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Norm attaining operators of finite rank (joint work with Vladimir Kadets, Gins Lpez, Miguel Martn) Dirk Werner Freie Universitt Berlin Madrid, 9.9.2019 Basic definitions and results , Dirk Werner, Norm attaining operators of finite

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Norm attaining operators of finite rank

(joint work with Vladimir Kadets, Ginés López, Miguel Martín)

Dirk Werner Freie Universität Berlin Madrid, 9.9.2019

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Basic definitions and results

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 2/10

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Basic definitions and results

  • x∗ ∈ X∗ is norm attaining (x∗ ∈ NA(X)):

∃x0: x0 = 1, x∗(x0) = sup{x∗(x): x ≤ 1} = x∗.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 2/10

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Basic definitions and results

  • x∗ ∈ X∗ is norm attaining (x∗ ∈ NA(X)):

∃x0: x0 = 1, x∗(x0) = sup{x∗(x): x ≤ 1} = x∗.

  • NA(X) = ∅ by the Hahn-Banach theorem.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 2/10

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Basic definitions and results

  • x∗ ∈ X∗ is norm attaining (x∗ ∈ NA(X)):

∃x0: x0 = 1, x∗(x0) = sup{x∗(x): x ≤ 1} = x∗.

  • NA(X) = ∅ by the Hahn-Banach theorem.
  • James: NA(X) = X∗ ⇐⇒ X reflexive.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 2/10

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Basic definitions and results

  • x∗ ∈ X∗ is norm attaining (x∗ ∈ NA(X)):

∃x0: x0 = 1, x∗(x0) = sup{x∗(x): x ≤ 1} = x∗.

  • NA(X) = ∅ by the Hahn-Banach theorem.
  • James: NA(X) = X∗ ⇐⇒ X reflexive.
  • Bishop-Phelps(-Bollobás): NA(X) is always dense; more precisely:

x = x∗ = 1, x∗(x) ≥ 1 − ϵ ⇒ ∃x0, x∗

0 :

x0 = x∗

0 = x∗ 0 (x0) = 1, x − x0 ≤ 2

  • ϵ, x∗ − x∗

0 ≤ 2

  • ϵ.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 2/10

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Basic definitions and results

  • x∗ ∈ X∗ is norm attaining (x∗ ∈ NA(X)):

∃x0: x0 = 1, x∗(x0) = sup{x∗(x): x ≤ 1} = x∗.

  • NA(X) = ∅ by the Hahn-Banach theorem.
  • James: NA(X) = X∗ ⇐⇒ X reflexive.
  • Bishop-Phelps(-Bollobás): NA(X) is always dense; more precisely:

x = x∗ = 1, x∗(x) ≥ 1 − ϵ ⇒ ∃x0, x∗

0 :

x0 = x∗

0 = x∗ 0 (x0) = 1, x − x0 ≤ 2

  • ϵ, x∗ − x∗

0 ≤ 2

  • ϵ.
  • Example: NA(c0) = c00 = all ℓ1-sequences of finite support.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 2/10

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On operators which attain their norm

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 3/10

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On operators which attain their norm

A linear operator T: X → Y is norm attaining (T ∈ NA(X, Y)): ∃x0: x0 = 1, Tx0 = sup{Tx: x ≤ 1} = T.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 3/10

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On operators which attain their norm

A linear operator T: X → Y is norm attaining (T ∈ NA(X, Y)): ∃x0: x0 = 1, Tx0 = sup{Tx: x ≤ 1} = T. Lindenstrauss 1963:

  • X reflexive ⇒ NA(X, Y) is always dense in L(X, Y).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 3/10

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On operators which attain their norm

A linear operator T: X → Y is norm attaining (T ∈ NA(X, Y)): ∃x0: x0 = 1, Tx0 = sup{Tx: x ≤ 1} = T. Lindenstrauss 1963:

  • X reflexive ⇒ NA(X, Y) is always dense in L(X, Y).
  • c0 ⊂ Y ⊂ ℓ∞ ⇒ NA(X, Y) is always dense in L(X, Y).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 3/10

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On operators which attain their norm

A linear operator T: X → Y is norm attaining (T ∈ NA(X, Y)): ∃x0: x0 = 1, Tx0 = sup{Tx: x ≤ 1} = T. Lindenstrauss 1963:

  • X reflexive ⇒ NA(X, Y) is always dense in L(X, Y).
  • c0 ⊂ Y ⊂ ℓ∞ ⇒ NA(X, Y) is always dense in L(X, Y).
  • There are X and Y such that NA(X, Y) is not dense in L(X, Y).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 3/10

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On operators which attain their norm

A linear operator T: X → Y is norm attaining (T ∈ NA(X, Y)): ∃x0: x0 = 1, Tx0 = sup{Tx: x ≤ 1} = T. Lindenstrauss 1963:

  • X reflexive ⇒ NA(X, Y) is always dense in L(X, Y).
  • c0 ⊂ Y ⊂ ℓ∞ ⇒ NA(X, Y) is always dense in L(X, Y).
  • There are X and Y such that NA(X, Y) is not dense in L(X, Y).

Bourgain 1977:

  • X RNP ⇒ NA(X, Y) is always dense in L(X, Y).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 3/10

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On operators which attain their norm

A linear operator T: X → Y is norm attaining (T ∈ NA(X, Y)): ∃x0: x0 = 1, Tx0 = sup{Tx: x ≤ 1} = T. Lindenstrauss 1963:

  • X reflexive ⇒ NA(X, Y) is always dense in L(X, Y).
  • c0 ⊂ Y ⊂ ℓ∞ ⇒ NA(X, Y) is always dense in L(X, Y).
  • There are X and Y such that NA(X, Y) is not dense in L(X, Y).

Bourgain 1977:

  • X RNP ⇒ NA(X, Y) is always dense in L(X, Y).

Gowers 1990:

  • NA(X, ℓ2) is not always dense in L(X, ℓ2).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 3/10

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Rank 2 operators into ℓ2

2

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 4/10

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Rank 2 operators into ℓ2

2

Question

  • Is NA(X, ℓ2) always nontrivially nonempty?

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 4/10

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Rank 2 operators into ℓ2

2

Question

  • Is NA(X, ℓ2) always nontrivially nonempty?
  • Is NA(X, ℓ2

2) always nontrivially nonempty?

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 4/10

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Rank 2 operators into ℓ2

2

Question

  • Is NA(X, ℓ2) always nontrivially nonempty?
  • Is NA(X, ℓ2

2) always nontrivially nonempty?

That is, is NA(2)(X, ℓ2

2) := {T ∈ NA(X, ℓ2 2): rank(T) = 2} = ∅ ?

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 4/10

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Rank 2 operators into ℓ2

2

Question

  • Is NA(X, ℓ2) always nontrivially nonempty?
  • Is NA(X, ℓ2

2) always nontrivially nonempty?

That is, is NA(2)(X, ℓ2

2) := {T ∈ NA(X, ℓ2 2): rank(T) = 2} = ∅ ?

Examples:

  • X = C(K), Tx = (x(t1), x(t2)):

T = T(1) =

  • 2

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 4/10

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Rank 2 operators into ℓ2

2

Question

  • Is NA(X, ℓ2) always nontrivially nonempty?
  • Is NA(X, ℓ2

2) always nontrivially nonempty?

That is, is NA(2)(X, ℓ2

2) := {T ∈ NA(X, ℓ2 2): rank(T) = 2} = ∅ ?

Examples:

  • X = C(K), Tx = (x(t1), x(t2)):

T = T(1) =

  • 2
  • X = L1[0, 1], Tx = (

1/2 x(t)dt, 1

1/2 x(t)dt):

T = T(1) = 1/

  • 2

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 4/10

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Mates

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2. Write Tx = (f(x), g(x)).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2. Write Tx = (f(x), g(x)). Suppose f = 1.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2. Write Tx = (f(x), g(x)). Suppose f = 1.

Lemma T ≤ 1 ⇐⇒ f + tg ≤

  • 1 + t2 for all t ∈ R.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2. Write Tx = (f(x), g(x)). Suppose f = 1.

Lemma T ≤ 1 ⇐⇒ f + tg ≤

  • 1 + t2 for all t ∈ R.

In this case g is called a mate of f.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2. Write Tx = (f(x), g(x)). Suppose f = 1.

Lemma T ≤ 1 ⇐⇒ f + tg ≤

  • 1 + t2 for all t ∈ R.

In this case g is called a mate of f. Proposition Let f = 1. (a) If, for some 0 = h ∈ BX∗, limsup

t→0

f + th − 1 t2 < ∞, then f has a mate (namely sh for some s ∈ (0, 1]).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2. Write Tx = (f(x), g(x)). Suppose f = 1.

Lemma T ≤ 1 ⇐⇒ f + tg ≤

  • 1 + t2 for all t ∈ R.

In this case g is called a mate of f. Proposition Let f = 1. (a) If, for some 0 = h ∈ BX∗, limsup

t→0

f + th − 1 t2 < ∞, then f has a mate (namely sh for some s ∈ (0, 1]). (b) If f is not an extreme point of the unit ball, then it has a mate.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Mates

Let T: X → ℓ2

2 of rank 2. Write Tx = (f(x), g(x)). Suppose f = 1.

Lemma T ≤ 1 ⇐⇒ f + tg ≤

  • 1 + t2 for all t ∈ R.

In this case g is called a mate of f. Proposition Let f = 1. (a) If, for some 0 = h ∈ BX∗, limsup

t→0

f + th − 1 t2 < ∞, then f has a mate (namely sh for some s ∈ (0, 1]). (b) If f is not an extreme point of the unit ball, then it has a mate. (c) There exists some f with a mate.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 5/10

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Existence results I

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 6/10

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Existence results I

Theorem NA(2)(X, ℓ2

2) = ∅ if and only if there exists f ∈ NA(X), f = 1, with a mate.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 6/10

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Existence results I

Theorem NA(2)(X, ℓ2

2) = ∅ if and only if there exists f ∈ NA(X), f = 1, with a mate.

Corollary

  • X not smooth ⇒ NA(2)(X, ℓ2

2) = ∅.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 6/10

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Existence results I

Theorem NA(2)(X, ℓ2

2) = ∅ if and only if there exists f ∈ NA(X), f = 1, with a mate.

Corollary

  • X not smooth ⇒ NA(2)(X, ℓ2

2) = ∅.

  • NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 6/10

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Existence results I

Theorem NA(2)(X, ℓ2

2) = ∅ if and only if there exists f ∈ NA(X), f = 1, with a mate.

Corollary

  • X not smooth ⇒ NA(2)(X, ℓ2

2) = ∅.

  • NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

Note: NA(2)(X, ℓ2

2) = ∅ ⇒ NA(2)(X, E) = ∅ whenever dimE ≥ 2.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 6/10

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Existence results I

Theorem NA(2)(X, ℓ2

2) = ∅ if and only if there exists f ∈ NA(X), f = 1, with a mate.

Corollary

  • X not smooth ⇒ NA(2)(X, ℓ2

2) = ∅.

  • NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

Note: NA(2)(X, ℓ2

2) = ∅ ⇒ NA(2)(X, E) = ∅ whenever dimE ≥ 2.

Question Does the Corollary cover all Banach spaces?

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 6/10

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The theorems of Read and Rmoutil

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 7/10

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The theorems of Read and Rmoutil

Theorem There exists a Banach space XR for which NA(XR) does not contain 2-dimensional linear subspaces.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 7/10

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The theorems of Read and Rmoutil

Theorem There exists a Banach space XR for which NA(XR) does not contain 2-dimensional linear subspaces.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 7/10

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The theorems of Read and Rmoutil

Theorem There exists a Banach space XR for which NA(XR) does not contain 2-dimensional linear subspaces.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 7/10

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The theorems of Read and Rmoutil

Theorem There exists a Banach space XR for which NA(XR) does not contain 2-dimensional linear subspaces. Charles Read (1958–2015), Martin Rmoutil, Bernardo Cascales (1958–2018)

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 7/10

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Read norms

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970)

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000)

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000) Known: A counterexample to (G) is a counterexample to (S).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000) Known: A counterexample to (G) is a counterexample to (S). Read (2013) constructs a renorming of c0 that is a counterexample to (S).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000) Known: A counterexample to (G) is a counterexample to (S). Read (2013) constructs a renorming of c0 that is a counterexample to (S). Rmoutil (2017) shows that this space, XR, is a counterexample to (G) as well.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000) Known: A counterexample to (G) is a counterexample to (S). Read (2013) constructs a renorming of c0 that is a counterexample to (S). Rmoutil (2017) shows that this space, XR, is a counterexample to (G) as

  • well. (Significant simplification of the proof by Kadets/López/Martín.)

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000) Known: A counterexample to (G) is a counterexample to (S). Read (2013) constructs a renorming of c0 that is a counterexample to (S). Rmoutil (2017) shows that this space, XR, is a counterexample to (G) as

  • well. (Significant simplification of the proof by Kadets/López/Martín.)

Call a norm p on a Banach space a Read norm if (X, p) is a counterexample to (G).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000) Known: A counterexample to (G) is a counterexample to (S). Read (2013) constructs a renorming of c0 that is a counterexample to (S). Rmoutil (2017) shows that this space, XR, is a counterexample to (G) as

  • well. (Significant simplification of the proof by Kadets/López/Martín.)

Call a norm p on a Banach space a Read norm if (X, p) is a counterexample to (G). KLMW (≥ 2019) provide a new approach to this circle of problems showing: Theorem Every separable Banach space containing a copy of c0 admits an equivalent Read norm.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Read norms

(S) Does every Banach space contain a 2-codimensional proximinal subspace? (I. Singer 1970) (G) Does NA(X) always contain a 2-dimensional linear subspace? (G. Godefroy 2000) Known: A counterexample to (G) is a counterexample to (S). Read (2013) constructs a renorming of c0 that is a counterexample to (S). Rmoutil (2017) shows that this space, XR, is a counterexample to (G) as

  • well. (Significant simplification of the proof by Kadets/López/Martín.)

Call a norm p on a Banach space a Read norm if (X, p) is a counterexample to (G). KLMW (≥ 2019) provide a new approach to this circle of problems showing: Theorem Every separable Banach space containing a copy of c0 admits an equivalent Read norm. Consequently this is also a counterexample to (S).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 8/10

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Existence results II

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 9/10

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Existence results II

Recall: NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 9/10

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Existence results II

Recall: NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

Main Theorem If NA(X) contains a nontrivial cone, i.e., some {sf + tg: s, t ≥ 0} with linearly independent f and g, then NA(2)(X, ℓ2

2) = ∅.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 9/10

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Existence results II

Recall: NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

Main Theorem If NA(X) contains a nontrivial cone, i.e., some {sf + tg: s, t ≥ 0} with linearly independent f and g, then NA(2)(X, ℓ2

2) = ∅.

Example: The original Read space XR is not smooth; hence NA(XR) contains a nontrivial cone (but not a nontrivial subspace).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 9/10

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Existence results II

Recall: NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

Main Theorem If NA(X) contains a nontrivial cone, i.e., some {sf + tg: s, t ≥ 0} with linearly independent f and g, then NA(2)(X, ℓ2

2) = ∅.

Example: The original Read space XR is not smooth; hence NA(XR) contains a nontrivial cone (but not a nontrivial subspace). There is a smooth renorming (XR, psm) with a smooth Read norm psm such that NA(XR) = NA((XR, psm)); hence NA((XR, psm)) contains a nontrivial cone (but not a nontrivial subspace).

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 9/10

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Existence results II

Recall: NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

Main Theorem If NA(X) contains a nontrivial cone, i.e., some {sf + tg: s, t ≥ 0} with linearly independent f and g, then NA(2)(X, ℓ2

2) = ∅.

Example: The original Read space XR is not smooth; hence NA(XR) contains a nontrivial cone (but not a nontrivial subspace). There is a smooth renorming (XR, psm) with a smooth Read norm psm such that NA(XR) = NA((XR, psm)); hence NA((XR, psm)) contains a nontrivial cone (but not a nontrivial subspace). Questions

  • Does the converse of the Main Theorem hold?

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 9/10

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SLIDE 57

Existence results II

Recall: NA(X) contains a 2-dimensional linear subspace ⇒ NA(2)(X, ℓ2

2) = ∅.

Main Theorem If NA(X) contains a nontrivial cone, i.e., some {sf + tg: s, t ≥ 0} with linearly independent f and g, then NA(2)(X, ℓ2

2) = ∅.

Example: The original Read space XR is not smooth; hence NA(XR) contains a nontrivial cone (but not a nontrivial subspace). There is a smooth renorming (XR, psm) with a smooth Read norm psm such that NA(XR) = NA((XR, psm)); hence NA((XR, psm)) contains a nontrivial cone (but not a nontrivial subspace). Questions

  • Does the converse of the Main Theorem hold?
  • Is the assumption of the Main Theorem always fulfilled?

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 9/10

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Density results

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Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . .

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

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SLIDE 60

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

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SLIDE 61

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.)

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

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SLIDE 62

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X:

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-63
SLIDE 63

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0,

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-64
SLIDE 64

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1,

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-65
SLIDE 65

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1, spaces having a shrinking monotone Schauder basis,

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-66
SLIDE 66

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1, spaces having a shrinking monotone Schauder basis, L1(μ),

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-67
SLIDE 67

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1, spaces having a shrinking monotone Schauder basis, L1(μ), C(K),

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-68
SLIDE 68

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1, spaces having a shrinking monotone Schauder basis, L1(μ), C(K), K(ℓ2),

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-69
SLIDE 69

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1, spaces having a shrinking monotone Schauder basis, L1(μ), C(K), K(ℓ2), c0-sums of reflexive spaces,

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-70
SLIDE 70

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1, spaces having a shrinking monotone Schauder basis, L1(μ), C(K), K(ℓ2), c0-sums of reflexive spaces, certain of their subspaces. . .

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10

slide-71
SLIDE 71

Density results

The characterisation of NA(2)(X, ℓ2

2) = ∅ in terms of mates provides a

characterisation of density of NA(2)(X, ℓ2

2), but this doesn’t seem to lead

  • anywhere. . . However:

Theorem If NA(X) contains a dense linear subspace, then NA(X, F) is dense in L(X, F) for every finite-dimensional F. (In fact, a weaker more technical assumption suffices.) Examples: The theorem applies to the following spaces X: c0, isometric preduals of ℓ1, spaces having a shrinking monotone Schauder basis, L1(μ), C(K), K(ℓ2), c0-sums of reflexive spaces, certain of their subspaces. . . Corollary In the setting of the Theorem, if X∗ has the AP, then every compact

  • perator with domain X can be approximated by finite rank norm

attaining operators.

, Dirk Werner, Norm attaining operators of finite rank, 9.9.2019 10/10