A new weak Hilbert space Jess Surez de la Fuente, UEx Workshop on - PowerPoint PPT Presentation

A new weak Hilbert space Jesús Suárez de la Fuente, UEx Workshop on Banach spaces and Banach lattices 10 de septiembre de 2019

Introduction 1. Every subspace of a Hilbert space is Hilbert. 2. Every subspace of a Hilbert space is complemented. Theorem (Komorowski and Tomczak-Jaegermann, Gowers) If every subspace of X is isomorphic to X , then X is Hilbert. Theorem (Lindenstrauss-Tzafriri) If every subspace of X is complemented, then X is Hilbert. 2 / 30

Introduction 1. Every subspace of a Hilbert space is Hilbert. 2. Every subspace of a Hilbert space is complemented. Theorem (Komorowski and Tomczak-Jaegermann, Gowers) If every subspace of X is isomorphic to X , then X is Hilbert. Theorem (Lindenstrauss-Tzafriri) If every subspace of X is complemented, then X is Hilbert. 3 / 30

Introduction 1. Every subspace of a Hilbert space is Hilbert. 2. Every subspace of a Hilbert space is complemented. Theorem (Komorowski and Tomczak-Jaegermann, Gowers) If every subspace of X is isomorphic to X , then X is Hilbert. Theorem (Lindenstrauss-Tzafriri) If every subspace of X is complemented, then X is Hilbert. 4 / 30

The notions of type and cotype n n n n 5 / 30 • X has type 2 if a 2 ( X ) = sup n ∈ N a 2 , n ( X ) < ∞ 1/2 � �   � � ∑ ∑ � � ∥ x j ∥ 2 ± x j ≤ a 2 , n ( X ) · . Average ± � �   � � j =1 j =1 � � • X has cotype 2 if c 2 ( X ) = sup n ∈ N c 2 , n ( X ) < ∞ 1/2   � � � � ∑ ∥ x j ∥ 2 ∑ � � ≤ c 2 , n ( X ) · Average ± ± x j . � �   � � j =1 j =1 � � • ℓ p has type min { p , 2 } and cotype max { p , 2 } .

The notions of weak type and cotype of Milman-Pisier X . and weak cotype X is a weak Hilbert space if it is both X weak type Defjnition (Pisier) . C P F with Weak type 2 for X : There is a projection P Theorem (Kwapień) Hilbert. -isomorphic to n -dimensional subspace, say F , that is C contains an , every n -dimensional subspace of X Weak cotype 2 for X : given 6 / 30 X is isomorphic to Hilbert if and only if X has type 2 and cotype 2 . Moreover, the isomorphism constant is bounded by a 2 ( X ) · c 2 ( X ) .

The notions of weak type and cotype of Milman-Pisier Theorem (Kwapień) Hilbert. Defjnition (Pisier) 7 / 30 X is isomorphic to Hilbert if and only if X has type 2 and cotype 2 . Moreover, the isomorphism constant is bounded by a 2 ( X ) · c 2 ( X ) . • Weak cotype 2 for X : given 0 < δ < 1 , every n -dimensional subspace of X contains an ( δ · n ) -dimensional subspace, say F , that is C ( δ ) -isomorphic to • Weak type 2 for X : There is a projection P : X → F with ∥ P ∥ ≤ C ( δ ) . X is a weak Hilbert space if it is both X weak type 2 and weak cotype 2 .

T ? And why this gives a weak Hilbert space? How to get such Kalton-Peck space. 8 / 30 A weak Hilbert space that is a twisted Hilbert space Z ( T 2 ) • T 2 is the prototype of a weak Hilbert space. • A twisted Hilbert space is a Banach space Z containing a copy of ℓ 2 such that Z / ℓ 2 ≈ ℓ 2 • Examples of twisted Hilbert spaces: Enfmo-Lindenstrauss-Pisier space, • Z ( T 2 ) = ℓ 2 ⊕ ℓ 2 with a quasi-norm ∥ ( x , y ) ∥ = ∥ x − Ω T 2 ( y ) ∥ + ∥ y ∥ .

Kalton-Peck space. 9 / 30 A weak Hilbert space that is a twisted Hilbert space Z ( T 2 ) • T 2 is the prototype of a weak Hilbert space. • A twisted Hilbert space is a Banach space Z containing a copy of ℓ 2 such that Z / ℓ 2 ≈ ℓ 2 • Examples of twisted Hilbert spaces: Enfmo-Lindenstrauss-Pisier space, • Z ( T 2 ) = ℓ 2 ⊕ ℓ 2 with a quasi-norm ∥ ( x , y ) ∥ = ∥ x − Ω T 2 ( y ) ∥ + ∥ y ∥ . • How to get such Ω T 2 ? And why this gives a weak Hilbert space?

T is...I have no clue! So then? Coming back to Kalton once more For X , we fjnd the Kalton-Peck map y y log y . The Kalton-Peck space is . For X T then 10 / 30 • Kalton: Given X , X ∗ there is always an Ω X induced by complex interpolation.

T is...I have no clue! So then? Coming back to Kalton once more For X T then 11 / 30 • Kalton: Given X , X ∗ there is always an Ω X induced by complex interpolation. • For X = ℓ 1 , we fjnd the Kalton-Peck map Ω ℓ 1 ( y ) = y log ( y ) . • The Kalton-Peck space is ℓ 2 ⊕ Ω ℓ 1 ℓ 2 .

Coming back to Kalton once more I have no clue! So then? 12 / 30 • Kalton: Given X , X ∗ there is always an Ω X induced by complex interpolation. • For X = ℓ 1 , we fjnd the Kalton-Peck map Ω ℓ 1 ( y ) = y log ( y ) . • The Kalton-Peck space is ℓ 2 ⊕ Ω ℓ 1 ℓ 2 . • For X = T 2 then Ω T 2 is...

Coming back to Kalton once more 13 / 30 • Kalton: Given X , X ∗ there is always an Ω X induced by complex interpolation. • For X = ℓ 1 , we fjnd the Kalton-Peck map Ω ℓ 1 ( y ) = y log ( y ) . • The Kalton-Peck space is ℓ 2 ⊕ Ω ℓ 1 ℓ 2 . • For X = T 2 then Ω T 2 is...I have no clue! So then?

U n X and similarly for U n X log U n X X u j U n X U n X U n X A key step: Castillo, Ferenczi and González j u j j n j n u j X n . u n u u j j n sup For example, the norm of n normalized and disjoint blocks. 14 / 30 • There is information on X and X ∗ that is refmected into Ω X even if you do not know the precise form of such Ω X . • What information?

A key step: Castillo, Ferenczi and González n u j n n 15 / 30 • There is information on X and X ∗ that is refmected into Ω X even if you do not know the precise form of such Ω X . • What information? For example, the norm of n normalized and disjoint blocks. j =1 u j ∥ : u 1 < ... < u n } and similarly for U n ( X ∗ ) . • U n ( X ) = sup {∥ ∑ n • � � � � Ω X ( u j ) − log U n ( X ) ∑ ∑ ∑ √ � � Ω X ( u j ) − ≤ 6 · U n ( X ) · U n ( X ∗ ) . � � U n ( X ∗ ) � � j =1 j =1 j =1 � �

log a n X X x j n X n X n X A random view j . a depends only of a x j j n x j a n j n x j j n X Average Our random version of the inequality using an idea of Corrêa. 16 / 30 • We are interested in a 2 , n ( X ) , a 2 , n ( X ∗ ) .

A random view n n n n 17 / 30 • We are interested in a 2 , n ( X ) , a 2 , n ( X ∗ ) . • Our random version of the inequality using an idea of Corrêa. 1/2 � �   � ± Ω X ( x j ) − log a 2 , n ( X ) � ∑ ∑ ∑ ∑ � � ∥ x j ∥ 2 Ω X ( ± x j ) − ± x j ≤ γ · . Average ± � �   a 2 , n ( X ∗ ) � � j =1 j =1 j =1 j =1 � � • γ depends only of a 2 , n ( X ) , a 2 , n ( X ∗ ) .

18 / 30 Conclusion for our twisted Hilbert Z ( T 2 ) • Our random version of the inequality gives that: a 2 , n ( Z ( T 2 )) ≤ C · max { a 2 , n ( T 2 ) , a 2 , n (( T 2 ) ∗ ) } → ∞ . • In particular, a 2 , n ( Z ( T 2 )) → ∞ very slowly. • Also, c 2 , n ( Z ( T 2 )) → ∞ very slowly (by simple duality).

n Z T n Z T n -dimensional subspaces of V n ARE HILBERTIAN!! Conclusion for certain n -codimensional subspaces V n . The same is a weak Hilbert space by a result of Johnson. Then Z T The argument shows that: Hilbert. We replace Z T -isomorphic to c are a The n -dimensional subspaces of Z T By Kwapień’s result: 19 / 30 • Therefore, a 2 , n ( Z ( T 2 )) · c 2 , n ( Z ( T 2 )) grows slowly to infjnity.

n -dimensional subspaces of V n ARE HILBERTIAN!! Conclusion Hilbert. We replace Z T for certain n -codimensional subspaces V n . The same argument shows that: The Then Z T is a weak Hilbert space by a result of Johnson. 20 / 30 • Therefore, a 2 , n ( Z ( T 2 )) · c 2 , n ( Z ( T 2 )) grows slowly to infjnity. • By Kwapień’s result: ▶ The n -dimensional subspaces of Z ( T 2 ) are a 2 , n ( Z ( T 2 )) · c 2 , n ( Z ( T 2 )) -isomorphic to

n -dimensional subspaces of V n ARE HILBERTIAN!! Conclusion Hilbert. argument shows that: The Then Z T is a weak Hilbert space by a result of Johnson. 21 / 30 • Therefore, a 2 , n ( Z ( T 2 )) · c 2 , n ( Z ( T 2 )) grows slowly to infjnity. • By Kwapień’s result: ▶ The n -dimensional subspaces of Z ( T 2 ) are a 2 , n ( Z ( T 2 )) · c 2 , n ( Z ( T 2 )) -isomorphic to • We replace Z ( T 2 ) for certain n -codimensional subspaces V n . The same

Conclusion Hilbert. argument shows that: 22 / 30 • Therefore, a 2 , n ( Z ( T 2 )) · c 2 , n ( Z ( T 2 )) grows slowly to infjnity. • By Kwapień’s result: ▶ The n -dimensional subspaces of Z ( T 2 ) are a 2 , n ( Z ( T 2 )) · c 2 , n ( Z ( T 2 )) -isomorphic to • We replace Z ( T 2 ) for certain n -codimensional subspaces V n . The same ▶ The 5 (5 n ) -dimensional subspaces of V n ARE HILBERTIAN!! • Then Z ( T 2 ) is a weak Hilbert space by a result of Johnson.

Consequences Z T is no isomorphic to a subspace or a quotient of the Kalton-Peck space or the E-L-P space. Neither the Kalton-Peck space nor the E-L-P space is isomorphic to a subspace or a quotient of Z T . 23 / 30 • Z ( T 2 ) is a new example of weak Hilbert space.

Consequences the E-L-P space. Neither the Kalton-Peck space nor the E-L-P space is isomorphic to a subspace or a quotient of Z T . 24 / 30 • Z ( T 2 ) is a new example of weak Hilbert space. • Z ( T 2 ) is no isomorphic to a subspace or a quotient of the Kalton-Peck space or