Disjoint sequences in Banach lattices Pedro Tradacete Mathematics - - PowerPoint PPT Presentation

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Disjoint sequences in Banach lattices Pedro Tradacete Mathematics - - PowerPoint PPT Presentation

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Disjoint sequences in Banach lattices Pedro Tradacete Mathematics Department, UC3M Based on joint work with J. Flores, F. Hern andez, E. Semenov, E. Spinu, V. Troitsky First Brazilian Workshop in Geometry of Banach Spaces 25-29 August 2014,

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Disjoint sequences in Banach lattices

Pedro Tradacete

Mathematics Department, UC3M Based on joint work with J. Flores, F. Hern´ andez, E. Semenov, E. Spinu, V. Troitsky

First Brazilian Workshop in Geometry of Banach Spaces 25-29 August 2014, Maresias

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Disjointly homogeneous Banach lattices:

Definition

E is disjointly homogeneous (DH) ⇔ ∀ (xn), (yn) normalized disjoint in E, ∃ (nk) such that

  • ∞
  • k=1

akxnk

  • ∼
  • ∞
  • k=1

akynk

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SLIDE 3

Disjointly homogeneous Banach lattices:

Definition

E is disjointly homogeneous (DH) ⇔ ∀ (xn), (yn) normalized disjoint in E, ∃ (nk) such that

  • ∞
  • k=1

akxnk

  • ∼
  • ∞
  • k=1

akynk

  • Examples: Lp, Lorentz spaces Lp,q, Λ(W , p), . . .
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SLIDE 4

Disjointly homogeneous Banach lattices:

Definition

E is disjointly homogeneous (DH) ⇔ ∀ (xn), (yn) normalized disjoint in E, ∃ (nk) such that

  • ∞
  • k=1

akxnk

  • ∼
  • ∞
  • k=1

akynk

  • Examples: Lp, Lorentz spaces Lp,q, Λ(W , p), . . .

Definition

E is p-disjointly homogeneous (p-DH) if every normalized disjoint sequence (xn) in E has a subsequence such that

  • ∞
  • k=1

akxnk

  • ∼

∞

  • k=1

|ak|p1/p ( sup

k

|ak| in case p = ∞)

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SLIDE 5

Applications of DH Banach lattices

Theorem

E DH with finite cotype and unconditional basis. T ∈ SS(E) ⇒ T 2 ∈ K(E)

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Applications of DH Banach lattices

Theorem

E DH with finite cotype and unconditional basis. T ∈ SS(E) ⇒ T 2 ∈ K(E)

Theorem

E 1-DH with finite cotype. T ∈ SS(E) ⇒ T ∈ DP(E)

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Applications of DH Banach lattices

Theorem

E DH with finite cotype and unconditional basis. T ∈ SS(E) ⇒ T 2 ∈ K(E)

Theorem

E 1-DH with finite cotype. T ∈ SS(E) ⇒ T ∈ DP(E)

Theorem

E 2-DH with finite cotype. T ∈ SS(E) ⇒ T ∈ K(E)

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Applications of DH Banach lattices

Theorem

E DH with finite cotype and unconditional basis. T ∈ SS(E) ⇒ T 2 ∈ K(E)

Theorem

E 1-DH with finite cotype. T ∈ SS(E) ⇒ T ∈ DP(E)

Theorem

E 2-DH with finite cotype. T ∈ SS(E) ⇒ T ∈ K(E)

Theorem

E discrete with a disjoint basis and DH. T ∈ SS(E) ⇒ T ∈ K(E)

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Duality

Question: Is the property DH stable by duality?

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Duality

Question: Is the property DH stable by duality? Known-facts:

◮ E ∞-DH ⇒ E ∗ 1-DH.

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Duality

Question: Is the property DH stable by duality? Known-facts:

◮ E ∞-DH ⇒ E ∗ 1-DH. ◮ Lp,1 is 1-DH but L∗ p,1 = Lp′,∞ is not DH.

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Duality

Question: Is the property DH stable by duality? Known-facts:

◮ E ∞-DH ⇒ E ∗ 1-DH. ◮ Lp,1 is 1-DH but L∗ p,1 = Lp′,∞ is not DH. ◮ Maybe for E reflexive?

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SLIDE 13

Duality

Question: Is the property DH stable by duality? Known-facts:

◮ E ∞-DH ⇒ E ∗ 1-DH. ◮ Lp,1 is 1-DH but L∗ p,1 = Lp′,∞ is not DH. ◮ Maybe for E reflexive?

We will see that in general the answer is negative

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Positive results

Definition

A Banach lattice E has property P if for every disjoint positive normalized sequence (fn) ⊂ E there exists a positive operator T : E → [fn], such that T ∗f ∗

n 0.

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Positive results

Definition

A Banach lattice E has property P if for every disjoint positive normalized sequence (fn) ⊂ E there exists a positive operator T : E → [fn], such that T ∗f ∗

n 0.

Theorem

Let E be a reflexive Banach lattice with property P. E ∗ DH ⇒ E DH

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Positive results

Definition

A Banach lattice E has property P if for every disjoint positive normalized sequence (fn) ⊂ E there exists a positive operator T : E → [fn], such that T ∗f ∗

n 0.

Theorem

Let E be a reflexive Banach lattice with property P. E ∗ DH ⇒ E DH E ∗ p − DH ⇒ E q − DH 1 p + 1 q = 1

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Positive results

Definition

A Banach lattice E has property P if for every disjoint positive normalized sequence (fn) ⊂ E there exists a positive operator T : E → [fn], such that T ∗f ∗

n 0.

Theorem

Let E be a reflexive Banach lattice with property P. E ∗ DH ⇒ E DH E ∗ p − DH ⇒ E q − DH 1 p + 1 q = 1

  • Corollary

Let E be a reflexive Banach lattice satisfying an upper p-estimate. E ∗ q − DH ⇒ E p − DH 1 p + 1 q = 1

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Orlicz spaces

Theorem

An Orlicz space Lϕ(0, 1) is p-DH ⇔ E ∞

ϕ ∼

= {tp}. E ∞

ϕ =

  • s>0

ϕ(r·) ϕ(r) : r ≥ s

  • .
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SLIDE 19

Orlicz spaces

Theorem

An Orlicz space Lϕ(0, 1) is p-DH ⇔ E ∞

ϕ ∼

= {tp}. E ∞

ϕ =

  • s>0

ϕ(r·) ϕ(r) : r ≥ s

  • .

l´ ım

t→∞

tϕ′(t) ϕ(t) = p ⇒ E ∞

ϕ ∼

= {tp}

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Orlicz spaces

Theorem

An Orlicz space Lϕ(0, 1) is p-DH ⇔ E ∞

ϕ ∼

= {tp}. E ∞

ϕ =

  • s>0

ϕ(r·) ϕ(r) : r ≥ s

  • .

l´ ım

t→∞

tϕ′(t) ϕ(t) = p ⇒ E ∞

ϕ ∼

= {tp} Remark: Lϕ(0, 1) is p-DH ⇔ L∗

ϕ(0, 1) is q-DH ( 1 p + 1 q = 1).

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Orlicz spaces

Theorem

An Orlicz space Lϕ(0, 1) is p-DH ⇔ E ∞

ϕ ∼

= {tp}. E ∞

ϕ =

  • s>0

ϕ(r·) ϕ(r) : r ≥ s

  • .

l´ ım

t→∞

tϕ′(t) ϕ(t) = p ⇒ E ∞

ϕ ∼

= {tp} Remark: Lϕ(0, 1) is p-DH ⇔ L∗

ϕ(0, 1) is q-DH ( 1 p + 1 q = 1).

Theorem

A separable Orlicz space Lϕ(0, ∞) is p-DH ⇔ Cϕ(0, ∞) ∼ = {tp}. Cϕ(0, ∞) = conv {F ∈ C(0, 1) | ∃s > 0, F(·) = ϕ(s·) ϕ(s) }.

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Counterexemples

Example

Given 1 < p < ∞ let ϕ(t) =    tp t < 1 tp log(1 + t) t ≥ 1 The Orlicz space Lϕ(0, ∞) is a reflexive p-DH Banach lattice whose dual is not DH.

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Counterexemples

Example

Given 1 < p < ∞ let ϕ(t) =    tp t < 1 tp log(1 + t) t ≥ 1 The Orlicz space Lϕ(0, ∞) is a reflexive p-DH Banach lattice whose dual is not DH.

Theorem (Knaust-Odell)

Let E be an atomic Banach lattice. If E is p-DH and E ∗ is p′-DH, then there is C > 0 such that every disjoint sequence in E has a subsequence C-equivalent to the basis of ℓp.

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Counterexemples

Example

Given 1 < p < ∞ let ϕ(t) =    tp t < 1 tp log(1 + t) t ≥ 1 The Orlicz space Lϕ(0, ∞) is a reflexive p-DH Banach lattice whose dual is not DH.

Theorem (Knaust-Odell)

Let E be an atomic Banach lattice. If E is p-DH and E ∗ is p′-DH, then there is C > 0 such that every disjoint sequence in E has a subsequence C-equivalent to the basis of ℓp.

Theorem (Johnson-Odell)

There is a p-DH atomic Banach lattice with no uniform constant

  • n the equivalence.
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Projections onto disjoint sequences

Question: Does every reflexive Banach lattice contain a complemented positive disjoint sequence?

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Projections onto disjoint sequences

Question: Does every reflexive Banach lattice contain a complemented positive disjoint sequence? It does provided:

  • 1. E contains infinitely many atoms (in particular, discrete),
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Projections onto disjoint sequences

Question: Does every reflexive Banach lattice contain a complemented positive disjoint sequence? It does provided:

  • 1. E contains infinitely many atoms (in particular, discrete),
  • 2. E is non-atomic and contains certain complemented

unconditional basic sequences (Casazza-Kalton),

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Projections onto disjoint sequences

Question: Does every reflexive Banach lattice contain a complemented positive disjoint sequence? It does provided:

  • 1. E contains infinitely many atoms (in particular, discrete),
  • 2. E is non-atomic and contains certain complemented

unconditional basic sequences (Casazza-Kalton),

  • 3. E is a rearrangement invariant space.
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Projections onto disjoint sequences

Theorem

Let E be a DH Banach lattice. E has property P if and only if E contains a complemented positive disjoint sequence.

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Projections onto disjoint sequences

Theorem

Let E be a DH Banach lattice. E has property P if and only if E contains a complemented positive disjoint sequence.

Theorem

If E is a separable non-reflexive DH Banach lattice, then every dis- joint sequence in E has a subsequence spanning a complemented subspace in E.

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Projections onto disjoint sequences

Theorem

Let E be a DH Banach lattice. E has property P if and only if E contains a complemented positive disjoint sequence.

Theorem

If E is a separable non-reflexive DH Banach lattice, then every dis- joint sequence in E has a subsequence spanning a complemented subspace in E.

Theorem

Let E be reflexive Banach lattice containing a complemented disjoint

  • sequence. If E and E ∗ are DH, then every disjoint sequence in E

has a subsequence spanning a complemented subspace in E.

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Projections onto disjoint sequences

Theorem

Let E be a DH Banach lattice. E has property P if and only if E contains a complemented positive disjoint sequence.

Theorem

If E is a separable non-reflexive DH Banach lattice, then every dis- joint sequence in E has a subsequence spanning a complemented subspace in E.

Theorem

Let E be reflexive Banach lattice containing a complemented disjoint

  • sequence. If E and E ∗ are DH, then every disjoint sequence in E

has a subsequence spanning a complemented subspace in E.

Theorem

Let E be a p-DH Banach lattice which is p-convex with 1 < p < ∞. Then every disjoint sequence in E has a subsequence spanning a complemented subspace in E.

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Disjoint sequences in Banach lattices

Pedro Tradacete

Mathematics Department, UC3M Based on joint work with J. Flores, F. Hern´ andez, E. Semenov, E. Spinu, V. Troitsky

First Brazilian Workshop in Geometry of Banach Spaces 25-29 August 2014, Maresias