Summability Properties and Factorization of Homogeneous Polynomials

Summability properties and factorization of homogeneous polynomials

Enrique A. S´anchez P´erez

U.P. Valencia

Valencia, 26/09/2015

Based on joint works with M. Mastylo and P. Rueda XIV Encuentros An´alisis Funcional Murcia Valencia: Homenaje a Manolo Maestre.

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 1 / 27 In this talk, we will present several factorization schemes associated to suitable modifications of the usual summability properties that are the natural polynomial versions of Pietsch’s and Pisier’s Factorization Theorems. We will show also some examples and applications.

Summability properties and factorization of homogeneous polynomials

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 2 / 27 Summability properties and factorization of homogeneous polynomials

Summing inequalities for multilinear operators often lead to norm-domination inequalities. However, sometimes domination inequalities do not allow us to construct factorization schemes for the corresponding maps. Recall that, in the most relevant linear cases (p-summing operators, (q, 1)-summing operators from C(K)-spaces), domination and factorization hold together; in fact they can be understood as the same property. In this talk, we will present several factorization schemes associated to suitable modiﬁcations of the usual summability properties that are the natural polynomial versions of Pietsch’s and Pisier’s Factorization Theorems. We will show also some examples and applications.

1 Introduction and basic deﬁnitions

2 Factorization theorem for the class of the factorable p-dominated polynomials

3 Factorization through Lorentz spaces: a Pisier’s Theorem

4 (q, p)-Concavity for polynomials

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 3 / 27 Pisier’s factorization theorem. An operator T from C(K) to Y is (q, p)-summing (q > p ≥ 1) iff there is a probability Borel measure µ on K such that T factors as

j S C(K) −→ Lq,1(µ) −→ Y ,

where Lq,1(µ) is a Lorentz space.

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

Pietsch’s Factorization Theorem. A linear operator T : X → Y is p-summing iﬀ it factors as

T X / Y . O i v

jp C(BX ∗ ) / S

for a subspace S ⊆ Lp(µ).

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 4 / 27 Summability properties and factorization of homogeneous polynomials Introduction and basic definitions

Pietsch’s Factorization Theorem. A linear operator T : X → Y is p-summing iﬀ it factors as

T X / Y . O i v

jp C(BX ∗ ) / S

for a subspace S ⊆ Lp(µ). Pisier’s factorization theorem. An operator T from C(K) to Y is (q, p)-summing (q > p ≥ 1) iﬀ there is a probability Borel measure µ on K such that T factors as

j S C(K) −→ Lq,1(µ) −→ Y ,

where Lq,1(µ) is a Lorentz space.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 4 / 27 n 1 n P p p k(xi )i=1kp = ( kxi k ) . i=1

n 1 n P ∗ p p k(xi )i=1kp,ω = sup ( |hxi , x i| ) . ∗ kx kX ∗ ≤1 i=1

n n T : X → Y is (q, p)-summing iﬀ k(T (xi ))i=1kq ≤ C k(xi )i=1kp,ω

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

Notation. Let X be a Banach space and x1, ..., xn ∈ X .

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 5 / 27 n 1 n P ∗ p p k(xi )i=1kp,ω = sup ( |hxi , x i| ) . ∗ kx kX ∗ ≤1 i=1

n n T : X → Y is (q, p)-summing iﬀ k(T (xi ))i=1kq ≤ C k(xi )i=1kp,ω

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

Notation. Let X be a Banach space and x1, ..., xn ∈ X .

n 1 n P p p k(xi )i=1kp = ( kxi k ) . i=1

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 5 / 27 n n T : X → Y is (q, p)-summing iff k(T (xi ))i=1kq ≤ C k(xi )i=1kp,ω

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

Notation. Let X be a Banach space and x1, ..., xn ∈ X .

n 1 n P p p k(xi )i=1kp = ( kxi k ) . i=1

n 1 n P ∗ p p k(xi )i=1kp,ω = sup ( |hxi , x i| ) . ∗ kx kX ∗ ≤1 i=1

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 5 / 27 Summability properties and factorization of homogeneous polynomials Introduction and basic definitions

Notation. Let X be a Banach space and x1, ..., xn ∈ X .

n 1 n P p p k(xi )i=1kp = ( kxi k ) . i=1

n 1 n P ∗ p p k(xi )i=1kp,ω = sup ( |hxi , x i| ) . ∗ kx kX ∗ ≤1 i=1

n n T : X → Y is (q, p)-summing iﬀ k(T (xi ))i=1kq ≤ C k(xi )i=1kp,ω

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 5 / 27 The projective s-tensor norm πs is

k k n X n X o πs (u) = inf |λj |kxj k ; k ∈ N, u = λj xj ⊗ · · · ⊗ xj . j=1 j=1

The injective s-tensor norm εs is

n k o X ∗ ∗ ∗ εs (u) = sup λj hxj , x i · · · hxj , x i ; x ∈ BX ∗ . j=1

m,s m,s m,s m,s ⊗b πs X and ⊗b εs X are the completions of ⊗πs X and ⊗εs X , respectively.

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

If X is a Banach space, the symmetric tensor product ⊗m,s X is the space generated by the elements x ⊗ · · · ⊗ x of the n-fold tensor product of X . Let u ∈ ⊗m,s X .

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 6 / 27 The injective s-tensor norm εs is

n k o X ∗ ∗ ∗ εs (u) = sup λj hxj , x i · · · hxj , x i ; x ∈ BX ∗ . j=1

m,s m,s m,s m,s ⊗b πs X and ⊗b εs X are the completions of ⊗πs X and ⊗εs X , respectively.

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

If X is a Banach space, the symmetric tensor product ⊗m,s X is the space generated by the elements x ⊗ · · · ⊗ x of the n-fold tensor product of X . Let u ∈ ⊗m,s X .

The projective s-tensor norm πs is

k k n X n X o πs (u) = inf |λj |kxj k ; k ∈ N, u = λj xj ⊗ · · · ⊗ xj . j=1 j=1

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 6 / 27 m,s m,s m,s m,s ⊗b πs X and ⊗b εs X are the completions of ⊗πs X and ⊗εs X , respectively.

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

If X is a Banach space, the symmetric tensor product ⊗m,s X is the space generated by the elements x ⊗ · · · ⊗ x of the n-fold tensor product of X . Let u ∈ ⊗m,s X .

The projective s-tensor norm πs is

k k n X n X o πs (u) = inf |λj |kxj k ; k ∈ N, u = λj xj ⊗ · · · ⊗ xj . j=1 j=1

The injective s-tensor norm εs is

n k o X ∗ ∗ ∗ εs (u) = sup λj hxj , x i · · · hxj , x i ; x ∈ BX ∗ . j=1

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 6 / 27 Summability properties and factorization of homogeneous polynomials Introduction and basic definitions

If X is a Banach space, the symmetric tensor product ⊗m,s X is the space generated by the elements x ⊗ · · · ⊗ x of the n-fold tensor product of X . Let u ∈ ⊗m,s X .

The projective s-tensor norm πs is

k k n X n X o πs (u) = inf |λj |kxj k ; k ∈ N, u = λj xj ⊗ · · · ⊗ xj . j=1 j=1

The injective s-tensor norm εs is

n k o X ∗ ∗ ∗ εs (u) = sup λj hxj , x i · · · hxj , x i ; x ∈ BX ∗ . j=1

m,s m,s m,s m,s ⊗b πs X and ⊗b εs X are the completions of ⊗πs X and ⊗εs X , respectively.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 6 / 27 The space of all continuous m-homogeneous polynomials from X to Y is denoted by P(mX ; Y ).

Given P ∈ P(mX ; Y ), the π-linearization of P is the unique linear m,s operator PL,s : ⊗b πs X → Y such that PL,s (x ⊗ · · · ⊗ x) = P(x) for all m m m,s x ∈ X . The space P( X ) := P( X , R) and the dual of ⊗b πs X are isometrically isomorphic via the correspondence P ↔ PL,s .

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

P : X → Y is an m-homogeneous polynomial if there is a symmetric m-linear mapping A: X × · · · × X → Y such that P(x) = A(x,..., x).

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 7 / 27 Given P ∈ P(mX ; Y ), the π-linearization of P is the unique linear m,s operator PL,s : ⊗b πs X → Y such that PL,s (x ⊗ · · · ⊗ x) = P(x) for all m m m,s x ∈ X . The space P( X ) := P( X , R) and the dual of ⊗b πs X are isometrically isomorphic via the correspondence P ↔ PL,s .

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

P : X → Y is an m-homogeneous polynomial if there is a symmetric m-linear mapping A: X × · · · × X → Y such that P(x) = A(x,..., x). The space of all continuous m-homogeneous polynomials from X to Y is denoted by P(mX ; Y ).

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 7 / 27 m m m,s The space P( X ) := P( X , R) and the dual of ⊗b πs X are isometrically isomorphic via the correspondence P ↔ PL,s .

Summability properties and factorization of homogeneous polynomials Introduction and basic deﬁnitions

Given P ∈ P(mX ; Y ), the π-linearization of P is the unique linear m,s operator PL,s : ⊗b πs X → Y such that PL,s (x ⊗ · · · ⊗ x) = P(x) for all x ∈ X .

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 7 / 27 Summability properties and factorization of homogeneous polynomials Introduction and basic definitions

1 Introduction and basic deﬁnitions

2 Factorization theorem for the class of the factorable p-dominated polynomials

3 Factorization through Lorentz spaces: a Pisier’s Theorem

4 (q, p)-Concavity for polynomials

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 8 / 27 m Pd,p( X ; Y ) is the space of all p-dominated m-homogeneous polynomials.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let p ≥ 1. A polynomial P ∈ P(mX ; Y ) is p-dominated if there is a n constant C > 0 such that for every (xj )j=1 ⊂ X we have m k(P(xj ))j kp/m ≤ C. [k(xj )j kp,w ] .

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 9 / 27 A polynomial P ∈ P(mX ; Y ) is factorable p-dominated if there is a i i constant C > 0 such that for every set of vectors xj ∈ X , and scalars λj , 1 ≤ j ≤ n, 1 ≤ i ≤ k, n, k ∈ N, we have k k X i i X i i m k λj P(xj ) j kp ≤ Ck λj hxj , ·i j kp,w . i=1 i=1 m PFd,p( X ; Y ) for the space of all factorable p-dominated m-homogeneous polynomials. Factorable p-dominated polynomials are pm-dominated polynomials, but the converse is not true.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let p ≥ 1. A polynomial P ∈ P(mX ; Y ) is p-dominated if there is a n constant C > 0 such that for every (xj )j=1 ⊂ X we have m k(P(xj ))j kp/m ≤ C. [k(xj )j kp,w ] .

m Pd,p( X ; Y ) is the space of all p-dominated m-homogeneous polynomials.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 9 / 27 Factorable p-dominated polynomials are pm-dominated polynomials, but the converse is not true.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let p ≥ 1. A polynomial P ∈ P(mX ; Y ) is p-dominated if there is a n constant C > 0 such that for every (xj )j=1 ⊂ X we have m k(P(xj ))j kp/m ≤ C. [k(xj )j kp,w ] .

m Pd,p( X ; Y ) is the space of all p-dominated m-homogeneous polynomials.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 9 / 27 Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let p ≥ 1. A polynomial P ∈ P(mX ; Y ) is p-dominated if there is a n constant C > 0 such that for every (xj )j=1 ⊂ X we have m k(P(xj ))j kp/m ≤ C. [k(xj )j kp,w ] .

m Pd,p( X ; Y ) is the space of all p-dominated m-homogeneous polynomials.

A polynomial P ∈ P(mX ; Y ) is factorable p-dominated if there is a i i constant C > 0 such that for every set of vectors xj ∈ X , and scalars λj , 1 ≤ j ≤ n, 1 ≤ i ≤ k, n, k ∈ N, we have k k X i i X i i m k λj P(xj ) j kp ≤ Ck λj hxj , ·i j kp,w . i=1 i=1 m PFd,p( X ; Y ) for the space of all factorable p-dominated m-homogeneous polynomials. Factorable p-dominated polynomials are pm-dominated polynomials, but the converse is not true. Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 9 / 27 Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Theorem Let p ≥ 1. An m-homogeneous polynomial P ∈ P(mX ; Y ) is factorable p-dominated if and only if there is a regular Borel probability measure µ on BX ∗ , endowed with the weak-star topology, such that

1 k k ! p X Z X k λi P(xi )k ≤ C | λi hxi , x∗im|pdµ i=1 BX ∗ i=1

for all x1, ..., xk ∈ X and scalars λ1, ..., λk .

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 10 / 27 Given µ a Borel measure on BX ∗ , let jp : C(BX ∗ ) → Lp(µ) be the canonical map that identifies each continuous function with itself when considered as an element of Lp(µ). Using this linear map, Y. Meléndezand A. Tonge (1999) proved a factorization theorem for dominated polynomials P of the form P = Q ◦ jp ◦ iX , where the map Q that closes the factorization is an homogeneous polynomial. Botelho, Pellegrino and Rueda (2007), (2014), proved other factorization theorem for dominated polynomials through renormed subspaces of Lp-spaces.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let iX : X → C(BX ∗ ) be the evaluation map given by iX (x) := hx, ·i, x ∈ X .

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 11 / 27 Using this linear map, Y. Meléndezand A. Tonge (1999) proved a factorization theorem for dominated polynomials P of the form P = Q ◦ jp ◦ iX , where the map Q that closes the factorization is an homogeneous polynomial. Botelho, Pellegrino and Rueda (2007), (2014), proved other factorization theorem for dominated polynomials through renormed subspaces of Lp-spaces.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let iX : X → C(BX ∗ ) be the evaluation map given by iX (x) := hx, ·i, x ∈ X .

Given µ a Borel measure on BX ∗ , let jp : C(BX ∗ ) → Lp(µ) be the canonical map that identiﬁes each continuous function with itself when considered as an element of Lp(µ).

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 11 / 27 Botelho, Pellegrino and Rueda (2007), (2014), proved other factorization theorem for dominated polynomials through renormed subspaces of Lp-spaces.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let iX : X → C(BX ∗ ) be the evaluation map given by iX (x) := hx, ·i, x ∈ X .

Given µ a Borel measure on BX ∗ , let jp : C(BX ∗ ) → Lp(µ) be the canonical map that identiﬁes each continuous function with itself when considered as an element of Lp(µ). Using this linear map, Y. Mel´endezand A. Tonge (1999) proved a factorization theorem for dominated polynomials P of the form P = Q ◦ jp ◦ iX , where the map Q that closes the factorization is an homogeneous polynomial.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 11 / 27 Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let iX : X → C(BX ∗ ) be the evaluation map given by iX (x) := hx, ·i, x ∈ X .

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 11 / 27 m Define jp : iX (X ) → Lp(µ) as

m m m jp (x) := (jp ◦ iX (x)) = hx, ·i , x ∈ X .

m Proposition. The m-homogeneous polynomial jp is factorable p-dominated and its factorable p-dominated norm is less or equal than 1.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

We propose a diﬀerent factorization theorem through a canonical factorable p-dominated m-homogeneous polynomial through which any other polynomial of the class must factor.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 12 / 27 m Proposition. The m-homogeneous polynomial jp is factorable p-dominated and its factorable p-dominated norm is less or equal than 1.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

We propose a diﬀerent factorization theorem through a canonical factorable p-dominated m-homogeneous polynomial through which any other polynomial of the class must factor. m Deﬁne jp : iX (X ) → Lp(µ) as

m m m jp (x) := (jp ◦ iX (x)) = hx, ·i , x ∈ X .

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 12 / 27 Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

m m m jp (x) := (jp ◦ iX (x)) = hx, ·i , x ∈ X .

m Proposition. The m-homogeneous polynomial jp is factorable p-dominated and its factorable p-dominated norm is less or equal than 1.

Theorem m (Rueda, SP, 2014) Let m ∈ N and p ≥ 1. A polynomial P ∈ P( X ; Y ) is factorable p-dominated if and only if there exist a regular Borel probability measure µ on BX ∗ —with the weak* topology—, a closed subspace S of Lp(µ) and a continuous linear operator v : S → Y such that the following diagram commutes P X / Y . O

iX v m jp iX (X ) / S

C(BX ∗ ) Lp(µ)

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 13 / 27 This example was used in Botelho, Pellegrino and Rueda (2007) to prove that p-dominated m-homogeneous polynomials could not be expected to factor through a subspace of an Lp-space.

The fact that we keep the Lp(µ)-norm on the subspace S, guarantees that m the m-homogeneous polynomial jp : iX (X ) → S is weakly compact. Hence, the class of factorable p-dominated polynomials does not coincide with the class of the pm-dominated polynomials.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let us consider 1 < p < ∞. Botelho proved in (2002) the existence of a p-dominated polynomial which is not weakly compact. Also Calı¸skan and Rueda (2012).

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 14 / 27 The fact that we keep the Lp(µ)-norm on the subspace S, guarantees that m the m-homogeneous polynomial jp : iX (X ) → S is weakly compact. Hence, the class of factorable p-dominated polynomials does not coincide with the class of the pm-dominated polynomials.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let us consider 1 < p < ∞. Botelho proved in (2002) the existence of a p-dominated polynomial which is not weakly compact. Also Calı¸skan and Rueda (2012).

This example was used in Botelho, Pellegrino and Rueda (2007) to prove that p-dominated m-homogeneous polynomials could not be expected to factor through a subspace of an Lp-space.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 14 / 27 Hence, the class of factorable p-dominated polynomials does not coincide with the class of the pm-dominated polynomials.

Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let us consider 1 < p < ∞. Botelho proved in (2002) the existence of a p-dominated polynomial which is not weakly compact. Also Calı¸skan and Rueda (2012).

This example was used in Botelho, Pellegrino and Rueda (2007) to prove that p-dominated m-homogeneous polynomials could not be expected to factor through a subspace of an Lp-space.

The fact that we keep the Lp(µ)-norm on the subspace S, guarantees that m the m-homogeneous polynomial jp : iX (X ) → S is weakly compact.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 14 / 27 Summability properties and factorization of homogeneous polynomials Factorization theorem for the class of the factorable p-dominated polynomials

Let us consider 1 < p < ∞. Botelho proved in (2002) the existence of a p-dominated polynomial which is not weakly compact. Also Calı¸skan and Rueda (2012).

This example was used in Botelho, Pellegrino and Rueda (2007) to prove that p-dominated m-homogeneous polynomials could not be expected to factor through a subspace of an Lp-space.

1 Introduction and basic deﬁnitions

2 Factorization theorem for the class of the factorable p-dominated polynomials

3 Factorization through Lorentz spaces: a Pisier’s Theorem

4 (q, p)-Concavity for polynomials

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 15 / 27 Theorem

m m,s The Banach spaces PF (q,p)( X ; Y ) and Πq,p(⊗b εs X , Y ) are isometrically isomorphic.

Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

Deﬁnition Let 1 ≤ q, p < ∞ and let X be a Banach space. An m-homogeneous polynomial P : X → Y is said to be factorable (q, p)-summing if for every positive integers M, N and all M × N matrices λjk jk and xjk jk , with λjk ∈ K and xjk ∈ X , we have

M N M N q 1/q p 1/p X X X X ∗ m λjk P(xjk ) ≤ C sup λjk hxjk , x i . ∗ j=1 k=1 x ∈BX ∗ j=1 k=1

m PF (q,p)( X ; Y ) is the space of all these polynomials.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 16 / 27 Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

M N M N q 1/q p 1/p X X X X ∗ m λjk P(xjk ) ≤ C sup λjk hxjk , x i . ∗ j=1 k=1 x ∈BX ∗ j=1 k=1

m PF (q,p)( X ; Y ) is the space of all these polynomials.

Theorem

m m,s The Banach spaces PF (q,p)( X ; Y ) and Πq,p(⊗b εs X , Y ) are isometrically isomorphic.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 16 / 27 Recall that the injective norm on the tensor product X1 ⊗ · · · ⊗ Xm is

n n X 1 m o ε(u) = sup hx , ϕ1i · · · hx , ϕmi ; ϕ1 ∈ BX ∗ , . . . , ϕm ∈ BX ∗ , j j 1 m j=1

Pn 1 m u ∈ X1 ⊗ · · · ⊗ Xm, where j=1 xj ⊗ · · · ⊗ xj is any representation of u.

Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

Fix m Banach spaces X1,..., Xm.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 17 / 27 Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

Fix m Banach spaces X1,..., Xm. Recall that the injective norm on the tensor product X1 ⊗ · · · ⊗ Xm is

n n X 1 m o ε(u) = sup hx , ϕ1i · · · hx , ϕmi ; ϕ1 ∈ BX ∗ , . . . , ϕm ∈ BX ∗ , j j 1 m j=1

Pn 1 m u ∈ X1 ⊗ · · · ⊗ Xm, where j=1 xj ⊗ · · · ⊗ xj is any representation of u.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 17 / 27 (i) P is factorable (q, p)-summing.

(ii) ∀ M, N ∈ N and ∀ M × N matrices (λjk ) and (fjk ) in K and C(K),

M N M N q 1/q p 1/p X X X X m λjk P(fjk ) ≤ C λjk (fjk ) . C(K) j=1 k=1 j=1 k=1

(iii) ∃ µ on K m and a continuous linear v such that P admits a factorization: P C(K) / Y , O m v j m q,1 C(K ) / Lq,1(µ) .

Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

Theorem (Rueda, Mastylo, SP 2015) 1 ≤ p < q < ∞. T.F.A.E. for P : C(K) → Y .

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 18 / 27 (ii) ∀ M, N ∈ N and ∀ M × N matrices (λjk ) and (fjk ) in K and C(K),

M N M N q 1/q p 1/p X X X X m λjk P(fjk ) ≤ C λjk (fjk ) . C(K) j=1 k=1 j=1 k=1

(iii) ∃ µ on K m and a continuous linear v such that P admits a factorization: P C(K) / Y , O m v j m q,1 C(K ) / Lq,1(µ) .

Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

Theorem (Rueda, Mastylo, SP 2015) 1 ≤ p < q < ∞. T.F.A.E. for P : C(K) → Y . (i) P is factorable (q, p)-summing.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 18 / 27 (iii) ∃ µ on K m and a continuous linear v such that P admits a factorization: P C(K) / Y , O m v j m q,1 C(K ) / Lq,1(µ) .

Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

Theorem (Rueda, Mastylo, SP 2015) 1 ≤ p < q < ∞. T.F.A.E. for P : C(K) → Y . (i) P is factorable (q, p)-summing.

(ii) ∀ M, N ∈ N and ∀ M × N matrices (λjk ) and (fjk ) in K and C(K),

M N M N q 1/q p 1/p X X X X m λjk P(fjk ) ≤ C λjk (fjk ) . C(K) j=1 k=1 j=1 k=1

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 18 / 27 Summability properties and factorization of homogeneous polynomials Factorization through Lorentz spaces: a Pisier’s Theorem

Theorem (Rueda, Mastylo, SP 2015) 1 ≤ p < q < ∞. T.F.A.E. for P : C(K) → Y . (i) P is factorable (q, p)-summing.

(ii) ∀ M, N ∈ N and ∀ M × N matrices (λjk ) and (fjk ) in K and C(K),

M N M N q 1/q p 1/p X X X X m λjk P(fjk ) ≤ C λjk (fjk ) . C(K) j=1 k=1 j=1 k=1

(iii) ∃ µ on K m and a continuous linear v such that P admits a factorization: P C(K) / Y , O m v j m q,1 C(K ) / Lq,1(µ) . Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 18 / 27 Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

1 Introduction and basic deﬁnitions

2 Factorization theorem for the class of the factorable p-dominated polynomials

3 Factorization through Lorentz spaces: a Pisier’s Theorem

4 (q, p)-Concavity for polynomials

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 19 / 27 Let X be a σ-order continuous and s-convex Banach lattice on (Ω1, µ), X0 a sublattice of X , and Y a t-concave Banach lattice on (Ω2, ν). Then for every m-homogeneous positive polynomial P : X0 → Y there are multiplication operators Mf and Mg as well as an m-homogeneous polynomial Q : Lr(µ) → Lr/m(ν) such that Mg QMf extends P from X0 to X as

Mg QMf X ⊆ X / Y . 0 O

Mf Mg Q Lr (µ) / L r (ν) m

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Theorem. (Defant, Mastylo, 2014) Let 1 ≤ r, s, t < ∞ and m ∈ N be such that 1 ≤ t ≤ r/m ≤ r ≤ s.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 20 / 27 Then for every m-homogeneous positive polynomial P : X0 → Y there are multiplication operators Mf and Mg as well as an m-homogeneous polynomial Q : Lr(µ) → Lr/m(ν) such that Mg QMf extends P from X0 to X as

Mg QMf X ⊆ X / Y . 0 O

Mf Mg Q Lr (µ) / L r (ν) m

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 20 / 27 Mg QMf extends P from X0 to X as

Mg QMf X ⊆ X / Y . 0 O

Mf Mg Q Lr (µ) / L r (ν) m

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Theorem. (Defant, Mastylo, 2014) Let 1 ≤ r, s, t < ∞ and m ∈ N be such that 1 ≤ t ≤ r/m ≤ r ≤ s. Let X be a σ-order continuous and s-convex Banach lattice on (Ω1, µ), X0 a sublattice of X , and Y a t-concave Banach lattice on (Ω2, ν). Then for every m-homogeneous positive polynomial P : X0 → Y there are multiplication operators Mf and Mg as well as an m-homogeneous polynomial Q : Lr(µ) → Lr/m(ν) such that

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 20 / 27 Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Mg QMf X ⊆ X / Y . 0 O

Mf Mg Q Lr (µ) / L r (ν) m

A quasi-Banach lattice X = (X , k · k) is said to be p-convex (0 < p < ∞), if there exists a constant C > 0 such that

n n 1/p 1/p X p X p |xk | ≤ C kxk k , k=1 k=1

for every choice of elements x1,..., xn ∈ X . The optimal constant C in this inequality is called the p-convexity constant of X , and is denoted, by M(p)(X ).

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 21 / 27 Let 1 ≤ q, p < ∞, X be a Banach lattice on a measure space space and Y a Banach space. We define the notion of (q, p)-concavity for polynomials in the spirit of the definition of Defant and Mastylo. Adapting the inequality to the factorable case, we say that an m-homogeneous polynomial P : X → Y is (q, p)-concave if for every positive integers M, N and all M × N matrices λjk jk and xjk jk , with λjk ∈ K and xjk ∈ X (µ), we have

M N M N q 1/q p 1/p X X X X m λjk P(xjk ) ≤ C λjk xjk . X m j=1 k=1 j=1 k=1

We call K(q,p)(P) the (q, p)-concavity constant of P, that is deﬁned to be the least constant C in the inequality above.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 22 / 27 Adapting the inequality to the factorable case, we say that an m-homogeneous polynomial P : X → Y is (q, p)-concave if for every positive integers M, N and all M × N matrices λjk jk and xjk jk , with λjk ∈ K and xjk ∈ X (µ), we have

M N M N q 1/q p 1/p X X X X m λjk P(xjk ) ≤ C λjk xjk . X m j=1 k=1 j=1 k=1

We call K(q,p)(P) the (q, p)-concavity constant of P, that is deﬁned to be the least constant C in the inequality above.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 22 / 27 for every positive integers M, N and all M × N matrices λjk jk and xjk jk , with λjk ∈ K and xjk ∈ X (µ), we have

M N M N q 1/q p 1/p X X X X m λjk P(xjk ) ≤ C λjk xjk . X m j=1 k=1 j=1 k=1

We call K(q,p)(P) the (q, p)-concavity constant of P, that is deﬁned to be the least constant C in the inequality above.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Let 1 ≤ q, p < ∞, X be a Banach lattice on a measure space space and Y a Banach space. We deﬁne the notion of (q, p)-concavity for polynomials in the spirit of the deﬁnition of Defant and Mastylo. Adapting the inequality to the factorable case, we say that an m-homogeneous polynomial P : X → Y is (q, p)-concave if

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 22 / 27 We call K(q,p)(P) the (q, p)-concavity constant of P, that is defined to be the least constant C in the inequality above.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

M N M N q 1/q p 1/p X X X X m λjk P(xjk ) ≤ C λjk xjk . X m j=1 k=1 j=1 k=1

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 22 / 27 Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

M N M N q 1/q p 1/p X X X X m λjk P(xjk ) ≤ C λjk xjk . X m j=1 k=1 j=1 k=1

We call K(q,p)(P) the (q, p)-concavity constant of P, that is deﬁned to be the least constant C in the inequality above.

Theorem (Rueda, Mastylo, SP 2015) Let 1 ≤ p < q < ∞ and let X be a Banach lattice on a measure space with the m-convexity constant 1, and let P : X → Y be an m-homogeneous polynomial. Then P is (q, p)-concave if and only if it is (q, 1)-concave.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 23 / 27 The (q, p)-concavity implies that k n k n X X X m X m λi P(xi ) − µi P(yi ) ≤ C λi xi − µi yi = 0. X m k=1 j=1 k=1 j=1

Pk m Pk The deﬁnition of PL given by PL( k=1 λi xi ) := k=1 λi P(xi ) is correct and linear in the linear span of the elements of {xm; x ∈ X } that coincides with X m. The factorization diagram is commutative, and the inequality of the (q, p)-concavity of P shows that PL is also (q, p)-concave. A direct computation gives the converse and ﬁnishes the proof of the claim.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Proof.

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 24 / 27 Pk m Pk The definition of PL given by PL( k=1 λi xi ) := k=1 λi P(xi ) is correct and linear in the linear span of the elements of {xm; x ∈ X } that coincides with X m. The factorization diagram is commutative, and the inequality of the (q, p)-concavity of P shows that PL is also (q, p)-concave. A direct computation gives the converse and finishes the proof of the claim.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Proof.

CLAIM: The linear operator PL is well-deﬁned as a consequence of the (q, p)-concavity of P. Suppose that there are two linear Pk m Pn m combinations i=1 λi xi and j=1 µi yi that are equal as elements of the m-th power X m of X (µ). The (q, p)-concavity implies that k n k n X X X m X m λi P(xi ) − µi P(yi ) ≤ C λi xi − µi yi = 0. X m k=1 j=1 k=1 j=1

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 24 / 27 Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Proof.

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 24 / 27 By the claim, there is a factorization as P = PL ◦ δm through a linear map m PL : X → Y that is (q, 1)-concave. m Since X is a Banach lattice, PL is also (q, p)-concave, as a consequence of the characterization of (q, p)-concave linear operators and Pisier’s Theorem. The claim above gives that PL is (q, p)-concave, and so we have the result. The converse implication comes directly from the inequality in the definition of (q, p)-concave polynomial.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Now, suppose that P is (q, 1)-concave.

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 25 / 27 m Since X is a Banach lattice, PL is also (q, p)-concave, as a consequence of the characterization of (q, p)-concave linear operators and Pisier’s Theorem. The claim above gives that PL is (q, p)-concave, and so we have the result. The converse implication comes directly from the inequality in the definition of (q, p)-concave polynomial.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Now, suppose that P is (q, 1)-concave. By the claim, there is a factorization as P = PL ◦ δm through a linear map m PL : X → Y that is (q, 1)-concave.

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 25 / 27 The claim above gives that PL is (q, p)-concave, and so we have the result. The converse implication comes directly from the inequality in the definition of (q, p)-concave polynomial.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 25 / 27 The converse implication comes directly from the inequality in the definition of (q, p)-concave polynomial.

Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

Now, suppose that P is (q, 1)-concave. By the claim, there is a factorization as P = PL ◦ δm through a linear map m PL : X → Y that is (q, 1)-concave. m Since X is a Banach lattice, PL is also (q, p)-concave, as a consequence of the characterization of (q, p)-concave linear operators and Pisier’s Theorem. The claim above gives that PL is (q, p)-concave, and so we have the result.

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 25 / 27 Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

G. Botelho, Weakly compact and absolutely summing polynomials. J. Math. Anal. Appl. 265 (2002), no. 2, 458–462. G. Botelho, D. Pellegrino, P. Rueda, Pietsch’s factorization theorem for dominated polynomials, J. Funct. Anal. 243 (2007), no. 1, 257-269. On Pietsch measures for summing operators and dominated polynomials. Lin. Mult. Alg. 62,7 (2014), 860-874. A. Defant, M. Mastylo, Factorization and extension of positive homogeneous polynomials, Studia Math. 221 (2014), 87-99. Y. Meléndezand A. Tonge, Polynomials and the Pietsch domination theorem, Proc. Roy. Irish Acad. Sect. A 99 (1999), 195–212. P. Rueda, M. Mastylo and E. A. SánchezPérez, Duality on spaces of summing polynomials and factorization through Lorentz spaces. Preprint. P. Rueda and E. A. SánchezPérez, Factorization of p-dominated polynomials through Lp-spaces, Michigan Math. J. 63,2 (2014), 345-354. E. Çalı¸skan and P. Rueda, On distinguished polynomials and their projections, Ann. Acad. Sci. Fenn. Math. 37,2 (2012), 595–603. Enrique A. SánchezPérez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 26 / 27 Summability properties and factorization of homogeneous polynomials (q, p)-Concavity for polynomials

FELICIDADESSSSSS!!!!!! and also from Mietek!

Enrique A. S´anchezP´erez (U.P. Valencia) Summability properties and factorization of homogeneousValencia, polynomials 26/09/2015 27 / 27