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.

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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.

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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.

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The notions of type and cotype

• X has type 2 if a2(X) = supn∈N a2,n(X) < ∞

Average±

• n

∑

j=1

±xj

• ≤ a2,n(X) ·

 

n

∑

j=1

∥xj∥2  

1/2

.

• X has cotype 2 if c2(X) = supn∈N c2,n(X) < ∞

 

n

∑

j=1

∥xj∥2  

1/2

≤ c2,n(X) · Average±

• n

∑

j=1

±xj

• .
• ℓp has type min{p, 2} and cotype max{p, 2}.

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The notions of weak type and cotype of Milman-Pisier

Theorem (Kwapień)

X is isomorphic to Hilbert if and only if X has type 2 and cotype 2. Moreover, the isomorphism constant is bounded by a2(X) · c2(X). Weak cotype 2 for X: given , every n-dimensional subspace of X contains an n -dimensional subspace, say F, that is C

• isomorphic to

Hilbert. Weak type 2 for X: There is a projection P X F with P C .

Defjnition (Pisier)

X is a weak Hilbert space if it is both X weak type and weak cotype .

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The notions of weak type and cotype of Milman-Pisier

Theorem (Kwapień)

X is isomorphic to Hilbert if and only if X has type 2 and cotype 2. Moreover, the isomorphism constant is bounded by a2(X) · c2(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 Hilbert.

• Weak type 2 for X: There is a projection P : X → F with ∥P∥ ≤ C(δ).

Defjnition (Pisier)

X is a weak Hilbert space if it is both X weak type 2 and weak cotype 2.

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A weak Hilbert space that is a twisted Hilbert space Z(T2)

• T2 is the prototype of a weak Hilbert space.
• A twisted Hilbert space is a Banach space Z containing a copy of ℓ2 such that

Z/ℓ2 ≈ ℓ2

• Examples of twisted Hilbert spaces: Enfmo-Lindenstrauss-Pisier space,

Kalton-Peck space.

• Z(T2) = ℓ2 ⊕ ℓ2 with a quasi-norm

∥(x, y)∥ = ∥x − ΩT2(y)∥ + ∥y∥. How to get such

T ? And why this gives a weak Hilbert space?

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A weak Hilbert space that is a twisted Hilbert space Z(T2)

• T2 is the prototype of a weak Hilbert space.
• A twisted Hilbert space is a Banach space Z containing a copy of ℓ2 such that

Z/ℓ2 ≈ ℓ2

• Examples of twisted Hilbert spaces: Enfmo-Lindenstrauss-Pisier space,

Kalton-Peck space.

• Z(T2) = ℓ2 ⊕ ℓ2 with a quasi-norm

∥(x, y)∥ = ∥x − ΩT2(y)∥ + ∥y∥.

• How to get such ΩT2? And why this gives a weak Hilbert space?

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Coming back to Kalton once more

• Kalton: Given X, X∗ there is always an ΩX induced by complex interpolation.

For X , we fjnd the Kalton-Peck map y y log y . The Kalton-Peck space is . For X T then

T is...I have no clue! So then?

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Coming back to Kalton once more

• Kalton: Given X, X∗ there is always an ΩX induced by complex interpolation.
• For X = ℓ1, we fjnd the Kalton-Peck map Ωℓ1(y) = y log(y).
• The Kalton-Peck space is ℓ2 ⊕Ωℓ1 ℓ2.

For X T then

T is...I have no clue! So then?

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Coming back to Kalton once more

• Kalton: Given X, X∗ there is always an ΩX induced by complex interpolation.
• For X = ℓ1, we fjnd the Kalton-Peck map Ωℓ1(y) = y log(y).
• The Kalton-Peck space is ℓ2 ⊕Ωℓ1 ℓ2.
• For X = T2 then ΩT2 is...

I have no clue! So then?

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Coming back to Kalton once more

• Kalton: Given X, X∗ there is always an ΩX induced by complex interpolation.
• For X = ℓ1, we fjnd the Kalton-Peck map Ωℓ1(y) = y log(y).
• The Kalton-Peck space is ℓ2 ⊕Ωℓ1 ℓ2.
• For X = T2 then ΩT2 is...I have no clue! So then?

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A key step: Castillo, Ferenczi and González

• There is information on X and X∗ that is refmected into ΩX even if you do not

know the precise form of such ΩX.

• What information?

For example, the norm of n normalized and disjoint blocks. Un X sup

n j

uj u un and similarly for Un X .

X n j

uj

n j X uj

log Un X Un X

n j

uj Un X Un X

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A key step: Castillo, Ferenczi and González

• There is information on X and X∗ that is refmected into ΩX even if you do not

know the precise form of such ΩX.

• What information? For example, the norm of n normalized and disjoint blocks.
• Un(X) = sup{∥ ∑n

j=1 uj∥ : u1 < ... < un} and similarly for Un(X∗).

• ΩX(

n

∑

j=1

uj) −

n

∑

j=1

ΩX(uj) − log Un(X) Un(X∗)

n

∑

j=1

uj

• ≤ 6 ·

√ Un(X) · Un(X∗).

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A random view

• We are interested in a2,n(X), a2,n(X∗).

Our random version of the inequality using an idea of Corrêa. Average

X n j

xj

n j X xj

log a

n X

a

n X n j

xj

n j

xj depends only of a

n X

a

n X

.

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A random view

• We are interested in a2,n(X), a2,n(X∗).
• Our random version of the inequality using an idea of Corrêa.

Average±

• ΩX(

n

∑

j=1

±xj) −

n

∑

j=1

±ΩX(xj) − log a2,n(X) a2,n(X∗)

n

∑

j=1

±xj

• ≤ γ·

 

n

∑

j=1

∥xj∥2  

1/2

.

• γ depends only of a2,n(X), a2,n(X∗).

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Conclusion for our twisted Hilbert Z(T2)

• Our random version of the inequality gives that:

a2,n(Z(T2)) ≤ C · max{a2,n(T2), a2,n((T2)∗)} → ∞.

• In particular, a2,n(Z(T2)) → ∞ very slowly.
• Also, c2,n(Z(T2)) → ∞ very slowly (by simple duality).

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Conclusion

• Therefore, a2,n(Z(T2)) · c2,n(Z(T2)) grows slowly to infjnity.

By Kwapień’s result: The n-dimensional subspaces of Z T are a

n Z T

c

n Z T

• isomorphic to

Hilbert. We replace Z T for certain n-codimensional subspaces Vn. The same argument shows that: The

n -dimensional subspaces of Vn ARE HILBERTIAN!!

Then Z T is a weak Hilbert space by a result of Johnson.

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Conclusion

• Therefore, a2,n(Z(T2)) · c2,n(Z(T2)) grows slowly to infjnity.
• By Kwapień’s result:

▶ The n-dimensional subspaces of Z(T2) are a2,n(Z(T2)) · c2,n(Z(T2))-isomorphic to Hilbert. We replace Z T for certain n-codimensional subspaces Vn. The same argument shows that: The

n -dimensional subspaces of Vn ARE HILBERTIAN!!

Then Z T is a weak Hilbert space by a result of Johnson.

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Conclusion

• Therefore, a2,n(Z(T2)) · c2,n(Z(T2)) grows slowly to infjnity.
• By Kwapień’s result:

▶ The n-dimensional subspaces of Z(T2) are a2,n(Z(T2)) · c2,n(Z(T2))-isomorphic to Hilbert.

• We replace Z(T2) for certain n-codimensional subspaces Vn. The same

argument shows that: The

n -dimensional subspaces of Vn ARE HILBERTIAN!!

Then Z T is a weak Hilbert space by a result of Johnson.

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Conclusion

• Therefore, a2,n(Z(T2)) · c2,n(Z(T2)) grows slowly to infjnity.
• By Kwapień’s result:

▶ The n-dimensional subspaces of Z(T2) are a2,n(Z(T2)) · c2,n(Z(T2))-isomorphic to Hilbert.

• We replace Z(T2) for certain n-codimensional subspaces Vn. The same

argument shows that: ▶ The 5(5n)-dimensional subspaces of Vn ARE HILBERTIAN!!

• Then Z(T2) is a weak Hilbert space by a result of Johnson.

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Consequences

• Z(T2) is a new example of weak Hilbert space.

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 .

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Consequences

• Z(T2) is a new example of weak Hilbert space.
• Z(T2) 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 .

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Consequences

• Z(T2) is a new example of weak Hilbert space.
• Z(T2) 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(T2).

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Some answers

• Z(T2) answers a question of Cuellar: Do there exists nontrivial twisted Hilbert

spaces that are weak Hilbert spaces? Yes. Z T solves the “weak type 2 converse” of Milman-Pisier (see Casazza-Shura, Castillo-Plichko,...): Does weak type 2 implies Maurey extension property? No. Z T satisfjes the J-L lemma and thus it also answers a question of Johnson-Naor: It was not known if weak Hilbert spaces with no unconditional basis may satisfy the J-L lemma.

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Some answers

• Z(T2) answers a question of Cuellar: Do there exists nontrivial twisted Hilbert

spaces that are weak Hilbert spaces? Yes. Z T solves the “weak type 2 converse” of Milman-Pisier (see Casazza-Shura, Castillo-Plichko,...): Does weak type 2 implies Maurey extension property? No. Z T satisfjes the J-L lemma and thus it also answers a question of Johnson-Naor: It was not known if weak Hilbert spaces with no unconditional basis may satisfy the J-L lemma.

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Some answers

• Z(T2) answers a question of Cuellar: Do there exists nontrivial twisted Hilbert

spaces that are weak Hilbert spaces? Yes.

• Z(T2) solves the “weak type 2 converse” of Milman-Pisier (see Casazza-Shura,

Castillo-Plichko,...): Does weak type 2 implies Maurey extension property? No. Z T satisfjes the J-L lemma and thus it also answers a question of Johnson-Naor: It was not known if weak Hilbert spaces with no unconditional basis may satisfy the J-L lemma.

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Some answers

• Z(T2) answers a question of Cuellar: Do there exists nontrivial twisted Hilbert

spaces that are weak Hilbert spaces? Yes.

• Z(T2) solves the “weak type 2 converse” of Milman-Pisier (see Casazza-Shura,

Castillo-Plichko,...): Does weak type 2 implies Maurey extension property? No.

• Z(T2) satisfjes the J-L lemma and thus it also answers a question of

Johnson-Naor: It was not known if weak Hilbert spaces with no unconditional basis may satisfy the J-L lemma.

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References

• 1. J. Suárez, A weak Hilbert space that is a twisted Hilbert space. J. Inst. Math.

Jussieu (to appear).

• 2. J. Suárez, A space with no unconditional basis that satisfjes the J-L lemma.

Results in Mathematics, 74 (2019), no. 3, Art. 126.

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