Some Algebra Ideals Of Regular Operators

Erdal Bayram, Witold Wnuk ^∗

Some Algebra Ideals Of Regular Operators

In honour of professor Julian Musielak, on the occasion of his 85th birthday.

Abstract. We investigate relationships joining the order continuity of a norm in a Banach lattice and some composition properties of L-weakly and M-weakly compact operators. Our results improve Proposition 3.6.16 from [5].

2000 Mathematics Subject Classification: 46B42, 47B60, 47B65.

Key words and phrases: L-weakly compact operator, M-weakly compact operator, order continuous norm.

1. Introduction. Let E be a Banach lattice and let W be a linear subset of the space L (E) of all bounded linear operators on E. If for every S ∈ W and for every T ∈ L (E) the compositions ST and T S belong to W then W is called two sided ideal in L (E). For example, weakly compact operators, compact operators, Dunford-Pettis operators are two sided ideals in L (E). Furthermore, semicompact regular operators, AM-compact regular operators are two sided ideals in the space of regular operators L r (E) . It is well known that L-weakly and M-weakly compact operators are closed subspaces of L (E) ([5], Proposition 3.6.15). However, L-weakly and M-weakly compact operators need not to be two sided ideals in L (E) (see [5], Example on page 215). So, what can be said for regular operators?

Example 1.1 Let us consider the inclusion operator i : L ² [0, 1] → L ¹ [0, 1] and the operator S : ` 1 → L <sup>2</sup> [0, 1] defined by S (e n ) = r n where r n is the n−th Rademacher function and {e n : n ∈ N} is the natural basis of ` ¹ . Clearly, i is a positive M- weakly compact operator ([5], Proposition 3.6.20) and S is a regular operator ([3], Example 4.14). On the other hand the composition iS is not M-weakly compact because kiS (e n ) k L

[0,1] = kr ⁿ k L

[0,1] = 1 does not convergent to zero although (e n ) is a norm bounded disjoint sequence of ` 1 .

∗

This research was partially supported by The National Center of Science, Poland, Grant no.

N N 201 605340.

Our next example shows that in general L-weakly and M-weakly compact re- gular operators need not to be two sided ideals in L r (E) for a Banach lattice E.

Example 1.2 Let us consider the Banach lattice E = ` 1 ⊕L ² [0, 1] ⊕L ¹ [0, 1] and the operators U (x, y, z) = (0, S (x) , 0) and V (x, y, z) = (0, 0, i (y)) defined on E where S and i are operators in Example 1. U is a regular operator and V is a regular M -weakly compact operator but the composition V U is not M-weakly compact.

Accordingly, M-weakly compact regular operators need not to be two sided ideals in L r (E). Moreover the composition U ^∗ V ^∗ is not L-weakly compact although the adjoint operator V ^∗ is a regular L-weakly compact and U ^∗ is a regular operator.

Hence we obtain that L-weakly compact regular operators need not to be two sided ideals in L r (E ^∗ ).

A purpose of our short paper is to present some conditions implying that L- weakly and M-weakly compact regular operators are two sided ideals in regular operators L r (E). Firstly, we will formulate two results that characterize Banach lattices with order continuous norms. We prove that a Banach lattice F has an order continuous norm if and only if for all Banach lattices E and G the composition operator ST is L-weakly compact whenever T : E → G is a regular L-weakly compact operator and S : G → F is regular. Dually, we show that norm dual space E ^∗ has an order continuous norm if and only if for all Banach lattices F and G the composition operator T S is M-weakly compact whenever T : E → G is a regular M -weakly compact operator and S : G → F is regular. Applying these results and using similar techniques we obtain that the collection of all L-weakly compact (M- weakly compact) regular operators on E forms a closed two sided ideal in L r (E) if and only if E (resp. E ^∗ ) has order continuous norm.

Throughout this paper, the notion “an operator” will denote a linear norm bo- unded operator. An operator T : E → F is called positive if T (E ⁺ ) ⊆ F ⁺ and we shall denote by L + (E, F ) the collection of all positive operators from the Banach lattice E into the Banach lattice F . Moreover L r (E, F ) = L + (E, F ) − L ⁺ (E, F ) will be the family of all regular operators from E into F . Let us recall that a non- empty norm bounded set A ⊂ E is called L-weakly compact whenever kx ⁿ k → 0 for every disjoint sequence (x n ) contained in the solid hull of A. Operators T : X → E transferring the unit ball in a Banach space X into L-weakly compact sets form the family of L-weakly compact operators. If an operator T : E → X maps norm bounded disjoint sequences onto norm convergent sequences, then T is said to be M-weakly compact. The classes of operators mentioned above were introduced by P. Meyer-Nieberg in [4]. It is worth to remember that L-weakly compact as well as M-weakly compact operators are weakly compact and that the adjoint of an L-weakly compact (M-weakly compact) operator is M-weakly compact (L-weakly compact).

A Banach lattice E has an order continuous norm if x α ↓ 0 in E implies kx ^α k ↓ 0.

All separable σ-Dedekind complete Banach lattices have order continuous norm but

` _∞ and c (the space of convergent sequences with the sup norm) are the best known

examples of Banach lattices without order continuous norms. Every Banach lattice

E contains the maximal closed order ideal E A on which the induced norm is order

continuous, i.e., E A = {x ∈ E : |x| > x ^α ↓ 0 ⇒ kx ^α k → 0}. It is interesting that an L-weakly compact subset in E is contained in E A (see [5], page 92 and page 212).

We refer the reader to [?], [2], [5] for any unexplained terms from the Banach lattice theory and operator theory and [6] for order continuous norms.

2. Auxiliary Results. We start with a few simple lemmas and observations.

In our results we will use the following notations.

X is a Banach space while E, F denote Banach lattices.

L (X, E) = {T : X → E : T is linear and continuous}

F (X, E) = {T ∈ L (X, E) : T is a finite rank operator}

L sc (X, E) = {T ∈ L (X, E) : T is a semi-compact operator}

L Lwc (X, E) = {T ∈ L (X, E) : T is an L-weakly compact operator}

L M wc (E, X) = {T ∈ L (E, X) : T is an M-weakly compact operator}

If X = E then we will shortly write L (E), F (E), L sc (E), L Lwc (E), L M wc (E).

If T ∈ F (X, E) and u 1 , ..., u n is a basis in the range of T , then T has a unique representation (see [1], Lemma 4.2, p.124)

( ∗) T = P n

i=1 f i ⊗ u ⁱ where f 1 , ..., f n ∈ X ^∗ are linearly independent.

Lemma 2.1 Let T ∈ F(X, E) (T ∈ F(E, X)) and let P n

i=1 f i ⊗ u ⁱ , where f i ∈ X ^∗ , u i ∈ E ( f ⁱ ∈ E ^∗ , u i ∈ X) be a representation of T in the form (∗). The follo- wing statements (a 1 ), (a 2 ), (a 3 ) as well as statements (b 1 ), (b 2 ), (b 3 ) are equivalent.

(a 1 ) T ∈ L Lwc (X, E).

(a 2 ) T (X) ⊂ E ^A .

(a 3 ) u i ∈ E ^A for i = 1, . . . , n.

(b 1 ) T ∈ L M sc (E, X).

(b 2 ) T ^∗ (X ^∗ ) ⊂ E ^{∗ A} , where T ^∗ is the adjoint operator of T . (b 3 ) f i ∈ E ^{∗ A} for i = 1, . . . , n.

Proof (a 1 ) ⇒ (a ² ) Obvious because T (B X ) ⊂ E ^A by [5] (see a remark on p. 212 made after Definition 3.6.1).

(a 2 ) ⇒ (a ³ ) The linear independence of f i ’s means that T n

i=1,i6=j Ker f i * Ker f j . Therefore there exists x ∈ X such that f ⁱ (x) = 0 for i 6= j and f ^j (x) 6= 0.

Hence 0 6= f ^j (x) u j = T (x) ∈ E ^A , i.e., u j ∈ E ^A . (a 3 ) ⇒ (a ¹ ) Put v = P n

i=1 kf ⁱ k W n

i=1 |u ⁱ |. Clearly v ∈ E ^A and T (B X ) ⊂ [−v, v], and so sol T (B X ) ⊂ [−v, v] ⊂ E ^A , i.e., T (B X ) is an L-weakly compact set ([2], Theorem 12.12).

Suppose now that T ∈ F (E, X) and let κ : X → X ^∗∗ denote the canonical embedding. The adjoint T ^∗ is of the form T = P n

i=1 κ (u i ) ⊗ f ⁱ .

(b 1 ) ⇒ (b ² ) By [5], Proposition 3.6.11 T ^∗ ∈ L ^Lwc (X ^∗ , E ^∗ ). Using the implica- tion (a 1 ) ⇒ (a ² ) we obtain T ^∗ (X ^∗ ) ⊂ (E ^∗ ) _A .

(b 2 ) ⇒ (b ³ ) is a consequence of (a 2 ) ⇒ (a ³ ) (with T ^∗ instead of T )

(b 3 ) ⇒ (b ¹ ) is an immediate corollary of (a 3 ) ⇒ (a ¹ ) and [5], Proposition 3.6.11. _

Lemma 2.2 For a Banach lattice E the following statements are equivalent.

(a) E has order continuous norm.

(b) L sc (X, E) = L Lwc (X, E) for an arbitrary Banach space X.

(c) L sc (E) = L Lwc (E).

(d) F(E) ⊂ L Lwc (E).

(e) F(X, E) ⊂ L Lwc (X, E) for an arbitrary Banach space X.

(f) Every continuous rank one operator T : E → E is L-weakly compact.

(g) Every continuous E-valued rank one operator is L-weakly compact.

Proof (a) ⇒ (b) According to [6] Proposition 1.3 families of almost order bounded and L-weakly compact subsets of E coincide.

(b) ⇒ (e) If T = P n

i=1 f i ⊗ u ⁱ for f i ∈ X ^∗ and u i ∈ E, then for v = P n

i=1 kf ⁱ k W n

i=1 |u ⁱ | there holds |T (x)| ≤ v whenever kxk ≤ 1, i.e., T ∈ L ^sc (X, E), and so T ∈ L ^Lwc (X, E) . The same arguments show (c) ⇒ (d).

(b) ⇒ (c), (e) ⇒ (g), (d) ⇒ (f), (g) ⇒ (f) are obvious.

(f ) ⇒ (a) Let x ∈ E \{0}. Fix f ∈ E ^∗ with f (x) = 1. The operator T = f ⊗x is L-weakly compact and _kxk ^x = T 

x kxk

 ∈ T (B ^E ) ⊂ E ^A by Lemma 2.1, i.e., x ∈ E ^A .

We have just proved E = E A . 

Our succeeding Lemma is a simple consequence of the Hahn-Banach extension theorem.

Lemma 2.3 Let X, Y be non zero Banach spaces. For arbitrary non zero x 0 ∈ X, y 0 ∈ Y there exists a continuous rank one operator T : X → Y such that T (x ⁰ ) = y 0 . If X, Y are Banach lattices and x 0 , y 0 are positive, then T can be chosen positive.

3. Main Results. We start with some operator characterizations of the order continuity of a norm.

Theorem 3.1 For a Banach lattice F the following statements are equivalent.

(a) F has an order continuous norm.

(b) For all Banach lattices E and G the composition operator ST : E → F is L-weakly compact for every T ∈ L ^Lwc (E, G) and every S ∈ L ^r (G, F ) . (c) For every Banach lattice E and all Banach lattices G with order continuous

norm the composition operator ST : E → F is L-weakly compact for every

T ∈ L ^Lwc (E, G) and every S ∈ L ^r (G, F ) .

(d) For every Banach lattice E and all discrete Banach lattices G with order continuous norm the composition operator ST : E → F is L-weakly compact for every T ∈ L Lwc (E, G) and every S ∈ L r (G, F ).

(e) For every Banach lattice E 6= {0} the composition operator ST : E → F is L-weakly compact for every positive T ∈ L Lwc (E, ` <sup>1</sup> ) and every positive S : ` ¹ → F .

Proof (a) ⇒ (b) Since L Lwc (E, G) ⊂ L ^sc (E, G) by [5] Proposition 3.6.10, then the regularity of S and Lemma 2.2 imply ST ∈ L Lwc (E, F ).

(b) ⇒ (c) ⇒ (d) ⇒ (e) are obvious.

(e) ⇒ (a) Suppose F 6= F ^A . Choose x ∈ E, kxk = 1, 0 6= y ∈ ` <sup>1</sup> , z ∈ F \ F <sup>A</sup> . By Lemma 2.3 there are rank one operators S : E → ` ¹ and T : ` ¹ → F such that S(x) = y, T (y) = z. According to Lemma 2.1 S is L-weakly compact and our assumption implies T S is L-weakly compact too. Using Lemma 2.1 again we obtain

T S (x) = z ∈ F ^A , a contradiction. 

Since the dual of an L-weakly compact operator is M-weakly compact, we obtain the following theorem.

Theorem 3.2 For a Banach lattice E the following statements are equivalent.

(a) The dual E ^∗ has an order continuous norm.

(b) For all Banach lattices F and G the composition operator ST : E → F is M-weakly compact for every T ∈ L r (E, G) and every S ∈ L M wc (G, F ).

(c) For every Banach lattice E and all Banach lattices G with order continuous norm the composition operator ST : E → F is M-weakly compact for every T ∈ L ^r (E, G) and every S ∈ L M wc (G, F ).

(d) For every Banach lattice E and all discrete Banach lattices G with order continuous norm the composition operator ST : E → F is M-weakly compact for every T ∈ L r (E, G) and every S ∈ L M wc (G, F ).

(e) For every Banach lattice F 6= {0} the composition operator ST : E → F is M-weakly compact for every positive S ∈ L M wc (c 0 , F ) and every positive T : E → c ⁰ .

Proof (a) ⇒ (b) According to [5] Proposition 3.6.11 the adjoint S ^∗ is L-weakly compact and using Theorem 3.1 we obtain T ^∗ S ^∗ = (ST ) ^∗ is L-weakly compact.

Applying [5] Proposition 3.6.11 again we confirm that ST is M-weakly compact.

(b) ⇒ (c) ⇒ (d) ⇒ (e) are obvious.

(e) ⇒ (a) Suppose that there exists a positive f ∈ E ^∗ \E ^{∗ A} . Choose 0 < w ∈ c ⁰ , 0 < g ∈ c ^∗ 0 with g (w) = 1, and 0 < y ∈ F . The operators T = f ⊗ w, S = g ⊗ y are positive and S is M-weakly compact by Lemma 2.1. Therefore ST = f ⊗y should be M -weakly compact, but using Lemma 2.1 again we obtain a contradiction because

f / ∈ E ^{∗ A} . 

Succeeding two results contain sufficient and necessary conditions for L-weakly and M-weakly compact operators to form two sided ideals in regular operators.

Theorem 3.3 For a Banach lattice E with E A 6= {0} the following statements are equivalent.

(a) L Lwc (E) ∩ L ^r (E) is a two sided ideal in L r (E).

(b) ST ∈ L Lwc (E) for every T ∈ L Lwc (E) ∩ L ^r (E) and every S ∈ F(E).

(c) ST ∈ L Lwc (E) for every T ∈ L Lwc (E) ∩ L ^r (E) and every continuous rank one operator S : E → E.

(d) E has an order continuous norm.

Proof (a) ⇒ (b) ⇒ (c) are obvious.

(c) ⇒ (d) Suppose there exists z ∈ E ⁺ \ E ^A . Fix 0 < y ∈ E A and 0 < x ∈ E.

By Lemma 2.3 there exists positive rank one operators T, S : E → E such that T (x) = y and S (y) = z. Lemma 2.1 implies that T is L-weakly compact, and so ST has to be L-weakly compact which is impossible by the same Lemma because z ∈ (ST ) (E) \ E ^A .

(d) ⇒ (a) If T ∈ L ^Lwc (E) ∩L ^r (E) and S ∈ L r (E) , then clearly T S ∈ L Lwc (E) ∩ L r (E) and ST ∈ L Lwc (E) ∩ L ^r (E) by Theorem 3.1. _

Next theorem is a dual version of Theorem 3.3.

Theorem 3.4 For a Banach lattice E with E ^∗ A 6= {0} the following statements are equivalent.

(a) L M wc (E) ∩ L ^r (E) is a two sided ideal in L r (E).

(b) T S ∈ L ^{M wc} (E) for every T ∈ L ^{M wc} (E) ∩ L ^r (E) and every S ∈ F(E).

(c) T S ∈ L ^{M wc} (E) for every T ∈ L ^{M wc} (E) ∩ L ^r (E) and every continuous rank one operator S : E → E.

(d) The dual E ^∗ has an order continuous norm.

Proof (a) ⇒ (b) ⇒ (c) are obvious.

(c) ⇒ (d) Suppose there exists f ∈ E + ^∗ \ E ^{∗ A} and let 0 < g ∈ E ^{∗ A} . Choose x ∈ E ⁺ \ Ker f, y ∈ E ⁺ , g (y) = 1 and consider operators T = g ⊗ x, S = f ⊗ y.

Using Lemma 2.1 we obtain that T is M-weakly compact, and so T S = f ⊗x has to be M-weakly compact, a contradiction because according to Lemma 2.1 it should be f ∈ E ^∗ A .

(d) ⇒ (a) If T ∈ L ^{M wc} (E) ∩ L ^r (E) and S ∈ L ^r (E) , then clearly ST ∈

L M wc (E) ∩ L ^r (E) and T S ∈ L ^{M wc} (E) ∩ L ^r (E) by Theorem 3.2. 

Remark 3.5 The equality E A = {0} (resp. E ^{∗ A} = {0}) is equivalent to the fact that T = 0 is a unique E-valued L-weakly compact (resp. M-weakly compact) operator, and so considering such type of operators it is naturally to assume E A 6=

{0} (resp. E ^{∗ A} 6= {0}) in Theorem 3.3 (resp. Theorem 3.4).

References

[1] Y.A. Abramovich, C.D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics vol 50, Amer. Math. Soc., 2002. Zbl 1022.47001

[2] C.D. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press, New York-London, 1985.

Zbl 0608.47039

[3] P.G. Dodds, D.H. Fremlin, Compact operators in Banach lattices, Israel J. Math. 34 (1979), 287-320. Zbl 0438.47042

[4] P. Meyer-Nieberg, Über klassen schwach kompakter operatoren in Banachverbänden, Math.

Z. 138 (1974), 145-159. Zbl 0291.47020

[5] P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin Heidelberg New York, 1991. Zbl 0743.46015

[6] W. Wnuk, Banach Lattices with Order Continuous Norms, Advenced Topics in Mathematics, Polish Scientific Publishers PWN, Warsaw, 1999. Zbl 0948.46017

Erdal Bayram

Namik Kemal University, Faculty of Science and Arts, Mathematics Department Tekirdag, Turkey

E-mail: ebayram@nku.edu.tr Witold Wnuk

Faculty of Mathematics and Computer Science, A. Mickiewicz University Umultowska 87, 61–614 Poznań, Poland

E-mail: wnukwit@amu.edu.pl

(Received: 9.10.2013)