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Atomic decompositions in locally convex spaces

Jornadas de Analisis Matematico

Alicante

Juan Miguel Ribera Puchades

Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 1 / 22

Our aim is discuss atomic decompositions on no normable Frechet Spaces.

Atomic Decompositions have been investigated by Carando, Casazza,Grochenig, Lassalle, Korobeınik, Schmidberg, Taskinen and others.

Our aim is discuss atomic decompositions on no normable Frechet Spaces.

Atomic Decompositions have been investigated by Carando, Casazza,Grochenig, Lassalle, Korobeınik, Schmidberg, Taskinen and others.

Introduction

Outline

1 Introduction

2 Atomic Decompositions

3 Duality

4 Unconditional atomic decompostions

5 Example

Introduction

Definition

Let E be a Hausdorff locally convex space.

Definition

Let {xj}∞j=1 ⊂ E and let {x ′j}∞j=1 ⊂ E ′, we say that({x ′j}, {xj}

)is an

atomic decomposition of E if

x =∞∑j=1

x ′j (x) xj , for all x ∈ E ,

the series converging in E .

We denote by ω the space KN endowed by the product topology.A sequence space K(N) ⊂

∧⊂ KN is a lcs such that there exists a

continuous inclusion in ω.

Introduction

Definition

)is an

x =∞∑j=1

Introduction

Definition

)is an

x =∞∑j=1

We denote by ω the space KN endowed by the product topology.

A sequence space K(N) ⊂∧⊂ KN is a lcs such that there exists a

Introduction

Definition

)is an

x =∞∑j=1

Introduction

Examples

Let {ei}2i=1 be an orthonormal basis of a two-dimensional vectorspace V with inner product, then

x1 = e1, x2 = e1 − e2, x3 = e1 + e2

x ′1 =1

3e1, x

′2 =

3e1 −

2e2, x

′3 =

is an atomic decomposition of V .

Let {ei}∞i=1 an orthonormal basis of Hilbert space H, then

{xj}∞j=1 = {e1, e1, e2, e3, . . .}

{x ′j}∞j=1 =

2e1, e2, e3, . . .

}is an atomic decomposition of H.

Introduction

Examples

Let {ei}2i=1 be an orthonormal basis of a two-dimensional vectorspace V with inner product, then

x1 = e1, x2 = e1 − e2, x3 = e1 + e2

x ′1 =1

3e1, x

′2 =

3e1 −

2e2, x

′3 =

is an atomic decomposition of V .

Let {ei}∞i=1 an orthonormal basis of Hilbert space H, then

{xj}∞j=1 = {e1, e1, e2, e3, . . .}

{x ′j}∞j=1 =

2e1, e2, e3, . . .

}is an atomic decomposition of H.

Introduction

Examples

Let E be a lcs with a Schauder basis {ej}∞j=1 ⊂ E , the coefficient

functionals will be denoted by {e ′j}∞j=1 ⊂ E ′. If({e ′j}, {ej}

Schauder basis in a lcs E , then it is an atomic decomposition for E .

Let E be a lcs and let P : E → E be a continuous linear projection. If({x ′j}, {xj}

)is an atomic decomposition for E , then(

{P ′(

x ′j

)}, {P (xj)}

)is an atomic decomposition for P (E ).

Introduction

Examples

Let E be a lcs with a Schauder basis {ej}∞j=1 ⊂ E , the coefficient

functionals will be denoted by {e ′j}∞j=1 ⊂ E ′. If({e ′j}, {ej}

Schauder basis in a lcs E , then it is an atomic decomposition for E .

Let E be a lcs and let P : E → E be a continuous linear projection. If({x ′j}, {xj}

)is an atomic decomposition for E , then(

{P ′(

x ′j

)}, {P (xj)}

)is an atomic decomposition for P (E ).

Atomic Decompositions

Outline

1 Introduction

3 Duality

5 Example

Barrelled locally convex spaces are those satisfying the uniformboundedness principle (Banach-Steinhaus’ Theorem).

Example

Frechet spaces (complete and metrizable locally convex spaces).

Characterization

Let E be a barrelled and complete lcs, the following are equivalent:

1 E admits an atomic decomposition.

2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as Schauder basis.

3 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.

Example

Characterization

Example

Characterization

Example

Characterization

Space structure

Let {xj} ⊂ E be a sequence of non-zero elements.

α = (αj)j ∈ ω :∞∑j=1

αjxj is convergent in E

⊂ ω,

((αj)j

):= sup

n∑j=1

, for all p ∈ P

Theorem

If E is a complete lcs then (∧,Q) is a complete lcs.

Space structure

α = (αj)j ∈ ω :∞∑j=1

⊂ ω,Q =

((αj)j

):= sup

n∑j=1

, for all p ∈ P

Theorem

Space structure

α = (αj)j ∈ ω :∞∑j=1

⊂ ω,Q =

((αj)j

):= sup

n∑j=1

, for all p ∈ P

Theorem

Definition

A lcs E has the bounded approximation property (BAP) if there exists(Aj)j∈J ⊂ L (E ,E ) a net which is equicontinuous whit dim(Aj (E )) <∞for every j ∈ J and limj∈J Aj (x) = x for every x ∈ E . In other words,Aj → I in Ls (E ).

Characterization (Bessaga, Pe lczynski)

Let E be a separable Frechet space, the following are equivalent:

1 The bounded approximation property holds in E .

Definition

Duality

Outline

1 Introduction

3 Duality

5 Example

Duality

Shrinking Atomic Decompositions

We define a linear operator T : E → E as Tn (x) :=∑∞

j=n+1 x ′j (x) xj .

Definition

An atomic decomposition({x ′j}, {xj}

)is shrinking if for all x ′ ∈ E ′,

limn→∞ x ′ ◦ Tn = 0 ( the limit is taken in the topologyE ′β = (E ′, β (E ′,E )).

Characterization

The following are equivalent:

({x ′j}, {xj}

)is a shrinking atomic decomposition.

({xj}, {x ′j}

)is an atomic decomposition for E ′β.

3 For all x ′ ∈ E ′,∑∞

j=1 x ′ (xj) x ′j is convergent in E ′β.

Duality

j=n+1 x ′j (x) xj .

Definition

Characterization

({x ′j}, {xj}

({xj}, {x ′j}

3 For all x ′ ∈ E ′,∑∞

Duality

j=n+1 x ′j (x) xj .

Definition

Characterization

({x ′j}, {xj}

({xj}, {x ′j}

3 For all x ′ ∈ E ′,∑∞

Duality

j=n+1 x ′j (x) xj .

Definition

Characterization

({x ′j}, {xj}

({xj}, {x ′j}

3 For all x ′ ∈ E ′,∑∞

Duality

Boundedly Complete Atomic Decompositions

Definition

)is boundedly complete if for all

x ′′ ∈ E ′′β , the series∑∞

j=1 x ′j (x ′′) xj converges in E .

Proposition

Let E be a barrelled lcs with a Schauder basis {xj}j ;({x ′j}, {xj}

boundedly complete atomic decomposition if and only if for every

(αj)j ⊂ K such that(∑k

j=1 αjxj

is bounded, then∑∞

j=1 αjxj is

convergent.

Proposition

If({x ′j}, {xj}

)is a boundedly complete atomic decomposition for a

barrelled and complete lcs E with E ′′β barrelled, then E is complemented inits bidual E ′′β .

Duality

Definition

Proposition

j=1 αjxj

j=1 αjxj is

convergent.

Proposition

If({x ′j}, {xj}

Duality

Definition

Proposition

j=1 αjxj

j=1 αjxj is

convergent.

Proposition

If({x ′j}, {xj}

Duality

Properties

Proposition

Let E be a lcs and let({x ′j}, {xj}

)be a shrinking atomic decomposition

of E . Then({xj}, {x ′j}

)is a boundedly complete atomic decomposition of

E ′.

Theorem

Let({x ′j}, {xj}

)be an atomic decomposition of a (barrelled) lcs E which

is shrinking and boundedly complete, then E is (semi-)reflexive.

Duality

Properties

Proposition

Let E be a lcs and let({x ′j}, {xj}

)be a shrinking atomic decomposition

of E . Then({xj}, {x ′j}

)is a boundedly complete atomic decomposition of

E ′.

Theorem

Let({x ′j}, {xj}

)be an atomic decomposition of a (barrelled) lcs E which

is shrinking and boundedly complete, then E is (semi-)reflexive.

Unconditional atomic decompostions

Outline

1 Introduction

3 Duality

5 Example

Definition

)for a lcs E is said to be

unconditional if for every x ∈ E , its series expansion∑∞

j=1 x ′j (x) xjconverges unconditionally, that is, x =

∑∞j=1 x ′j (x) xj with unconditional

convergence.

McArthur, Retherford, 1969

If a series∑∞

j=1 xj converges unconditionally, then, for every boundedsequence of scalars {aj}, the series

∑∞j=1 ajxj converges and the operator

`∞ −→ E{aj} −→

∑∞j=1 ajxj ;

is a continuous linear operator.

Definition

)for a lcs E is said to be

unconditional if for every x ∈ E , its series expansion∑∞

j=1 x ′j (x) xjconverges unconditionally, that is, x =

∑∞j=1 x ′j (x) xj with unconditional

convergence.

McArthur, Retherford, 1969

If a series∑∞

j=1 xj converges unconditionally, then, for every boundedsequence of scalars {aj}, the series

∑∞j=1 ajxj converges and the operator

`∞ −→ E{aj} −→

∑∞j=1 ajxj ;

is a continuous linear operator.

Characterization

1 E admits an unconditional atomic decomposition.

2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as unconditional Schauder basis.

3 E is isomorphic to a complemented subspace of a complete sequencespace with unconditional Schauder basis.

Characterization

Example

Outline

1 Introduction

3 Duality

5 Example

Example

An atomic decomposition on the space C∞ (K )

Let K be a compact set in Rp, p ≥ 1, with◦K 6= ∅ such that K =

◦K .

Let C∞ (K ) := {f ∈ C∞(◦K ) : f and all its partial derivatives admit

continuous extension to K}.The Frechet space topology in C∞ (K ) is defined by the seminorms:

qn (f ) := sup{∣∣∣f (α) (x)

∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.

Remark

No system of exponencials can be a basis in C∞ ([0, 1]).

Theorem

Let K ⊂ Rp be a compact set which is the closure of its interior and let({uj}, {e ix ·λ

be an unconditional atomic decomposition of C∞(K ),then there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp).

Example

◦K .

continuous extension to K}.

The Frechet space topology in C∞ (K ) is defined by the seminorms:

qn (f ) := sup{∣∣∣f (α) (x)

∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.

Remark

Theorem

Example

◦K .

qn (f ) := sup{∣∣∣f (α) (x)

∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.

Remark

Theorem

Example

◦K .

qn (f ) := sup{∣∣∣f (α) (x)

∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.

Remark

Theorem

Example

◦K .

qn (f ) := sup{∣∣∣f (α) (x)

∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.

Remark

Theorem

Example

Theorem

Let us assume that there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp). Then there are sequences

(λj)⊂ Rp and

(uj) ∈ C∞ (K )′ such that({uj} ,

{e2πix ·λ

is an atomic decomposition

for C∞ (K ).

Let M > 0 such that K ⊂ [−M,M]p

Choosing φ ∈ D ([−2M, 2M]p) such that φ ≡ 1 on a neighborhood of[−M,M]p.

Let f ∈ C∞ (K ) we define Hf = φ (T (f )) ∈ D (]−2M, 2M[p).

Then we take u (f ) = aj (Hf ).

The atomic decomposition of C∞ (K ) is shrinking and boundedlycomplete since C∞ (K ) is a Montel space.

Example

Theorem

(λj)⊂ Rp and

{e2πix ·λ

for C∞ (K ).

Example

Theorem

(λj)⊂ Rp and

{e2πix ·λ

for C∞ (K ).

Example

Theorem

(λj)⊂ Rp and

{e2πix ·λ

for C∞ (K ).

Example

Theorem

(λj)⊂ Rp and

{e2πix ·λ

for C∞ (K ).

Example

Theorem

(λj)⊂ Rp and

{e2πix ·λ

for C∞ (K ).

Bibliography

1 Bonet, Jose and Perez Carreras, Pedro Barrelled locally convexspaces, North-Holland Publishing Co. (1987).

2 Carando, Daniel and Lassalle, Silvia Duality, reflexivity and atomicdecompositions in Banach spaces, Studia Math. 191 (2009.1), 67–80.

3 Casazza, Pete and Christensen, Ole and Stoeva, Diana T.Frame expansions in separable Banach spaces, J. Math. Anal. Appl.307 (2005.2), 710–723.

4 Korobeınik, Yu. F. On absolutely representing systems in spaces ofinfinitely differentiable functions, J. Math. Anal. Appl. 139 (2000.2),175–188.

5 Ribera, Juan M. Atomic Decompositions in Locally Convex Spaces,2012 (Preprint) .

Thanks for your attention

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