Spaceability in Banach and quasi-Banach spaces of vector-valued sequences

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Spaceability in Banach and quasi-Banach spaces of vector-valued sequences

Vin´ıcius Vieira F´avaro

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

G. Botelho, V. V. F´avaro BWB - Maresias-SP 08/25/2014

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Introduction

In this work we continue the research initiated in [1, 3] on the existence of infinite dimensional closed subspaces of Banach or quasi-Banach sequence spaces formed by sequences with special properties.

Given a Banach spaceX, in [3] the authors introduce a large class of Banach or quasi-Banach spaces formed by X-valued sequences, calledinvariant sequences spaces, which encompasses several classical sequences spaces as particular cases.

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Introduction

In this work we continue the research initiated in [1, 3] on the existence of infinite dimensional closed subspaces of Banach or quasi-Banach sequence spaces formed by sequences with special properties.

Given a Banach spaceX, in [3] the authors introduce a large class of Banach or quasi-Banach spaces formed by X-valued sequences, calledinvariant sequences spaces, which encompasses several classical sequences spaces as particular cases.

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Introduction

Roughly speaking, the main results of [1, 3] prove that, for every invariant sequence space E of X-valued sequences and every subset Γ of (0,∞], there exists a closed infinite dimensional subspace ofE formed, up to the null vector, by sequences not belonging to S

q∈Γ

`_q(X);as well as a closed infinite dimensional subspace of E formed, up to the null vector, by sequences not belonging to c₀(X).

In other words we can say thatE− S

q∈Γ

`_q(X) and

E−c0(X) are spaceable. Remember that a subsetA of a topological vector space V is spaceableifA∪ {0}contains a closed infinite dimensional subspace of V.

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Introduction

q∈Γ

`_q(X); as well as a closed infinite dimensional subspace of E formed, up to the null vector, by sequences not belonging to c₀(X).

q∈Γ

`_q(X) and

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Introduction

q∈Γ

`_q(X) and

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Introduction

q∈Γ

`_q(X) and

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Introduction

In this talk we consider the following much more general situation: given Banach spaces X and Y, a map

f:X−→Y, a set Γ⊆(0,+∞] and an invariant sequence space E ofX-valued sequences, we investigate the existence of closed infinite dimensional subspaces of E formed, up to the origin, by sequences (xj)^∞_j=1∈E such that either

(f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) or (f(x_j))^∞_j=1∈/ S

q∈Γ

`^w_q(Y) or (f(x_j))^∞_j=1∈/ c₀(Y).

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Introduction

(f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) or (f(x_j))^∞_j=1∈/ S

q∈Γ

`^w_q(Y) or (f(x_j))^∞_j=1∈/ c₀(Y).

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Introduction

(f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) or (f(x_j))^∞_j=1∈/ S

q∈Γ

`^w_q(Y) or (f(x_j))^∞_j=1∈/ c₀(Y).

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Introduction

(f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) or (f(x_j))^∞_j=1∈/ S

q∈Γ

`^w_q(Y) or (f(x_j))^∞_j=1∈/ c₀(Y).

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Introduction

As usual, `_p(X) and `^w_p(X) are the Banach spaces

(p-Banach spaces if 0< p <1) of p-summable and weakly p-summable X-valued sequences, respectively,and c₀(X) is the Banach space of norm null X-valued sequences. Letting f be the identity on X, the cases of sequences (xj)^∞_j=1∈E such that (f(x_j))^∞_j=1∈/ S

q∈Γ

`_q(Y) or (f(x_j))^∞_j=1∈/ c₀(Y) recover the situation investigated in [1, 3]. So, the results of this talk generalize the previous results in two directions:

we consider f belonging to a large class of functions and we consider spaces formed by sequences (xj)^∞_j=1∈E such that (f(x_j))^∞_j=1 does not belong to S

q∈Γ

`^w_q(Y), a condition much more restrictive than not to belong to S

q∈Γ

`q(Y).

In this talk X and Y are always Banach spaces.

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Introduction

(p-Banach spaces if 0< p <1) of p-summable and weakly p-summable X-valued sequences, respectively, and c₀(X) is the Banach space of norm null X-valued sequences. Letting f be the identity on X, the cases of sequences (xj)^∞_j=1∈E such that (f(x_j))^∞_j=1∈/ S

q∈Γ

`q(Y).

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Introduction

q∈Γ

`q(Y).

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Introduction

q∈Γ

`q(Y).

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Introduction

q∈Γ

`q(Y).

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Main result

Definition LetX6={0}.

(a) Givenx∈X^N, by x⁰ we mean the zerofree version of x, that is: ifx has only finitely many non-zero coordinates, then

x⁰ = 0;otherwise,x⁰= (x_j)^∞_j=1 wherex_j is the j-th non-zero coordinate ofx.

(b) By aninvariant sequence space over X we mean an infinite-dimensional Banach or quasi-Banach spaceE of X-valued sequences enjoying the following conditions:

(b1) Forx∈X^Nsuch that x⁰ 6= 0,x∈E if and only if x⁰ ∈E, and in this casekxk_E ≤Kkx⁰k_E for some constantK

depending only onE.

(b2)kx_jk_X ≤ kxk_E for everyx= (xj)^∞_j=1 ∈E and every j∈N. Aninvariant sequence space is an invariant sequence space over some Banach spaceX.

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Main result

depending only onE.

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Main result

depending only onE.

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Main result

x⁰ = 0; otherwise,x⁰= (x_j)^∞_j=1 wherex_j is the j-th non-zero coordinate ofx.

depending only onE.

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Main result

depending only onE.

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Main result

depending only onE.

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Main result

depending only onE.

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Main result

Example

(a) For 0< p≤ ∞,`p(X),`^w_p(X), `^u_p(X) (unconditionally p-summable X-valued sequences) and `_m(s;p)(X) (mixed

sequence space)are invariant sequence spaces over X with their respective usual norms (p-norms if 0< p <1).

(b) The Lorentz sequence spaces, Orlicz sequence space, Nakano sequence spaces, . . . .

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Main result

Example

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Main result

Example

(b) The Lorentz sequence spaces,Orlicz sequence space, Nakano sequence spaces, . . . .

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Main result

Example

(b) The Lorentz sequence spaces,Orlicz sequence space,Nakano sequence spaces, . . . .

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Main result

Example

sequence space) are invariant sequence spaces over X with their respective usual norms (p-norms if 0< p <1).

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Main result

Example

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Main result

Example

(b) The Lorentz sequence spaces, Orlicz sequence space,Nakano sequence spaces, . . . .

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Main result

Example

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Main result

Definition

LetE be an invariant sequence space over X, Γ⊆(0,+∞] and f:X−→Y be a function. We define the sets:

C(E, f,Γ) = (

(xj)^∞_j=1∈E: (f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) )

,

C^w(E, f,Γ) = (

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`^w_q(Y) )

and

C(E, f,0) =

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1∈/ c₀(Y) .

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Main result

Definition

C(E, f,Γ) = (

(xj)^∞_j=1∈E: (f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) )

,

C^w(E, f,Γ) = (

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`^w_q(Y) )

and

C(E, f,0) =

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1∈/ c₀(Y) .

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Main result

Definition

C(E, f,Γ) = (

(xj)^∞_j=1∈E: (f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) )

,

C^w(E, f,Γ) = (

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`^w_q(Y) )

and

C(E, f,0) =

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1∈/ c₀(Y) .

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Main result

Definition

C(E, f,Γ) = (

(xj)^∞_j=1∈E: (f(xj))^∞_j=1 ∈/ S

q∈Γ

`q(Y) )

,

C^w(E, f,Γ) = (

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`^w_q(Y) )

and

C(E, f,0) =

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1∈/ c₀(Y) .

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Main result

Definition

A mapf:X −→Y is said to be:

(a)Non-contractiveiff(0) = 0 and for every scalarα6= 0 there is a constantK(α)>0 such that

kf(αx)k_Y ≥K(α)· kf(x)k_Y

for everyx∈X.

(b)Strongly non-contractiveiff(0) = 0 and for every scalar α6= 0 there is a constantK(α)>0 such that

|ϕ(f(αx))| ≥K(α)· |ϕ(f(x))|

for allx∈X and ϕ∈Y⁰.

By the Hahn–Banach theorem, strongly non-contractive functions are non-contractive.

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Main result

Definition

kf(αx)k_Y ≥K(α)· kf(x)k_Y for everyx∈X.

|ϕ(f(αx))| ≥K(α)· |ϕ(f(x))|

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Main result

Definition

|ϕ(f(αx))| ≥K(α)· |ϕ(f(x))|

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Main result

Definition

|ϕ(f(αx))| ≥K(α)· |ϕ(f(x))|

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Main result

Example

Subhomogeneous functions (with f(0) = 0) are non-contractive;

bounded and unbounded linear operators are strongly non-contractive (hence non-contractive); and

homogeneous polynomials (continuous or not) are strongly non-contractive (hence non-contractive).

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Main result

Example

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Main result

Example

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Main result

Theorem

Let E be an invariant sequence space overX, f:X−→Y be a function and Γ⊆(0,+∞].

(a)If f is non-contractive, then C(E, f,Γ) and C(E, f,0) are either empty or spaceable.

(b)If f is strongly non-contractive, then C^w(E, f,Γ)is either empty or spaceable.

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Main result

Theorem

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Main result

Theorem

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Main result

Theorem

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Main result

Theorem

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Main result

Sketch of the proof for the case C (E, f, Γ).

Remember that C(E, f,Γ) =

(

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`_q(Y) )

. Let us fix a notation. For α= (αn)^∞_n=1∈K^N and w∈X we denote

w⊗α=α⊗w:= (α_nw)^∞_n=1∈X^N.

Assume thatC(E, f,Γ) is non-empty and choose x∈C(E, f,Γ).

SinceE is an invariant sequence space, thenx⁰ ∈E and the conditionf(0) = 0 guarantees thatx⁰∈C(E, f,Γ). Writing x⁰ = (xj)^∞_j=1 we have thatxj 6= 0 for every j.

Step 1: SplitNinto countably many infinite pairwise disjoint subsets (Ni)^∞_i=1. For everyi∈Nset Ni={i₁ < i₂ < . . .} and define

yi =

∞

X

j=1

xj ⊗eij ∈X^N.

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Main result

Sketch of the proof for the case C (E, f, Γ).

(

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`_q(Y) )

. Let us fix a notation. For α= (αn)^∞_n=1∈K^N and w∈X we denote

w⊗α=α⊗w:= (α_nw)^∞_n=1∈X^N.

yi =

∞

X

j=1

xj ⊗eij ∈X^N.

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Main result

Sketch of the proof for the case C (E, f, Γ).

(

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`_q(Y) )

. Let us fix a notation. Forα= (αn)^∞_n=1∈K^N and w∈X we denote

w⊗α=α⊗w:= (α_nw)^∞_n=1∈X^N.

yi =

∞

X

j=1

xj ⊗eij ∈X^N.

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Main result

Sketch of the proof for the case C (E, f, Γ).

(

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`_q(Y) )

w⊗α=α⊗w:= (α_nw)^∞_n=1∈X^N.

yi =

∞

X

j=1

xj ⊗eij ∈X^N.

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Main result

Sketch of the proof for the case C (E, f, Γ).

(

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`_q(Y) )

w⊗α=α⊗w:= (α_nw)^∞_n=1∈X^N.

yi =

∞

X

j=1

xj ⊗eij ∈X^N.

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Main result

Sketch of the proof for the case C (E, f, Γ).

(

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`_q(Y) )

w⊗α=α⊗w:= (α_nw)^∞_n=1∈X^N.

yi =

∞

X

j=1

xj ⊗eij ∈X^N.

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Main result

Sketch of the proof for the case C (E, f, Γ).

(

(x_j)^∞_j=1∈E: (f(x_j))^∞_j=1 ∈/ S

q∈Γ

`_q(Y) )

w⊗α=α⊗w:= (α_nw)^∞_n=1∈X^N.

yi =

∞

X

j=1

xj ⊗eij ∈X^N.

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Main result

Sketch of the proof for the case C (E, f, Γ).

Observe thaty⁰_i =x⁰, so 06=y_i⁰∈E, hencey_i ∈E for everyi becauseE is an invariant sequence space. For q∈Γ,q <+∞, we have

∞

P

j=1

kf(xj)k^q_Y = +∞ because x⁰ ∈C(E, f,Γ). If +∞ ∈Γ, by the same reason we have sup

i

kf(x_i)k_Y = +∞. It follows that eachy_i∈C(E, f,Γ).

Step 2: Define ˜s= 1 ifE is a Banach space and ˜s=sifE is an s-Banach space, 0< s <1. We need to prove that the operator

T:`_˜_s−→E , T((a_i)^∞_i=1) =

∞

X

i=1

a_iy_i,

is well defined. It is possible to prove that

∞

P

i=1

ka_iy_ik_E <+∞ if E is a Banach space and

∞

P

i=1

ka_iy_ik^s_E <+∞ ifE is ans-Banach space, 0< s <1.

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Main result

Sketch of the proof for the case C (E, f, Γ).

Observe thaty⁰_i =x⁰, so 06=y_i⁰∈E, hencey_i ∈E for everyi becauseE is an invariant sequence space. Forq∈Γ,q <+∞, we have

∞

P

j=1

i

T:`_˜_s−→E , T((a_i)^∞_i=1) =

∞

X

i=1

a_iy_i,

∞

P

i=1

∞

P

i=1

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Main result

Sketch of the proof for the case C (E, f, Γ).

∞

P

j=1

i

T:`_˜_s−→E , T((a_i)^∞_i=1) =

∞

X

i=1

a_iy_i,

∞

P

i=1

∞

P

i=1

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Main result

Sketch of the proof for the case C (E, f, Γ).

∞

P

j=1

i

T:`_˜_s−→E , T((a_i)^∞_i=1) =

∞

X

i=1

a_iy_i,

∞

P

i=1

∞

P

i=1

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Main result

Sketch of the proof for the case C (E, f, Γ).

∞

P

j=1

i

T:`_˜_s−→E , T((a_i)^∞_i=1) =

∞

X

i=1

a_iy_i,

∞

P

i=1

∞

P

i=1

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Main result

Sketch of the proof for the case C (E, f, Γ).

∞

P

j=1

i

T:`_˜_s−→E , T((a_i)^∞_i=1) =

∞

X

i=1

a_iy_i,

∞

P

i=1

∞

P

i=1

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Main result

Sketch of the proof for the case C (E, f, Γ).

∞

P

j=1

i

T:`_˜_s−→E , T((a_i)^∞_i=1) =

∞

X

i=1

a_iy_i,

∞

P

i=1

∞

P

i=1

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Main result

Sketch of the proof for the case C (E, f, Γ).

In both cases the series

∞

P

i=1

aiyi converges in E, hence the operator is well defined. It is easy to see thatT is linear and injective. ThusT(`_s_˜) is a closed infinite-dimensional subspace ofE.

Step 3: We have to show that ifz= (zn)^∞_n=1 ∈T(`˜s), z6= 0, then (f(z_n))^∞_n=1∈/ S

q∈Γ

`_q(Y). Given such a z, there are sequences

a^(k)_i

∞

i=1∈`˜s,k∈N, such that z= limk→∞T

a^(k)_i

∞ i=1

inE.

Using the convergence below and the fact thatf is

non-contractive, it is possible to prove (hardwork) that there is a subsequence zmj

∞

j=1 of z= (zn)^∞_n=1 satisfying:

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Main result

Sketch of the proof for the case C (E, f, Γ).

In both cases the series

∞

P

i=1

aiyi converges in E, hence the operator is well defined. It is easy to see thatT is linear and injective. ThusT(`_s_˜) is a closed infinite-dimensional subspace ofE.

Step 3: We have to show that ifz= (zn)^∞_n=1 ∈T(`˜s), z6= 0, then (f(z_n))^∞_n=1∈/ S

q∈Γ

`_q(Y). Given such a z, there are sequences

a^(k)_i

∞

i=1∈`˜s,k∈N, such that z= limk→∞T

a^(k)_i

∞ i=1

inE.

Using the convergence below and the fact thatf is

non-contractive, it is possible to prove (hardwork) that there is a subsequence zmj

∞

j=1 of z= (zn)^∞_n=1 satisfying:

Spaceability in Banach and quasi-Banach spaces of vector-valued sequences