Obstructions to embedding subsets of Schatten classes in L p spaces Gideon Schechtman Joint work with Assaf Naor Berkeley, November 2017 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Bi-Lipschitz embedding A metric space ( X , d X ) is said to admit a bi-Lipschitz embedding into a metric space ( Y , d Y ) if there exist s ∈ ( 0 , ∞ ) , D ∈ [ 1 , ∞ ) and a mapping f : X → Y such that ∀ x , y ∈ X , sd X ( x , y ) ≤ d Y ( f ( x ) , f ( y )) ≤ Dsd X ( x , y ) . When this happens we say that ( X , d X ) embeds into ( Y , d Y ) with distortion at most D . We denote by c Y ( X ) the infinum over such D ∈ [ 1 , ∞ ] . When Y = L p we use the shorter notation c L p ( X ) = c p ( X ) . We are interested in bounding from below the distortion of embedding certain metric spaces into L p . I’ll concentrate on embedding certain grids in Schatten p-classes into L p . Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Bi-Lipschitz embedding A metric space ( X , d X ) is said to admit a bi-Lipschitz embedding into a metric space ( Y , d Y ) if there exist s ∈ ( 0 , ∞ ) , D ∈ [ 1 , ∞ ) and a mapping f : X → Y such that ∀ x , y ∈ X , sd X ( x , y ) ≤ d Y ( f ( x ) , f ( y )) ≤ Dsd X ( x , y ) . When this happens we say that ( X , d X ) embeds into ( Y , d Y ) with distortion at most D . We denote by c Y ( X ) the infinum over such D ∈ [ 1 , ∞ ] . When Y = L p we use the shorter notation c L p ( X ) = c p ( X ) . We are interested in bounding from below the distortion of embedding certain metric spaces into L p . I’ll concentrate on embedding certain grids in Schatten p-classes into L p . Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes Given a (finite or infinite, real or complex) matrix A and 1 ≤ p < ∞ ∞ � A � p = ( trace ( A ∗ A ) p / 2 ) 1 / 2 = ( � λ p i ) 1 / p i = 1 where the λ i -s are the singular values of A . � A � ∞ = � A : ℓ 2 → ℓ 2 � . S n p is the space of all n × n matrices equipped with the norm � · � p . e ij denotes the matrix with 1 in the ij place and zero elsewhere. This is a good basis in a certain order but, except if p = 2, NOT a good unconditional basis. Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes Given a (finite or infinite, real or complex) matrix A and 1 ≤ p < ∞ ∞ � A � p = ( trace ( A ∗ A ) p / 2 ) 1 / 2 = ( � λ p i ) 1 / p i = 1 where the λ i -s are the singular values of A . � A � ∞ = � A : ℓ 2 → ℓ 2 � . S n p is the space of all n × n matrices equipped with the norm � · � p . e ij denotes the matrix with 1 in the ij place and zero elsewhere. This is a good basis in a certain order but, except if p = 2, NOT a good unconditional basis. Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes Given a (finite or infinite, real or complex) matrix A and 1 ≤ p < ∞ ∞ � A � p = ( trace ( A ∗ A ) p / 2 ) 1 / 2 = ( � λ p i ) 1 / p i = 1 where the λ i -s are the singular values of A . � A � ∞ = � A : ℓ 2 → ℓ 2 � . S n p is the space of all n × n matrices equipped with the norm � · � p . e ij denotes the matrix with 1 in the ij place and zero elsewhere. This is a good basis in a certain order but, except if p = 2, NOT a good unconditional basis. Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes Given a (finite or infinite, real or complex) matrix A and 1 ≤ p < ∞ ∞ � A � p = ( trace ( A ∗ A ) p / 2 ) 1 / 2 = ( � λ p i ) 1 / p i = 1 where the λ i -s are the singular values of A . � A � ∞ = � A : ℓ 2 → ℓ 2 � . S n p is the space of all n × n matrices equipped with the norm � · � p . e ij denotes the matrix with 1 in the ij place and zero elsewhere. This is a good basis in a certain order but, except if p = 2, NOT a good unconditional basis. Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes Recall that { x i } m i = 1 ⊂ X is a K -unconditional if for all (say real) scalars { a i } m i = 1 and signs { ε i } m i = 1 , � � � a i x i � ≤ K � ε i a i x i � . Here is a simple way to show that e ij is not a good unconditional basis. For simplicity, p = 1. Claim n � ε ij e ij � 1 ≈ n 3 / 2 , E ε ij = ± 1 � i , j = 1 While n � � e ij � 1 = n . i , j = 1 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes Recall that { x i } m i = 1 ⊂ X is a K -unconditional if for all (say real) scalars { a i } m i = 1 and signs { ε i } m i = 1 , � � � a i x i � ≤ K � ε i a i x i � . Here is a simple way to show that e ij is not a good unconditional basis. For simplicity, p = 1. Claim n � ε ij e ij � 1 ≈ n 3 / 2 , E ε ij = ± 1 � i , j = 1 While n � � e ij � 1 = n . i , j = 1 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes Recall that { x i } m i = 1 ⊂ X is a K -unconditional if for all (say real) scalars { a i } m i = 1 and signs { ε i } m i = 1 , � � � a i x i � ≤ K � ε i a i x i � . Here is a simple way to show that e ij is not a good unconditional basis. For simplicity, p = 1. Claim n � ε ij e ij � 1 ≈ n 3 / 2 , E ε ij = ± 1 � i , j = 1 While n � � e ij � 1 = n . i , j = 1 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes The ≥ side in the first equivalence follows easily from duality between S n 1 and S n ∞ and the not-hard fact that n � ε ij e ij � ∞ � n 1 / 2 . E ε ij = ± 1 � i , j = 1 � � n Note also that for all ε i , δ j = ± 1 i , j = 1 ε i δ j e ij � 1 = n . So, the best constant K in the inequality n n � � E ε ij = ± 1 � ε ij x ij � 1 ≤ K E ε i ,δ j = ± 1 � ε i δ j x ij � 1 i , j = 1 i , j = 1 holding for all { x ij } in S 1 is at least of order n 1 / 2 . On the other hand, it follows from Khinchine’s inequality that for all { x ij } in L 1 , n n � � E ε ij = ± 1 � ε ij x ij � 1 � E ε i ,δ j = ± 1 � ε i δ j x ij � 1 . ( upper property α ) i , j = 1 i , j = 1 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes The ≥ side in the first equivalence follows easily from duality between S n 1 and S n ∞ and the not-hard fact that n � ε ij e ij � ∞ � n 1 / 2 . E ε ij = ± 1 � i , j = 1 � � n Note also that for all ε i , δ j = ± 1 i , j = 1 ε i δ j e ij � 1 = n . So, the best constant K in the inequality n n � � E ε ij = ± 1 � ε ij x ij � 1 ≤ K E ε i ,δ j = ± 1 � ε i δ j x ij � 1 i , j = 1 i , j = 1 holding for all { x ij } in S 1 is at least of order n 1 / 2 . On the other hand, it follows from Khinchine’s inequality that for all { x ij } in L 1 , n n � � E ε ij = ± 1 � ε ij x ij � 1 � E ε i ,δ j = ± 1 � ε i δ j x ij � 1 . ( upper property α ) i , j = 1 i , j = 1 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes The ≥ side in the first equivalence follows easily from duality between S n 1 and S n ∞ and the not-hard fact that n � ε ij e ij � ∞ � n 1 / 2 . E ε ij = ± 1 � i , j = 1 � � n Note also that for all ε i , δ j = ± 1 i , j = 1 ε i δ j e ij � 1 = n . So, the best constant K in the inequality n n � � E ε ij = ± 1 � ε ij x ij � 1 ≤ K E ε i ,δ j = ± 1 � ε i δ j x ij � 1 i , j = 1 i , j = 1 holding for all { x ij } in S 1 is at least of order n 1 / 2 . On the other hand, it follows from Khinchine’s inequality that for all { x ij } in L 1 , n n � � E ε ij = ± 1 � ε ij x ij � 1 � E ε i ,δ j = ± 1 � ε i δ j x ij � 1 . ( upper property α ) i , j = 1 i , j = 1 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
Schatten classes The ≥ side in the first equivalence follows easily from duality between S n 1 and S n ∞ and the not-hard fact that n � ε ij e ij � ∞ � n 1 / 2 . E ε ij = ± 1 � i , j = 1 � � n Note also that for all ε i , δ j = ± 1 i , j = 1 ε i δ j e ij � 1 = n . So, the best constant K in the inequality n n � � E ε ij = ± 1 � ε ij x ij � 1 ≤ K E ε i ,δ j = ± 1 � ε i δ j x ij � 1 i , j = 1 i , j = 1 holding for all { x ij } in S 1 is at least of order n 1 / 2 . On the other hand, it follows from Khinchine’s inequality that for all { x ij } in L 1 , n n � � E ε ij = ± 1 � ε ij x ij � 1 � E ε i ,δ j = ± 1 � ε i δ j x ij � 1 . ( upper property α ) i , j = 1 i , j = 1 Gideon Schechtman Obstructions to embedding subsets of Schatten classes in L p spaces
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