Projections onto spaces of polynomials Tommaso Russo (Joint work in - PowerPoint PPT Presentation

Faculty of Electrical Engineering Czech Technical University in Prague Projections onto spaces of polynomials Tommaso Russo (Joint work in progress with P. Hájek) Workshop on Banach spaces and Banach lattices Madrid, Spain September 9–13, 2019

International Mobility of Researchers in CTU Project number: CZ.02.2.69/0.0/0.0/16_027/0008465

1 The starting point Theorem (Godefroy and Kalton, 2003) Problem (Godefroy) T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials ▶ A Banach space X has the AP if for every compact set K ⊆ X and ε > 0 there exists a fjnite-rank, bounded linear operator T : X → X such that ∥ Tx − x ∥ < ε ( x ∈ K ); ▶ X has λ -BAP if additionally ∥ T ∥ ⩽ λ . A Banach space X has the λ -BAP if and only if F ( X ) has the λ -BAP. ▶ In particular, F ( ℓ 2 ) has the MAP ( ≡ 1 -BAP). Does Lip 0 ( ℓ 2 ) have the AP? ▶ Grothendieck (1955). The AP passes from X ∗ to X ; ▶ Hence, this would be a stronger result.

1 The starting point Theorem (Godefroy and Kalton, 2003) Problem (Godefroy) T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials ▶ A Banach space X has the AP if for every compact set K ⊆ X and ε > 0 there exists a fjnite-rank, bounded linear operator T : X → X such that ∥ Tx − x ∥ < ε ( x ∈ K ); ▶ X has λ -BAP if additionally ∥ T ∥ ⩽ λ . A Banach space X has the λ -BAP if and only if F ( X ) has the λ -BAP. ▶ In particular, F ( ℓ 2 ) has the MAP ( ≡ 1 -BAP). Does Lip 0 ( ℓ 2 ) have the AP? ▶ Grothendieck (1955). The AP passes from X ∗ to X ; ▶ Hence, this would be a stronger result.

1 The starting point Theorem (Godefroy and Kalton, 2003) Problem (Godefroy) T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials ▶ A Banach space X has the AP if for every compact set K ⊆ X and ε > 0 there exists a fjnite-rank, bounded linear operator T : X → X such that ∥ Tx − x ∥ < ε ( x ∈ K ); ▶ X has λ -BAP if additionally ∥ T ∥ ⩽ λ . A Banach space X has the λ -BAP if and only if F ( X ) has the λ -BAP. ▶ In particular, F ( ℓ 2 ) has the MAP ( ≡ 1 -BAP). Does Lip 0 ( ℓ 2 ) have the AP? ▶ Grothendieck (1955). The AP passes from X ∗ to X ; ▶ Hence, this would be a stronger result.

1 The starting point Theorem (Godefroy and Kalton, 2003) Problem (Godefroy) T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials ▶ A Banach space X has the AP if for every compact set K ⊆ X and ε > 0 there exists a fjnite-rank, bounded linear operator T : X → X such that ∥ Tx − x ∥ < ε ( x ∈ K ); ▶ X has λ -BAP if additionally ∥ T ∥ ⩽ λ . A Banach space X has the λ -BAP if and only if F ( X ) has the λ -BAP. ▶ In particular, F ( ℓ 2 ) has the MAP ( ≡ 1 -BAP). Does Lip 0 ( ℓ 2 ) have the AP? ▶ Grothendieck (1955). The AP passes from X ∗ to X ; ▶ Hence, this would be a stronger result.

1 The starting point Theorem (Godefroy and Kalton, 2003) Problem (Godefroy) T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials ▶ A Banach space X has the AP if for every compact set K ⊆ X and ε > 0 there exists a fjnite-rank, bounded linear operator T : X → X such that ∥ Tx − x ∥ < ε ( x ∈ K ); ▶ X has λ -BAP if additionally ∥ T ∥ ⩽ λ . A Banach space X has the λ -BAP if and only if F ( X ) has the λ -BAP. ▶ In particular, F ( ℓ 2 ) has the MAP ( ≡ 1 -BAP). Does Lip 0 ( ℓ 2 ) have the AP? ▶ Grothendieck (1955). The AP passes from X ∗ to X ; ▶ Hence, this would be a stronger result.

1 The starting point Theorem (Godefroy and Kalton, 2003) Problem (Godefroy) T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials ▶ A Banach space X has the AP if for every compact set K ⊆ X and ε > 0 there exists a fjnite-rank, bounded linear operator T : X → X such that ∥ Tx − x ∥ < ε ( x ∈ K ); ▶ X has λ -BAP if additionally ∥ T ∥ ⩽ λ . A Banach space X has the λ -BAP if and only if F ( X ) has the λ -BAP. ▶ In particular, F ( ℓ 2 ) has the MAP ( ≡ 1 -BAP). Does Lip 0 ( ℓ 2 ) have the AP? ▶ Grothendieck (1955). The AP passes from X ∗ to X ; ▶ Hence, this would be a stronger result.

2 Our approach on X . T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials For a Banach space X , P ( 2 X ) is a subspace of Lip 0 ( X ) . ▶ P ( 2 X ) is the Banach space of bounded 2 -homogeneous polynomials ▶ P ∈ P ( 2 X ) if there is a bounded bilinear map M : X × X → R such that P ( x ) = M ( x , x ) ; ▶ ∥ P ∥ P = sup x ∈ B X | P ( x ) | . ▶ But polynomials are not Lipschitz functions! ▶ However, they are Lipschitz on the unit ball. ▶ Therefore, P ( 2 X ) is a natural subspace of Lip 0 ( B X ) ; ▶ Moreover, ∥ · ∥ P is equivalent to ∥ · ∥ Lip . ▶ Consequently, P ( 2 X ) is naturally isomorphic to a subspace of Lip 0 ( B X ) , via the restriction map P �→ P ↾ B X .

2 Our approach on X . T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials For a Banach space X , P ( 2 X ) is a subspace of Lip 0 ( X ) . ▶ P ( 2 X ) is the Banach space of bounded 2 -homogeneous polynomials ▶ P ∈ P ( 2 X ) if there is a bounded bilinear map M : X × X → R such that P ( x ) = M ( x , x ) ; ▶ ∥ P ∥ P = sup x ∈ B X | P ( x ) | . ▶ But polynomials are not Lipschitz functions! ▶ However, they are Lipschitz on the unit ball. ▶ Therefore, P ( 2 X ) is a natural subspace of Lip 0 ( B X ) ; ▶ Moreover, ∥ · ∥ P is equivalent to ∥ · ∥ Lip . ▶ Consequently, P ( 2 X ) is naturally isomorphic to a subspace of Lip 0 ( B X ) , via the restriction map P �→ P ↾ B X .

2 Our approach on X . T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials For a Banach space X , P ( 2 X ) is a subspace of Lip 0 ( X ) . ▶ P ( 2 X ) is the Banach space of bounded 2 -homogeneous polynomials ▶ P ∈ P ( 2 X ) if there is a bounded bilinear map M : X × X → R such that P ( x ) = M ( x , x ) ; ▶ ∥ P ∥ P = sup x ∈ B X | P ( x ) | . ▶ But polynomials are not Lipschitz functions! ▶ However, they are Lipschitz on the unit ball. ▶ Therefore, P ( 2 X ) is a natural subspace of Lip 0 ( B X ) ; ▶ Moreover, ∥ · ∥ P is equivalent to ∥ · ∥ Lip . ▶ Consequently, P ( 2 X ) is naturally isomorphic to a subspace of Lip 0 ( B X ) , via the restriction map P �→ P ↾ B X .

2 Our approach on X . T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials For a Banach space X , P ( 2 X ) is a subspace of Lip 0 ( X ) . ▶ P ( 2 X ) is the Banach space of bounded 2 -homogeneous polynomials ▶ P ∈ P ( 2 X ) if there is a bounded bilinear map M : X × X → R such that P ( x ) = M ( x , x ) ; ▶ ∥ P ∥ P = sup x ∈ B X | P ( x ) | . ▶ But polynomials are not Lipschitz functions! ▶ However, they are Lipschitz on the unit ball. ▶ Therefore, P ( 2 X ) is a natural subspace of Lip 0 ( B X ) ; ▶ Moreover, ∥ · ∥ P is equivalent to ∥ · ∥ Lip . ▶ Consequently, P ( 2 X ) is naturally isomorphic to a subspace of Lip 0 ( B X ) , via the restriction map P �→ P ↾ B X .

2 Our approach on X . T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials For a Banach space X , P ( 2 X ) is a subspace of Lip 0 ( X ) . ▶ P ( 2 X ) is the Banach space of bounded 2 -homogeneous polynomials ▶ P ∈ P ( 2 X ) if there is a bounded bilinear map M : X × X → R such that P ( x ) = M ( x , x ) ; ▶ ∥ P ∥ P = sup x ∈ B X | P ( x ) | . ▶ But polynomials are not Lipschitz functions! ▶ However, they are Lipschitz on the unit ball. ▶ Therefore, P ( 2 X ) is a natural subspace of Lip 0 ( B X ) ; ▶ Moreover, ∥ · ∥ P is equivalent to ∥ · ∥ Lip . ▶ Consequently, P ( 2 X ) is naturally isomorphic to a subspace of Lip 0 ( B X ) , via the restriction map P �→ P ↾ B X .

2 Our approach on X . T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials For a Banach space X , P ( 2 X ) is a subspace of Lip 0 ( X ) . ▶ P ( 2 X ) is the Banach space of bounded 2 -homogeneous polynomials ▶ P ∈ P ( 2 X ) if there is a bounded bilinear map M : X × X → R such that P ( x ) = M ( x , x ) ; ▶ ∥ P ∥ P = sup x ∈ B X | P ( x ) | . ▶ But polynomials are not Lipschitz functions! ▶ However, they are Lipschitz on the unit ball. ▶ Therefore, P ( 2 X ) is a natural subspace of Lip 0 ( B X ) ; ▶ Moreover, ∥ · ∥ P is equivalent to ∥ · ∥ Lip . ▶ Consequently, P ( 2 X ) is naturally isomorphic to a subspace of Lip 0 ( B X ) , via the restriction map P �→ P ↾ B X .

2 Our approach on X . T. Russo (russotom@fel.cvut.cz) | Projections onto spaces of polynomials For a Banach space X , P ( 2 X ) is a subspace of Lip 0 ( X ) . ▶ P ( 2 X ) is the Banach space of bounded 2 -homogeneous polynomials ▶ P ∈ P ( 2 X ) if there is a bounded bilinear map M : X × X → R such that P ( x ) = M ( x , x ) ; ▶ ∥ P ∥ P = sup x ∈ B X | P ( x ) | . ▶ But polynomials are not Lipschitz functions! ▶ However, they are Lipschitz on the unit ball. ▶ Therefore, P ( 2 X ) is a natural subspace of Lip 0 ( B X ) ; ▶ Moreover, ∥ · ∥ P is equivalent to ∥ · ∥ Lip . ▶ Consequently, P ( 2 X ) is naturally isomorphic to a subspace of Lip 0 ( B X ) , via the restriction map P �→ P ↾ B X .