Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day...

Welcome!

This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.

If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.

We look forward to seeing you in Madrid. The organizers:

Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).

Speakers

Félix Cabello, Universidad de Extremadura.

María J. Carro, Universidad de Barcelona.

Ben de Pagter, Delft University of Technology (The Netherlands).

Antonio Fernández-Carrión, Universidad de Sevilla.

Manuel González, Universidad de Cantabria.

Miguel Martín, Universidad de Granada.

Yves Raynaud, CNRS/Université Paris VI (France).

José Rodríguez, Universidad de Murcia.

Luis Rodríguez-Piazza, Universidad de Sevilla.

Workshop Operators and Banach lattices

Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012

WELCOME PROGRAMME LOCAL INFO PARTICIPANTS

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interpolation

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interpolation integrable

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measure

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Interpolation of spaces of integrable functions with respect to a vector measure

Antonio Fernández (Universidad de Sevilla)


measure

The team (FQM–133).

The Bartle–Dunford–Schwartz integral.


The ingredients: set, sigma-algebra

( delta-ring), Banach space and

( ) vector measure.

⌦ ⌃

R X

m : ⌃ ! X ⌫ : R ! X




( ) vector measure.

The definition. A measurable function

⌦ ⌃

R X

m : ⌃ ! X ⌫ : R ! X

f : ⌦ �! K


Measurable. (sigma-algebra)

f : ⌦ �! K



f : ⌦ �! KRloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}



Null set.

f : ⌦ �! KRloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}



Null set.

f : ⌦ �! K

h⌫, x0i (A) := h⌫(A), x0i , A 2 RR ⌫�! X

x

0�! K, x

0 2 X

0

Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}



Null set.

f : ⌦ �! K

h⌫, x0i (A) := h⌫(A), x0i , A 2 R

k⌫k : A 2 Rloc �! k⌫k(A) 2 [0,1]

R ⌫�! X

x

0�! K, x

0 2 X

0

Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}

k⌫k(A) := sup {|h⌫, x0i| (A) : kx0k 1}



Null set.

A set is said to be null if

f : ⌦ �! K

A 2 Rloc

h⌫, x0i (A) := h⌫(A), x0i , A 2 R

k⌫k : A 2 Rloc �! k⌫k(A) 2 [0,1]

k⌫k(A) = 0.

R ⌫�! X

x

0�! K, x

0 2 X

0

Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}

k⌫k(A) := sup {|h⌫, x0i| (A) : kx0k 1}




( ) vector measure.


⌦ ⌃

R X

m : ⌃ ! X ⌫ : R ! X

f : ⌦ �! K




( ) vector measure.


is said to be integrable if:

I) for each x

0 2 X

0,

⌦ ⌃

R X

m : ⌃ ! X ⌫ : R ! X

f : ⌦ �! K

f 2 L

1 (|h⌫, x0i|)




( ) vector measure.


is said to be integrable if:

I) for each and

II) For every there exists

such that

x

0 2 X

0,

A 2 Rloc

⌦ ⌃

R X

m : ⌃ ! X ⌫ : R ! X

f : ⌦ �! K

f 2 L

1 (|h⌫, x0i|) Z

Afd⌫ 2 X

⌧Z

Afd⌫, x

0�

=

Z

Afd h⌫, x0i , x

0 2 X

0.

The function spaces.


L1w(⌫) := {f : ⌦ �! K : I)} ,

L1(⌫) := {f : ⌦ �! K : I)+ II)} .


kfk1 := sup

⇢Z

⌦|f | d |h⌫, x0i| : kx0k 1

�.

L1w(⌫) := {f : ⌦ �! K : I)} ,

L1(⌫) := {f : ⌦ �! K : I)+ II)} .


For

kfk1 := sup

⇢Z

⌦|f | d |h⌫, x0i| : kx0k 1

�.

1 p < 1

L1w(⌫) := {f : ⌦ �! K : I)} ,

L1(⌫) := {f : ⌦ �! K : I)+ II)} .

Lpw(⌫) :=

�f : ⌦ �! K : |f |p 2 L1

w(⌫) ,

Lp(⌫) :=�f : ⌦ �! K : |f |p 2 L1(⌫)

.


For

kfk1 := sup

⇢Z

⌦|f | d |h⌫, x0i| : kx0k 1

�.

kfkp := sup

(✓Z

⌦|f |p d |h⌫, x0i|

◆ 1p

: kx0k 1

).

1 p < 1

L1w(⌫) := {f : ⌦ �! K : I)} ,

L1(⌫) := {f : ⌦ �! K : I)+ II)} .

Lpw(⌫) :=

�f : ⌦ �! K : |f |p 2 L1

w(⌫) ,

Lp(⌫) :=�f : ⌦ �! K : |f |p 2 L1(⌫)

.

Basic properties.

Basic properties.

1) is a Banach lattice with o.c.n:

Lp(⌫)

Basic properties.


Lp(⌫)

0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.

Basic properties.


2) is a Banach lattice with the Fatou

property:

Lp(⌫)

0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.

Lpw(⌫)

Basic properties.



property:

Lp(⌫)

0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.

Lpw(⌫)

0 fn " f 2 L0(⌫),

supn kfnkp < 1

))

8<

:

f 2 Lpw(⌫),

limn

kfnkp = kfkp .

Basic properties.



property:

3) If then

Lp(⌫)

0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.

Lpw(⌫)

0 fn " f 2 L0(⌫),

supn kfnkp < 1

))

8<

:

f 2 Lpw(⌫),

limn

kfnkp = kfkp .

⌫ = µ, Lp(⌫) = Lpw(⌫) = Lp(µ).

Basic properties.



property:

3) If then

In general,

Lp(⌫)

0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.

Lpw(⌫)

0 fn " f 2 L0(⌫),

supn kfnkp < 1

))

8<

:

f 2 Lpw(⌫),

limn

kfnkp = kfkp .

⌫ = µ, Lp(⌫) = Lpw(⌫) = Lp(µ).

Lp(⌫) Lpw(⌫).

Basic properties.

4) and can be non reflexive, even if

Lp(⌫) Lpw(⌫)

p > 1.

Basic properties.


5) If then

Lp(⌫) Lpw(⌫)

p > 1.

1 p1 < p2 < 1,

Lp2w (m) ⇢ Lp1

w (m) ⇢ L1w(m) ⇢ L0(m)

[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)

Basic properties.


5) If then

Lp2w (m) ⇢ Lp1

w (m) ⇢ L1w(m) ⇢ L0(m)

[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)

Lp(⌫) Lpw(⌫)

p > 1.

1 p1 < p2 < 1,

Basic properties.


5) If then

6) In general,

Lp2w (m) ⇢ Lp1

w (m) ⇢ L1w(m) ⇢ L0(m)

[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)

Lp(⌫) Lpw(⌫)

p > 1.

1 p1 < p2 < 1,

L1(⌫) 6✓ L1(⌫).

Basic properties.


5) If then

6) In general,

7) and

Lp2w (m) ⇢ Lp1

w (m) ⇢ L1w(m) ⇢ L0(m)

[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)

Lp(⌫) Lpw(⌫)

Lp(m) Lp[0, 1] Lp(⌫) Lp (R)

p > 1.

1 p1 < p2 < 1,

L1(⌫) 6✓ L1(⌫).

Interpolation methods.


Let be a (compatible) couple of Banach

spaces.

(X0, X1)

X0 \X1,! F(X0, X1) ,!X0 +X1



spaces. An interpolation method

with the interpolation property:

X0 \X1 ,! F(X0, X1) ,! X0 +X1

F

(X0, X1)



spaces. An interpolation method

with the interpolation property:

X0T�! Y0

X1T�! Y1

9>=

>;) F(X0, X1)

T�! F(Y0, Y1)

X0 \X1 ,! F(X0, X1) ,! X0 +X1

F

(X0, X1)

Compatible couple of Banach spaces.


If and sigma-finite

1 p0 6= p1 < 1 ⌫ : R ! X

⌦ =[

n�1

⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,


If and sigma-finite

we can consider the following (compatible) couples

of Banach spaces:

1 p0 6= p1 < 1 ⌫ : R ! X

⌦ =[

n�1

⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,

(Lp0(⌫), Lp1(⌫)) (Lp0w (⌫), Lp1

w (⌫))

(Lp0w (⌫), Lp1(⌫)) (Lp0(⌫), Lp1

w (⌫))


If and sigma-finite

we can consider the following (compatible) couples

of Banach spaces:

The ambient space:

1 p0 6= p1 < 1 ⌫ : R ! X

L0(⌫).

⌦ =[

n�1

⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,

(Lp0(⌫), Lp1(⌫)) (Lp0w (⌫), Lp1

w (⌫))

(Lp0w (⌫), Lp1(⌫)) (Lp0(⌫), Lp1

w (⌫))

Complex methods. [Calderón (1964)]


[X0, X1][✓] ,! X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.


1) Calderón-Lozanowskiĭ product: X1�✓0 X✓

1 .

[X0, X1][✓] ,! X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.


1) Calderón-Lozanowskiĭ product:

2) If or has order continuous norm:

[X0, X1][✓] = X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.

X0 X1

X1�✓0 X✓

1 .

[X0, X1][✓] ,! X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.




3) If and have the Fatou property:

[X0, X1][✓] = X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.

[X0, X1][✓] ,! X1�✓0 X✓

1 = [X0, X1][✓] , 0 < ✓ < 1.

X0 X1

X1�✓0 X✓

1 .

X0 X1

[X0, X1][✓] ,! X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.




3) If and have the Fatou property:

4)

[Lp0(µ), Lp1(µ)][✓]=(Lp0(µ))1�✓ (Lp1(µ))✓

=[Lp0(µ), Lp1(µ)][✓]= Lp(µ).

1 p0, p1 1

1

p:=

1� ✓

p0+

✓

p1

[X0, X1][✓] = X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.

[X0, X1][✓] ,! X1�✓0 X✓

1 = [X0, X1][✓] , 0 < ✓ < 1.

X0 X1

X1�✓0 X✓

1 .

X0 X1

[X0, X1][✓] ,! X1�✓0 X✓

1 ,! [X0, X1][✓] , 0 < ✓ < 1.

Complex methods: the problem.

For sigma-finite, and

describe

0 < ✓ < 1 p0 6= p1 1⌫ : R ! X

[Lp0(⌫), Lp1(⌫)][✓] , [Lp0(⌫), Lp1(⌫)][✓] ,

[Lp0(⌫), Lp1w (⌫)][✓] , [Lp0(⌫), Lp1

w (⌫)][✓] ,

[Lp0w (⌫), Lp1(⌫)][✓] , [Lp0

w (⌫), Lp1(⌫)][✓] ,

[Lp0w (⌫), Lp1

w (⌫)][✓] , [Lp0w (⌫), Lp1

w (⌫)][✓] .

Complex method [.,.]θ (sigma-algebra).


[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]

If and

then

0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X



If and

then

0 < ✓ < 1 p0 6= p1 1

1

p:=

1� ✓

p0+

✓

p1

m : ⌃ ! X

[·, ·]✓ Lp0(m) Lp0w (m)

Lp1w (m)

Lp1(m)

Lp(m)

Lp(m) Lp(m)

Lp(m)



If and

then

0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X



If and

then

1

p:=

1� ✓

p0+

✓

p1

Lp0(m) Lp0w (m)

Lp1w (m)

Lp1(m)

[·, ·]✓

Lpw(m)Lp

w(m)

Lpw(m) Lp

w(m)

0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X

Complex methods (delta-ring).


[·, ·]✓ Lp0(m) Lp0w (m)

Lp1w (m)

Lp1(m)

Lp(m)

Lp(m) Lp(m)

Lp(m)

Lp0(m) Lp0w (m)

Lp1w (m)

Lp1(m)

[·, ·]✓

Lpw(m)Lp

w(m)

Lpw(m) Lp

w(m)(R)

(R)(R)

(R)(⌃)

(⌃)(⌃)

(⌃)


[·, ·]✓ Lp0(m) Lp0w (m)

Lp1w (m)

Lp1(m)

Lp(m)

Lp(m) Lp(m)

Lp(m)

Lp0(m) Lp0w (m)

Lp1w (m)

Lp1(m)

[·, ·]✓

Lpw(m)Lp

w(m)

Lpw(m) Lp

w(m)

(R)

(R)(⌃)

(⌃)(⌃)

(⌃)

Complex methods (delta-ring). Example.


Let be the delta-ring of finite subsets of

natural numbers, and consider the measure

Pf (N)

⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.




It is not difficult to show that

Pf (N)

⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.

Lp(⌫) = c0 ✓ Lpw(⌫) = `1, 1 p < 1.




It is not difficult to show that

Then, for 1 p0, p1 < 1,

Pf (N)

⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.

Lp(⌫) = c0 ✓ Lpw(⌫) = `1, 1 p < 1.

[Lp0w (⌫), Lp1

w (⌫)][✓] = [`1, `1][✓] = `1 = Lpw(⌫),

[Lp0(⌫), Lp1(⌫)][✓] = [c0, c0][✓] = c0 = Lp(⌫).

Intermediate spaces (sigma-algebra).


If and where

then

1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},

r < p < s,

Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp

w(m) ✓ Lr(m) ✓ Lrw(m).


If and where

then

[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]

1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},

r < p < s,




If and where

then


1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},

r < p < s,



Lp0(m) \ Lp1(m)

Lp0w (m) \ Lp1(m)

Lp0(m) \ Lp1w (m)

Lp0w (m) \ Lp1

w (m)


If and where

then


1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},

r < p < s,



Lp0(m) \ Lp1(m)

Lp0w (m) \ Lp1(m)

Lp0(m) \ Lp1w (m)

Lp0w (m) \ Lp1

w (m)

Lp0(m) + Lp1(m)

Lp0w (m) + Lp1(m)

Lp0(m) + Lp1w (m)

Lp0w (m) + Lp1

w (m)

Gain of integrability (sigma-algebra).



If then

r < s, Lsw(m) ✓ Lr(m).



If then

f 2 Lsw(m) ✓ Lr

w(m)f

⌦




If then

f�[fn] 2 Lr(m)

f 2 Lsw(m) ✓ Lr

w(m)f

⌦

n f�[fn]




If then

f�[fn] 2 Lr(m)

f 2 Lsw(m) ✓ Lr

w(m)

��f � f�[fn]

��Lr

w(m)! 0

f

⌦

n f�[fn]


Intermediate spaces (delta-ring).


If then

1 p0 < p < p1 < 1,

(i) Lp0w (⌫) \ Lp1

w (⌫) ✓ Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫).

(ii) Lp0(⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1(⌫).

(iii) Lp0(⌫) \ Lp1w (⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1

w (⌫).

(iv) Lp0w (⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0

w (⌫) + Lp1(⌫).


If then

Nevertheless,

1 p0 < p < p1 < 1,

(i) Lp0w (⌫) \ Lp1

w (⌫) ✓ Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫).

(ii) Lp0(⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1(⌫).

(iii) Lp0(⌫) \ Lp1w (⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1

w (⌫).

(iv) Lp0w (⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0

w (⌫) + Lp1(⌫).

Lp0w (⌫) \ Lp1

w (⌫) 6✓ Lp(⌫),

Lpw(⌫) 6✓ Lp0(⌫) + Lp1(⌫).

Locally strongly additive measure.


is said to be locally strongly additive if

for each disjoint sequence

such that

⌫ : R ! X

limn!1

k⌫(An)kX = 0

(An)n ✓ R k⌫k⇣S

n�1 An

⌘< 1.




such that

1) The Lebesgue measure is l.s.a.

⌫ : R ! X

limn!1

k⌫(An)kX = 0


n�1 An

⌘< 1.




such that


2) is not

locally strongly additive.

⌫ : R ! X

limn!1

k⌫(An)kX = 0


n�1 An

⌘< 1.

k⌫k(A) = 1,? 6= A 2 Pf (N), k⌫k(?) = 0

⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0




such that


2) is not

locally strongly additive.

3) Every is (locally) strongly additive.

⌫ : R ! X

limn!1

k⌫(An)kX = 0


n�1 An

⌘< 1.

m : ⌃ ! X

k⌫k(A) = 1,? 6= A 2 Pf (N), k⌫k(?) = 0

⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0


[del Campo, Mayoral, Naranjo & AF, 2012]

Let The following assertions are

equivalent:

1 p0 < p1 < 1.




equivalent:

1) is locally strongly additive.

⌫

1 p0 < p1 < 1.




equivalent:


⌫

1 p0 < p1 < 1.

�B 2 Rloc, �B 2 L1

w(⌫) ) �B 2 L1(⌫)�




equivalent:


2) for every (some)

⌫

1 p0 < p1 < 1.

p0 < p < p1.

Lp0w (⌫) \ Lp1

w (⌫) ✓ Lp(⌫)

�B 2 Rloc, �B 2 L1

w(⌫) ) �B 2 L1(⌫)�




equivalent:


2) for every (some)

3) for every (some)

⌫

1 p0 < p1 < 1.

p0 < p < p1.

p0 < p < p1.

Lp0w (⌫) \ Lp1

w (⌫) ✓ Lp(⌫)

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫)

�B 2 Rloc, �B 2 L1

w(⌫) ) �B 2 L1(⌫)�

Gain of integrability (delta-ring).



If is l.s.a. and then

⌫ p0 < p < p1,

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).




f

⌦

⌫ p0 < p < p1,

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).

Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫)




f

⌦

f�[fn]

n

⌫ p0 < p < p1,

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).

Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫)




f

⌦

n

1

n

⌫ p0 < p < p1,

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).

Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫)




f

⌦

n

1

n

f�[ 1nfn] 2 Lp0(⌫) + Lp1(⌫)

⌫ p0 < p < p1,

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).

Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫)




f

⌦

n

1

n

��f�[f>n]

��L

p0w (⌫)

! 0

⌫ p0 < p < p1,

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).

Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫)




⌫ p0 < p < p1,

f

⌦

n

1

n

��f�[f< 1n ]

��L

p1w (⌫)

! 0

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).

Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫)




f

⌦

n

1

n

��f�[f< 1n ]

��L

p1w (⌫)

! 0

f�[ 1nfn] 2 Lp0(⌫) + Lp1(⌫)

⌫ p0 < p < p1,

Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).

��f�[f>n]

��L

p0w (⌫)

! 0

Lpw(⌫) ✓ Lp0

w (⌫) + Lp1w (⌫)

Complex methods: equalities (R).



Let The following

assertions are equivalent:

0 < ✓ < 1 p0 < p1 < 1.



Let The following



⌫

0 < ✓ < 1 p0 < p1 < 1.



Let The following



2)

⌫

0 < ✓ < 1 p0 < p1 < 1.

[Lp0w (⌫), Lp1

w (⌫)][✓] = Lp(⌫).

1

p:=

1� ✓

p0+

✓

p1



Let The following



2)

3)

⌫

0 < ✓ < 1 p0 < p1 < 1.

[Lp0w (⌫), Lp1

w (⌫)][✓] = Lp(⌫).

[Lp0(⌫), Lp1(⌫)][✓] = [Lp0w (⌫), Lp1(⌫)][✓]

= [Lp0(⌫), Lp1w (⌫)][✓]

= Lpw(⌫).1

p:=

1� ✓

p0+

✓

p1


If is not locally strongly additive, and

⌫

0 < ✓ < 1 p0 < p1 < 1.



⌫

0 < ✓ < 1 p0 < p1 < 1.

1

p:=

1� ✓

p0+

✓

p1

Lp(⌫) [Lp0w (⌫), Lp1

w (⌫)][✓] Lpw(⌫)



⌫

0 < ✓ < 1 p0 < p1 < 1.

1

p:=

1� ✓

p0+

✓

p1

Lp(⌫) [Lp0(⌫), Lp1(⌫)][✓] Lpw(⌫)

Lp(⌫) [Lp0w (⌫), Lp1

w (⌫)][✓] Lpw(⌫)

The real method (.,.)θ,q. [Lions & Peetre (1964)]


Given: a couple of Banach spaces and

(X0, X1)

0 < ✓ < 1 q 1,



is the space of

those such that

is finite.

f 2 X0 +X1

(X0, X1)

0 < ✓ < 1 q 1, (X0, X1)✓,q

kfk✓,q :=

8>><

>>:

Z 1

0

�t�✓K(t, f)

�q dt

t

� 1q

, q < 1

supt>0

t�✓K(t, f), q = 1



is the space of

those such that

is finite.

f 2 X0 +X1

(X0, X1)

0 < ✓ < 1 q 1, (X0, X1)✓,q

kfk✓,q :=

8>><

>>:

Z 1

0

�t�✓K(t, f)

�q dt

t

� 1q

, q < 1

supt>0

t�✓K(t, f), q = 1

K(t, f) := inf {kf0kX0 + tkf1kX1 :

f0 2 X0, f1 2 X1, f = f0 + f1} .

The real method: problem/classical results.


For we want

to describe with 0 < ✓ < 1 q 1.(X0, X1)✓,q ,

Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,


For we want

to describe with

Let be a positive sigma-finite measure, and

Then:

0 < ✓ < 1 q 1.(X0, X1)✓,q ,

µ

0 < ✓ < 1 p0 6= p1 1.

Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,


For we want

to describe with


Then:

0 < ✓ < 1 q 1.(X0, X1)✓,q ,

µ

0 < ✓ < 1 p0 6= p1 1.

Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,

•�L1(µ), L1(µ)

�✓,q

= L1

1�✓ ,q(µ).

• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).

• Lp,p(µ) = Lp(µ), 1 p < 1.

• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.


For we want

to describe with


Then:

0 < ✓ < 1 q 1.(X0, X1)✓,q ,

µ

0 < ✓ < 1 p0 6= p1 1.

1

p:=

1� ✓

p0+

✓

p1

Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,

•�L1(µ), L1(µ)

�✓,q

= L1

1�✓ ,q(µ).

• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).

• Lp,p(µ) = Lp(µ), 1 p < 1.

• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.

Lorentz spaces of a vector measure.


Let and consider

hm,x

0i (A) := hm(A), x0i , A 2 ⌃.

⌃m�! X

x

0�! R, x

0 2 X

0,


Let and consider

If we say that if:

hm,x

0i (A) := hm(A), x0i , A 2 ⌃.

1 p, q 1, f 2 Lp,qw (m)

⌃m�! X

x

0�! R, x

0 2 X

0,


Let and consider

If we say that if:

i) measurable and

hm,x

0i (A) := hm(A), x0i , A 2 ⌃.

1 p, q 1, f 2 Lp,qw (m)

f : ⌦ �! R

⌃m�! X

x

0�! R, x

0 2 X

0,


Let and consider

If we say that if:

i) measurable and

ii)

hm,x

0i (A) := hm(A), x0i , A 2 ⌃.

1 p, q 1, f 2 Lp,qw (m)

f 2 L

p,q (|hm,x

0i|) , x

0 2 X

0.

f : ⌦ �! R

⌃m�! X

x

0�! R, x

0 2 X

0,


Let and consider

If we say that if:

i) measurable and

ii)

hm,x

0i (A) := hm(A), x0i , A 2 ⌃.

1 p, q 1, f 2 Lp,qw (m)

f 2 L

p,q (|hm,x

0i|) , x

0 2 X

0.

f : ⌦ �! R

kfkp,q

:= supn

kfkL

p,q(|hm,x

0i|) : kx0k 1

o

.

⌃m�! X

x

0�! R, x

0 2 X

0,


Let and consider

If we say that if:

i) measurable and

ii)

hm,x

0i (A) := hm(A), x0i , A 2 ⌃.

1 p, q 1, f 2 Lp,qw (m)

f 2 L

p,q (|hm,x

0i|) , x

0 2 X

0.

f : ⌦ �! R

kfkp,q

:= supn

kfkL

p,q(|hm,x

0i|) : kx0k 1

o

.

⌃m�! X

x

0�! R, x

0 2 X

0,

�L1w(m), L1(m)

�✓,q

S(⌃)Lp,q

w (m)✓ Lp,q

w (m).

Distribution and decreasing rearrangement.


Distribution function. measurable:

f : ⌦ �! Rkmkf : t 2 [0,1) �! kmkf (t) 2 [0,1)

kmkf (t) := kmk ({w 2 ⌦ : |f(w)| > t})

= supn

|hm,x

0i|f (t) : kx0k 1

o

.


Distribution function. measurable:

Decreasing rearrangement function.

kmkf : t 2 [0,1) �! kmkf (t) 2 [0,1)

kmkf (t) := kmk ({w 2 ⌦ : |f(w)| > t})

= supn

|hm,x

0i|f (t) : kx0k 1

o

.

f

⇤(s) := inf {t � 0 : kmkf

(t) s}= �kmkf

(s)

= sup {f⇤x

0(s) : kx0k 1}

f : ⌦ �! R

Lorentz spaces of the semivariation.


For the Lorentz space

consists of all measurable functions

such that where

1 p, q 1,

f : ⌦ �! RLp,q(kmk)

kfkLp,q(kmk) < 1,




such that where

1 p, q 1,

f : ⌦ �! R

1 p, q < 1

Lp,q(kmk)

kfkLp,q(kmk) :=

Z 1

0

⇣s

1p f⇤(s)

⌘q ds

s

� 1q

,

kfkLp,q(kmk) < 1,




such that where

1 p, q 1,

f : ⌦ �! R

1 p, q < 1

1 p < 1, q = 1

Lp,q(kmk)

kfkLp,q(kmk) :=

Z 1

0

⇣s

1p f⇤(s)

⌘q ds

s

� 1q

,

kfkLp,1(kmk) := supn

s1p f⇤(s) : s > 0

o

.

kfkLp,q(kmk) < 1,


1) is a quasi-Banach lattice.

Lp,q(kmk)kf + gkLp,q(kmk) C(p, q)

�kfkLp,q(kmk) + kgkLp,q(kmk)

�



2) has separating dual:

a) b) and

Lp,q(kmk)

Lp,q(kmk)p = q = 1, p > 1 1 q 1.

kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)

�




a) b) and

Lp,q(kmk)

Lp,q(kmk)p = q = 1, p > 1 1 q 1.


�

Lp,q(kmk) ✓ Lp,q(µ), 1 p, q 1




a) b) and

3) Inclusions.

Lp,q(kmk)

Lp,q(kmk)p = q = 1, p > 1 1 q 1.


�



✓✓

✓✓

Lp1,1(kmk)

Lp1,p1(kmk)

Lp1,1(kmk)

...

...

1 p1 < p2 < 1


✓ ✓✓

✓✓✓

✓✓

Lp2,1(kmk)

Lp2,p2(kmk)

Lp2,1(kmk) Lp1,1(kmk)

Lp1,p1(kmk)

Lp1,1(kmk)

...

......

...

1 p1 < p2 < 1


L1(m)✓

✓ ✓✓

✓✓✓

✓✓

Lp2,1(kmk)

Lp2,p2(kmk)


Lp1,p1(kmk)

Lp1,1(kmk)

...

......

...

1 p1 < p2 < 1


L1(m)✓

✓

✓✓

✓✓

✓✓✓

✓✓

Lp2,1(kmk)

Lp2,p2(kmk)


Lp1,p1(kmk)

Lp1,1(kmk)

L1,1(kmk)

L1,1(kmk)

...

...

......

...

1 p1 < p2 < 1


L1(m)✓

✓

✓✓

✓✓

✓✓✓

✓✓

Lp2,1(kmk)

Lp2,p2(kmk)


Lp1,p1(kmk)

Lp1,1(kmk)

L1,1(kmk)

L1,1(kmk)

...

...

......

... Lp(m)

Lpw(m)

1 p1 < p2 < 1




a) b) and

3) Inclusions.

4) For

Lp,q(kmk)

Lp,q(kmk)p = q = 1, p > 1 1 q 1.


�


1 p < 1,

Lp,p(kmk) Lp(m) Lpw(m) Lp,1(kmk).

Estimates for the K−functional.


If and then

f 2 L1w(m) t > 0,


If and then

a)

f 2 L1w(m) t > 0,

tf⇤(t) K�t, f, L1

w(m), L1(m)�.


If and then

a)

b)

f 2 L1w(m) t > 0,


w(m), L1(m)�.

K�t, f, L1

w(m), L1(m)�

Z t

0f⇤(s)ds.


If and then

a)

b)

c)

f 2 L1w(m) t > 0,


w(m), L1(m)�.

K�t, f, L1

w(m), L1(m)�

Z t

0f⇤(s)ds.

f 2 L1(m) 6)Z t

0f⇤(s)ds < 1,

6)Z 1

0f⇤(s)ds < 1.

The real method. [Mayoral, Naranjo & AF, 2011]


1) For

with equivalence of norm−quasinorm.

0 < ✓ < 1 q 1,�L1(m), L1(m)

�✓,q

=�L1w(m), L1(m)

�✓,q

= L1

1�✓ ,q(kmk),


1) For

with equivalence of norm−quasinorm.

2) For

0 < ✓ < 1 q 1,

1 p0 6= p1 1; 0 < ✓ < 1 q 1,

1

p=

1� ✓

p0+

✓

p1.

�L1(m), L1(m)

�✓,q

=�L1w(m), L1(m)

�✓,q

= L1

1�✓ ,q(kmk),

(Lp0(m), Lp1(m))✓,q = (Lp0w (m), Lp1

w (m))✓,q

= Lp,q(kmk),

The real method: consequences.


Consequence 1. If the space is

normable.

Lp,q(kmk)p > 1,



normable.

Concequence 2. If the space

is reflexive.

Lp,q(kmk)p > 1,

1 < p, q < 1,

Lp,q(kmk)



normable.


is reflexive.

[Beauzamy 1978] If and the

inclusion is weakly compact,

then is reflexive.

Lp,q(kmk)p > 1,

1 < p, q < 1,

Lp,q(kmk)0 < ✓ < 1 < q < 1

X0 \X1 ✓ X0 +X1

(X0, X1)✓,q



normable.


is reflexive.

[Beauzamy 1978] If and the

inclusion is weakly compact,

then is reflexive.


If then is

weakly compact.

Lp,q(kmk)p > 1,

1 < p, q < 1,

Lp,q(kmk)0 < ✓ < 1 < q < 1

X0 \X1 ✓ X0 +X1

(X0, X1)✓,q

1 p0 < p1 1, Lp1w (m) ✓ Lp0(m)

Welcome!





Speakers













Welcome!





Speakers













Interpolation of spaces of integrable functions with respect to a vector measure

Antonio Fernández (Universidad de Sevilla)

Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day...

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Transcript of Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day...