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. S´anchezP´erez (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 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. 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. 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 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 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. S´anchezP´erez (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;! = sup ( jhxi ; x ij ) : ∗ 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;! Summability properties and factorization of homogeneous polynomials Introduction and basic definitions Notation. Let X be a Banach space and x1; :::; xn 2 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;! = sup ( jhxi ; x ij ) : ∗ 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;! Summability properties and factorization of homogeneous polynomials Introduction and basic definitions Notation. Let X be a Banach space and x1; :::; xn 2 X . n 1 n P p p k(xi )i=1kp = ( kxi k ) : i=1 Enrique A. S´anchezP´erez (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;! Summability properties and factorization of homogeneous polynomials Introduction and basic definitions Notation. Let X be a Banach space and x1; :::; xn 2 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;! = sup ( jhxi ; x ij ) : ∗ kx kX ∗ ≤1 i=1 Enrique A. S´anchezP´erez (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 2 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;! = sup ( jhxi ; x ij ) : ∗ 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;! 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 jλj jkxj k ; k 2 N; u = λj xj ⊗ · · · ⊗ xj : j=1 j=1 The injective s-tensor norm "s is n k o X ∗ ∗ ∗ "s (u) = sup λj hxj ; x i · · · hxj ; x i ; x 2 BX ∗ : j=1 m;s m;s m;s m;s ⊗b πs X and ⊗b "s X are the completions of ⊗πs X and ⊗"s 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 2 ⊗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 "s is n k o X ∗ ∗ ∗ "s (u) = sup λj hxj ; x i · · · hxj ; x i ; x 2 BX ∗ : j=1 m;s m;s m;s m;s ⊗b πs X and ⊗b "s X are the completions of ⊗πs X and ⊗"s 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 2 ⊗m;s X . The projective s-tensor norm πs is k k n X n X o πs (u) = inf jλj jkxj k ; k 2 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 "s X are the completions of ⊗πs X and ⊗"s 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 2 ⊗m;s X . The projective s-tensor norm πs is k k n X n X o πs (u) = inf jλj jkxj k ; k 2 N; u = λj xj ⊗ · · · ⊗ xj : j=1 j=1 The injective s-tensor norm "s is n k o X ∗ ∗ ∗ "s (u) = sup λj hxj ; x i · · · hxj ; x i ; x 2 BX ∗ : j=1 Enrique A. S´anchezP´erez (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 2 ⊗m;s X . The projective s-tensor norm πs is k k n X n X o πs (u) = inf jλj jkxj k ; k 2 N; u = λj xj ⊗ · · · ⊗ xj : j=1 j=1 The injective s-tensor norm "s is n k o X ∗ ∗ ∗ "s (u) = sup λj hxj ; x i · · · hxj ; x i ; x 2 BX ∗ : j=1 m;s m;s m;s m;s ⊗b πs X and ⊗b "s X are the completions of ⊗πs X and ⊗"s 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 ).