Universal operators between separable Banach spaces Wiesaw Kubi s - PowerPoint PPT Presentation

Universal operators between separable Banach spaces Wiesław Kubi´ s Institute of Mathematics, Academy of Sciences of the Czech Republic http://www.math.cas.cz/kubis/ Integration, Vector Measures and Related Topics Be ¸dlewo, June 15 – 21, 2014 W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 1 / 15

Co-authors: F . Cabello Sánchez (University of Extremadura, Spain) 1 J. Garbuli´ nska-We ¸grzyn (Jan Kochanowski University in Kielce, 2 Poland) W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 2 / 15

Motivation Question: Fix a separable space F . Does there exist a non-expansive linear operator P : E → F “containing" all non-expansive linear operators from separable spaces into F ? W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 3 / 15

Motivation Question: Fix a separable space F . Does there exist a non-expansive linear operator P : E → F “containing" all non-expansive linear operators from separable spaces into F ? Some history: Caradus (1969): Universal operators in the Hilbert space Lindenstrauss and Pełczy´ nski (1968): Isomorphically universal operator for non-compact operators W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 3 / 15

Definition Let E , F be separable spaces. An operator P : E → F is left-universal if for every operator T : X → F with X separable and with � T � � � P � there exists an isometric embedding e : X → E such that T = P ◦ e . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 4 / 15

Definition Let E , F be separable spaces. An operator P : E → F is left-universal if for every operator T : X → F with X separable and with � T � � � P � there exists an isometric embedding e : X → E such that T = P ◦ e . An operator P : E → F is universal if for every operator T : X → Y with X , Y separable and with � T � � � P � there exist isometric embeddings i : X → E , j : Y → F such that P ◦ i = j ◦ T . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 4 / 15

� � � � Definition Let E , F be separable spaces. An operator P : E → F is left-universal if for every operator T : X → F with X separable and with � T � � � P � there exists an isometric embedding e : X → E such that T = P ◦ e . An operator P : E → F is universal if for every operator T : X → Y with X , Y separable and with � T � � � P � there exist isometric embeddings i : X → E , j : Y → F such that P ◦ i = j ◦ T . P P � F � F E E e j i T � Y X X T W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 4 / 15

Quasi-Banach spaces Definition Fix p ∈ ( 0 , 1 ] . A p -normed space is a vector space X endowed with a p -norm � · � , that is, � x � � 0 and � x � = 0 ⇐ ⇒ x = 0, � λ x � = | λ | · � x � , � x + y � p � � x � p + � y � p . A quasi-Banach space is a p -Banach space for some p ∈ ( 0 , 1 ] . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 5 / 15

Quasi-Banach spaces Definition Fix p ∈ ( 0 , 1 ] . A p -normed space is a vector space X endowed with a p -norm � · � , that is, � x � � 0 and � x � = 0 ⇐ ⇒ x = 0, � λ x � = | λ | · � x � , � x + y � p � � x � p + � y � p . A quasi-Banach space is a p -Banach space for some p ∈ ( 0 , 1 ] . Canonical example: ℓ p W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 5 / 15

Main result I Theorem Fix p ∈ ( 0 , 1 ] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator P F : E → F W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Main result I Theorem Fix p ∈ ( 0 , 1 ] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator P F : E → F with the following property: (G) Given ε > 0 , finite-dimensional spaces A ⊆ B, an isometric embedding e : A → E, and a non-expansive operator T : B → F such that T ↾ A = P F ◦ e, there exists an ε -isometric embedding f : B → E such that � P F ◦ f − T � � ε . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Main result I Theorem Fix p ∈ ( 0 , 1 ] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator P F : E → F with the following property: (G) Given ε > 0 , finite-dimensional spaces A ⊆ B, an isometric embedding e : A → E, and a non-expansive operator T : B → F such that T ↾ A = P F ◦ e, there exists an ε -isometric embedding f : B → E such that � P F ◦ f − T � � ε . Furthermore, this property describes P F uniquely, up to isometry. W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Main result I Theorem Fix p ∈ ( 0 , 1 ] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator P F : E → F with the following property: (G) Given ε > 0 , finite-dimensional spaces A ⊆ B, an isometric embedding e : A → E, and a non-expansive operator T : B → F such that T ↾ A = P F ◦ e, there exists an ε -isometric embedding f : B → E such that � P F ◦ f − T � � ε . Furthermore, this property describes P F uniquely, up to isometry. Corollary The operator P F is a projection. Proof. Apply left-universality to id F . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Fix p ∈ ( 0 , 1 ] . Let 0 denote the trivial p -Banach space. Definition The domain of the left-universal operator P 0 is called the p -Gurari˘ ı space, denoted by G p . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 7 / 15

Fix p ∈ ( 0 , 1 ] . Let 0 denote the trivial p -Banach space. Definition The domain of the left-universal operator P 0 is called the p -Gurari˘ ı space, denoted by G p . Fact The p-Gurari˘ ı space is the unique separable p-Banach space G satisfying the following condition: ( G 0 ) For every ε > 0 , for every finite-dimensional spaces A ⊆ B, for every isometric embedding e : A → G there exists an ε -isometric embedding f : B → G such that f ↾ A = e. For p = 1, the space G p was constructed by Gurari˘ ı in 1966. Uniqueness was proved by Lusky in 1976. W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 7 / 15

Fact The space G p is isometrically universal in the class of all separable p-Banach spaces. W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 8 / 15

Fact The space G p is isometrically universal in the class of all separable p-Banach spaces. Fact If p < 1 then the dual of G p is trivial. W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 8 / 15

Fact The space G p is isometrically universal in the class of all separable p-Banach spaces. Fact If p < 1 then the dual of G p is trivial. Theorem For every separable p-Banach space F, the kernel of P F is linearly isometric to G p . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 8 / 15

Almost injective spaces Definition A p -Banach space F is locally almost 1-injective if for every finite-dimensional spaces A ⊆ B , for every bounded linear operator S : A → F , for every ε > 0 there exists a linear operator T : B → F such that and � T � � ( 1 + ε ) � S � . T ↾ A = S W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 9 / 15

Almost injective spaces Definition A p -Banach space F is locally almost 1-injective if for every finite-dimensional spaces A ⊆ B , for every bounded linear operator S : A → F , for every ε > 0 there exists a linear operator T : B → F such that and � T � � ( 1 + ε ) � S � . T ↾ A = S Fact A separable Banach space is locally almost 1-injective ⇐ ⇒ it is a Lindenstrauss space, i.e. an isometric L 1 predual. W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 9 / 15

Lemma Every 1 -complemented subspace of G p is locally almost 1-injective. W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 10 / 15

Lemma Every 1 -complemented subspace of G p is locally almost 1-injective. Theorem For a separable p-Banach space F, the following properties are equivalent: 1 F is locally almost 1-injective. dom ( P F ) is linearly isometric to G p . 2 W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 10 / 15

Lemma Every 1 -complemented subspace of G p is locally almost 1-injective. Theorem For a separable p-Banach space F, the following properties are equivalent: 1 F is locally almost 1-injective. dom ( P F ) is linearly isometric to G p . 2 Corollary G p ≈ G p ⊕ G p . W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 10 / 15