A note on lineability
aa r X i v : . [ m a t h . F A ] M a y A NOTE ON LINEABILITY
G. BOTELHO, D. DINIZ, D. PELLEGRINO AND E. TEIXEIRA
Abstract.
In this note we answer a question concerning lineability of the set ofnon-absolutely summing operators. Introduction and main result
A subset A of an infinite-dimensional vector space V is µ -lineable if A ∪ { } containsan infinite-dimensional subspace of dimension µ . Let ℵ be the countable cardinalityand ℵ be the cardinality of R . From now on E and F denote Banach spaces, the spaceof absolutely p -summing linear operators from E to F will be denoted by Π p ( E ; F ) , thespace of bounded linear operators from E to F will be represented by L ( E ; F ) and thespace of compact operators from E to F is represented by K ( E ; F ) . For details on thetheory of absolutely summing operators we refer to [3].In recent papers [1, 5] it was shown that under certain circumstances L ( E ; F ) (cid:31) Π p ( E ; F )is ℵ -lineable. In [1] there is a question from the anonymous referee, asking about thepossibility of proving that the set is µ -lineable, for µ > ℵ . Our next result shows thatan adaptation of the proof of [1] answers this question in the positive:
Theorem 1.1.
Let p ≥ and E be superreflexive. If E contains a complementedinfinite-dimensional subspace with unconditional basis or F contains an infinite un-conditional basic sequence then K ( E ; F ) (cid:31) Π p ( E ; F ) (hence L ( E ; F ) (cid:31) Π p ( E ; F ) ) is ℵ -lineable.Proof. Assume that E contains a complemented infinite-dimensional subspace E withunconditional basis ( e n ) ∞ n =1 . First consider(1.1) N = A ∪ A ∪ · · · a decomposition of N into infinitely many infinite pairwise disjoint subsets ( A j ) ∞ j =1 .Since { e n ; n ∈ N } is an unconditional basis, it is well known that { e n ; n ∈ A j } is anunconditional basic sequence for every j ∈ N . Let us denote by E j the closed span of { e n ; n ∈ A j } . As a subspace of a superreflexive space, E j is superreflexive as well, sofrom [2, Theorem] it follows that for each j there is an operator u j : E j −→ F belonging to K ( E j ; F ) (cid:31) Π p ( E j ; F ). From the proof of [1] we know that each projection P i : E −→ E i is continuous and has norm ≤ ̺ (the constant of the unconditional basis of E ). This also implies that each E i is a complemented subspace of E . If π : E −→ E denotes the projection onto E , for each j ∈ N we can define de operator e u j : E −→ F , e u j := u j ◦ P j ◦ π . Since ( P j ◦ π )( x ) = x for every x ∈ E j , it is plain that e u j belongs to K ( E ; F ) (cid:31) Π p ( E ; F ).There is no loss of generality in supposing k e u j k = 1 for every j . Now, consider the map T : ℓ → K ( E ; F ) T (( a n ) ∞ n =1 ) = ∞ X j =1 a j e u j Since the supports of the f u n are disjoint it is clear that T is an injective linear operator,such that T ( ℓ ) ⊂ ( K ( E ; F ) (cid:31) Π p ( E ; F )) ∪ { } . And therefore ( K ( E ; F ) (cid:31) Π p ( E ; F )) ∪ { } contains a vector space with the same dimen-sion of ℓ (and it is well-known that dim ℓ = ℵ ).Now, suppose that F contains a subspace G with unconditional basis { e n ; n ∈ N } with unconditional basis constant ̺ . Still considering the subsets ( A n ) of N as above,define F j as the closed span of { e n ; n ∈ A j } and let P j : G −→ F j be the correspondingprojections. Proceeding as above we conclude that k P j k ≤ ̺ . From [2, Theorem] weknow that for each j there is an operator u j : E −→ F j belonging to K ( E ; F j ) (cid:31) Π p ( E ; F j ) . Now by e u j we mean the composition of u j with theinclusion from F j to F . Once again consider the map T : ℓ → K ( E ; F ) T (( a n ) ∞ n =1 ) = ∞ X j =1 a j e u j . Since the projections P i : G −→ F i are continuous and have norm ≤ ̺ , it follows that(1.2) k T (( a n ) ∞ n =1 ) ( x ) k ≥ ρ − k α j e u j ( x ) k for every j ∈ N . It is clear that T is a linear and injective. It also follows from (1.2)that T ( ℓ ) ⊂ ( K ( E ; F ) (cid:31) Π p ( E ; F )) ∪ { } . (cid:3) Remark 1.2.
It is not difficult to show that dim L ( ℓ p ; ℓ q ) = ℵ so, for example, for E = ℓ p ( p > and F = ℓ q the result of the previous theorem isoptimal, i.e., we cannot improve the result to µ -lineable for µ > ℵ . NOTE ON LINEABILITY 3 Lineability of the set of norm attaining-operators
Next we show that the same idea of the proof of Theorem 1.1 can be adapted toextend a result from [4] concerning norm-attaining operators.In what follows NA x ( E ; F ) denotes the set of continuous linear operators from E to F that attain their norms at x . Proposition 2.1.
Let E and F be Banach spaces so that E contains an isometric copyof ℓ q for some ≤ q < ∞ , and let x ∈ S E . Then NA x ( E ; F ) is ℵ -lineable in L ( E ; F ) .Proof. The beginning of the proof follows the lines of the similar result from [4]. Itsuffices to prove for F = ℓ q . We can write the set of positive integers N as N = ∞ [ k =1 A k , where each(2.1) A k := { a ( k )1 < a ( k )2 < ... } has the same cardinality as N and the sets A k are pairwise disjoint. For each positiveinteger k , we define ℓ ( k ) q := { x ∈ ℓ q : x j = 0 if j / ∈ A k } . For each k we can find operators u ( k ) on NA x ( E ; ℓ ( k ) q ) . By composing these operatorswith the inclusion of ℓ ( k ) q into ℓ q we get a vector (and we maintain the same notationfor the sake of simplicity) on NA x ( E ; ℓ q ). Consider the map T : ℓ → NA x ( E ; ℓ q ) T (( a n ) ∞ n =1 ) = ∞ X j =1 a j u ( j ) . It is clear that T is linear and injective. We also have that (due the disjoint supportsof the u ( j ) ) T ( ℓ ) ⊂ NA x ( E ; ℓ q ) . Since T is injective, it follows that T ( ℓ ) is an infinite-dimensional space and its basishas the same cardinality of the basis of ℓ . Recall that dim( ℓ ) = ℵ . (cid:3) References [1] G. Botelho, D. Diniz and D. Pellegrino, Lineability of the set of bounded linear non-absolutelysumming operators, J. Math. Anal. Appl. (2009), 171-175.[2] W.J. Davis and W.B. Johnson, Compact non-nuclear operators, Studia Math. (1974), 81-85.[3] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge Studies in Ad-vanced Mathematics 43, 1995.[4] D. Pellegrino and E. Teixeira, Norm optimization problem for linear operators in classical Banachspaces, to appear in Bull. Braz. Math. Soc. G. BOTELHO, D. DINIZ, D. PELLEGRINO AND E. TEIXEIRA [5] D. Puglisi, J. B. Seoane-Sep´ulveda, Bounded linear non-absolutely summing operators, J. Math.Anal. Appl. (2008), 292-298.(Geraldo Botelho)(2008), 292-298.(Geraldo Botelho)