, we have the following useful inequality in the Archimedean vector lattices.

Lemma 2. Suppose E is a vector lattice. Then for x1,..., xn and y1,..., yn in E, the following inequality holds

xi V ... V xn - yi V ... V yn ^ (xi - yi) V ... V (xn - yn).

< We proceed the proof by induction. For n = 2, we have

xi V x2 - yi V y2 = (xi - (yi V y2)) V (x2 - (yi V y2)) = (xi + ((-yi) A (-y2))) V (x2 + ((-yi) A (-y2))) = ((xi - yi) A (xi - y2)) V ((x2 - yi) A (x2 - y2)) ^ (xi - yi) V (x2 - y2).

Now, suppose for n = k, the statement is valid. We need prove it for n = k + 1. By using validness of the result for n = 2 and n = k, we have

xi V ... V xfc V xfc+i - yi V ... V yk V yfc+i ^ ((xi V ... V xfc) - (yi V ... V yfc)) V (xfc+i - yfc+i) ^ (xi - yi) V ... V (xfc - yfc) V (xfc+i - yfc+i). >

Recall that a subset B of a topological vector space X is said to be totally bounded, if for each arbitrary zero neighborhood V C X there is a finite set F such that B C F + V; for more information, see [1]. Another proof of the following result with a different technique has been obtained in [6, Corollary 4.3.5].

Lemma 3. Suppose X is a topologically complete locally solid AM-space. If B C X is totally bounded, then so is Bv. In particular, sup B exists in X and sup£> € Bv.

< Choose arbitrary solid zero neighborhood U C X .By the assumption, there exists a finite set F C X such that B C F + U. Assume that F = {zi,...,zm}. We claim that Bv C Fv + U; note that Fv, the set of all finite suprema of elements of F, is clearly finite. Given any xi,..., xn € B. There are some zi,..., zn (possibly with the repetition), such that xj - Zi € U for all i = 1,..., n. Therefore, by using Lemma 2 and Lemma 1, we have

xi V ... V xn - zi V ... V zn ^ (—zi + xi) V ... V (—zn + xn) € U V ... V U = U. Since U is solid, similarly, we have

zi V ... V zn - xi V ... V xn ^ (zi - xi) V ... V (zn - xn) € U V ... V U = U.

This means that (xi V ... V xn) - (zi V ... V zn) € U so that xi V ... V xn € Fv + U.

Now, assume that D is the set of all finite subsets of B directed by the inclusion C. For each a € D, put ga = sup a. Observe that {ga} C Bv satisfies ga t- By compactness of Bv, there exists a subnet of (ga) that converges to some g € Bv. Therefore, sup£> = sup£>v = sup{ga} = g. >

Now, we are able to consider a version of the Krengle's theorem [1, Theorem 5.7] for each class of compact operators between locally solid vector lattices. First, we recall some preliminaries which are needed in the sequel.

Suppose X and Y are locally solid vector lattices and T : X - Y is a linear operator. T is called nb-bounded, if there is a zero neighborhood U C X, such that T(U) is also bounded in Y ; T is said to be bb-bounded, if it maps bounded sets into bounded sets.

Moreover, a linear operator T : X —>• Y is said to be n6-compact provided that there is a zero neighborhood IJ C X, such that T(U) is compact in Y; T is 66-compact if for every

bounded set B C X, T(B) is compact in Y. It is obvious that every n6-compact operator is nb-bounded and every bb-compact operator in bb-bounded. These classes of operators enjoy some topological and lattice structures; for a detailed exposition as well as related notions about bounded and compact operators see [2, 7, 8].

Krengel has proved that when the range of a compact operator T between the Banach lattices is an AM-space, then the modulus of T exists and is also compact (see [1, Theorem 5.7]). In the following, we prove this remarkable result for nb-compact operators as well as for bb-compact operators, when the range space is a locally solid AM-space.

Theorem 1. Suppose X is a locally solid vector lattice, Y is a topologically complete locally solid AM-space and T : X ^ Y is a bb-compact operator. Then the modulus of T exists and is also bb-compact.

< Fix a bounded set B C X such that T(B) is totally bounded in Y; by replacing B with Sol(B), if necessary, we may assume that B is solid. Observe that for each x € B+, T[—x,x] is totally bounded in Y so that by Lemma 3, the supremum |T|(x) = sup{|Ty| : |y| ^ x} = supT[—x,x] exists in Y. Thus, by [1, Theorem 1.14], the modulus of T exists. According to Lemma 3, T(B)V is also compact and \T\(x) € T(B)V. Therefore, |T\(B+) C T(B)V. Since B C B+ — B+, we have the desired result. >

Corollary 1. Suppose X is a locally solid vector lattice, Y is a topologically complete locally solid AM-space and T : X ^ Y is an nb-compact operator. Then the modulus of T exists and is also nb-compact.

< Observe that every nb-compact operator is bb-compact. Therefore, by Theorem 1, the modulus of T exists. We need to show that it is also nb-compact. There exists a zero neighborhood IJ C X, such that T(U) is totally bounded in Y. Note that according to Lemma 3, T(U)V is also compact and \T\(x) € T(U)V. Therefore, \T\(U+) C T(U)V. Since U C U+ — U+, the proof would be complete. >

References

1. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, Springer, 2006.

2. Zabeti, O. AM-Spaces from a Locally Solid Vector Lattice Point of View with Applications, Bulletin of the Iranian Mathematical Society, 2021, vol. 47, pp. 1559-1569. DOI: 10.1007/s41980-020-00458-7.

3. Aliprantis, C. D. and Burkinshaw, O. Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, vol. 105, Providence, American Mathematical Society, 2003.

4. Jameson, G. J. O. Topological M-Spaces, Mathematische Zeitschrift, 1968, vol. 103, pp. 139-150. DOI: 10.1007/BF01110626.

5. Zabeti, O. The Banach-Saks Property from a Locally Solid Vector Lattice Point of View, Positivity, 2021, vol. 25, pp. 1579-1583. DOI: 10.1007/s11117-021-00830-9.

6. Jameson, G. J. O. Ordered Linear Spaces, Lecture Notes in Mathematics, vol. 141, Springer, 1970.

7. Erkursun-Ozcan, N., Anil Gezer, N. and Zabeti, O. Spaces of ur-Dunford-Pettis and ur-Compact Operators on Locally Solid Vector Lattices, Matematicki Vesnik, 2019, vol. 71, no. 4, pp. 351—358.

8. Troitsky, V. G. Spectral Radii of Bounded Operators on Topological Vector Spaces, PanAmerican Mathematical Journal, 2001, vol. 11, no. 3, pp. 1-35.

Received August 5, 2022 Omid Zabeti

Department of Mathematics, Faculty of Mathematics, Statistics,

and Computer Science, University of Sistan and Baluchestan,

Zahedan, P.O. Box 98135-674, Iran,

Associate Professor

E-mail: o.zabeti@gmail.com

https://orcid.org/0000-0001-6875-2418

Владикавказский математический журнал 2023, Том 25, Выпуск 3, С. 76-80

ТИП ТЕОРЕМЫ КРЕНГЕЛЯ ДЛЯ КОМПАКТНЫХ ОПЕРАТОРОВ МЕЖДУ ЛОКАЛЬНО ПЛОТНЫМИ ВЕКТОРНЫМИ РЕШЕТКАМИ

Забети О.1

1 Университет Систана и Белуджистана, Иран, 98155-987, Захедан E-mail: o.zabeti@gmail.com

Аннотация. Предположим, что X и Y — локально плотные векторные решетки. Линейный оператор Г : X —У Y называется пЪ-компактным, если существует нулевая окрестность U С X такая, что оператор T(U) компактен в Y. Оператор Г 66-компактен, если для любого ограниченного множества B С X T(B) компактно. Эти понятия далеко не равнозначны, вообще говоря. В этой статье мы вводим понятие локально плотного AM-пространства как расширения для AM-пространств в банаховых решетках. С помощью этого понятия устанавливается вариант известной теоремы Кренгеля для различных типов компактных операторов между локально плотными векторными решетками. Эта теорема распространяется [1, Теорема 5.7] (установленную для компактных операторов между банаховыми решетками) на различные классы компактных операторов между локально телесными векторными решетками.

Ключевые слова: компактный оператор, теорема Кренгеля, локально плотное AM-пространство.

AMS Subject Classification: 46B42, 47B65.

Образец цитирования: Zabeti O. A Type of the Krengel Theorem for Compact Operators between Locally Solid Vector Lattices // Владикавк. мат. журн.—2023.—Т. 25, № 3.—C. 76-80 (in English). DOI: 10.46698/g6863-7709-2981-j.