The convex hull of a Banach-Saks set Pedro Tradacete Universidad - PowerPoint PPT Presentation

The convex hull of a Banach-Saks set Pedro Tradacete Universidad Carlos III de Madrid Joint work with J. Lopez-Abad (CSIC) and C. Ruiz-Bermejo (UCM) Congreso de la RSME 22 de enero de 2013 P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 1 / 1

Convergent sequences Let ( X , � · � ) be a Banach space. ( x n ) n ⊂ X x n − → x ⇔ � x n − x � → 0 n � 1 � � Cesaro � x n − → x ⇔ x j − x � → 0 � � n j = 1 x ∗ ( x n − x ) → 0 , ∀ x ∗ ∈ X ∗ w x n − → x ⇔ Cesaro w x n − → x ⇒ x n − → x ⇒ x n − → x P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 2 / 1

Definition A subset A ⊆ X of a Banach space is called Banach-Saks if every sequence in A has a Ces` aro convergent subsequence. Examples: The unit basis of c 0 , ℓ p , p > 1 are Banach-Saks (and weakly-null) 1 The unit basis of ℓ 1 is not (and it is not weakly-null) 2 The unit basis of the Schreier space X S is not, but it is weakly-null. 3 Recall X S is the completion of c 00 under the norm given by � � ( a n ) � X S = sup | a n | , E ∈S n ∈ E where S is the class of finite sets of the form { n 1 < n 2 < · · · < n k } with k ≤ n 1 . P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 3 / 1

Convex hulls A compact ⇒ A Banach-Saks ⇒ A weakly-compact. Given A ⊂ X , � � � � co ( A ) := closure λ i x i : λ i ≥ 0 , λ i ≤ 1 , x i ∈ A i i A compact ⇒ co ( A ) compact (Mazur). A weakly-compact ⇒ co ( A ) weakly-compact (Krein-Smulian) Question: A Banach-Saks ⇒ co ( A ) Banach-Saks? P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 4 / 1

Positive results A Banach space has the weak Banach-Saks property if every weakly convergent sequence has a Ces` aro convergent subsequence. Examples: L p (1 ≤ p < ∞ ), c 0 , . . . Proposition Let X have the weak Banach-Saks property. A ⊂ X is Banach-Saks if and only if co ( A ) is Banach-Saks. P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 5 / 1

Positive results A sequence ( x n ) n in a Banach space X is weakly uniformly convergent to x ∈ X if for every ε > 0, there is n ( ε ) ∈ N such that for every x ∗ ∈ X ∗ # { n ∈ N : | x ∗ ( x n − x ) | ≥ ε } ≤ n ( ε ) . Theorem (Mercourakis) Cesaro ( x n ) n converges uniformly weakly to x ⇔ ∀ ( x n k ) k , x n k − → x. Proposition If ( x n ) n is uniformly weakly convergent ⇒ co ( { x n } ) is Banach-Saks. P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 6 / 1

Schreier spaces Theorem (Gonz´ alez-Rodr´ ıguez) A ⊆ X S is Banach-Saks if and only if co ( A ) is Banach-Saks. The Schreier family S can be extended by induction � � S 2 = S⊗S = s 1 ∪· · ·∪ s n : s i ∈ S , s 1 < · · · < s n , { min ( s 1 ) , . . . , min ( s n ) }∈ S S 3 = S 2 ⊗ S . . . S α can be defined for any countable ordinal α . Theorem A ⊆ X S α is Banach-Saks if and only if co ( A ) is Banach-Saks. P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 7 / 1

Towards a counterexample Let F ⊆ [ N ] <ω be a compact family. � � ( a n ) � X F = sup | a n | E ∈F n ∈ E Definition A family F is large in M when for every n ∈ N and N ⊆ M there is s ∈ F such that #( s ∩ N ) ≥ n . Definition A T -family is a compact and hereditary family F on N such that: (1) F is never large in any M ⊆ N . (2) There is a partition � n I n = N in finite sets I n and δ > 0 such that G δ ( F ) := { t ⊆ N : there is s ∈ F with #( s ∩ I n ) ≥ δ # I n for all n ∈ t } is large. P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 8 / 1

Theorem There is a T-family. In fact, for every ε > 0 there is a family F such that F is not 4-large in any M. 1 G 1 − ε ( F ) = S . 2 Therefore, in X F every subsequence of the unit basis ( u n ) n has a subsequence 4-equivalent to the unit basis of c 0 . In particular, ( u n ) n is Banach-Saks. 1 � j ∈ I n u j ∈ co ( { u n } ) is equivalent to the While, the sequence x n = # I n unit basis of Schreier space X S . Thus, ( x n ) n is not Banach-Saks. P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 9 / 1

Proof of the Theorem: Please, go to http://arxiv.org/abs/1209.4851 P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 10 / 1

Proposition (Gillis) For every ε > 0 , δ > 0 and m ∈ N , there is n := n ( ε, δ, m ) such that for every probability space (Ω , F , µ ) and every sequence ( A i ) i < n with µ ( A i ) ≥ ε for all i < n, there is s ⊂ { 1 , . . . , n } with ♯ s = m such that � A i ) ≥ ( 1 − δ ) ε m . µ ( i ∈ s A key idea in the proof of our theorem is the following construction by Erd˝ os and Hajnal: Let r , n ∈ N . Given i < j < n , let A i , j := { ( a k ) k < n ∈ r n : a i � = a j } . Clearly # A i , j = r n − 1 ( r − 1 ) . Now if s ⊆ n has cardinality ≥ r + 1, then � A i , j = ∅ . { i , j }∈ [ s ] 2 This provides a counterexample for double-indexed sequences of the expected generalization of Gillis’ result. P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 11 / 1

Thank you very much for your attention. P . Tradacete (UC3M) The convex hull of a Banach-Saks set RSME 2013 12 / 1