Measuring noncompactness and discontinuity Ond rej F.K. Kalenda - - PowerPoint PPT Presentation

Measuring noncompactness and discontinuity

Ondˇ rej F.K. Kalenda

Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University in Prague

Twelfth Symposium on General Topology

and its Relations to Modern Analysis and Algebra

July 25–29, 2016, Prague

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators Weak non-compactness Two approaches to weak noncompactness Comparison of the two approaches Weak compactness and continuity

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators Weak non-compactness Two approaches to weak noncompactness Comparison of the two approaches Weak compactness and continuity Mackey non-compactness Measuring Mackey non-compactness Mackey compactness and continuity

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators Weak non-compactness Two approaches to weak noncompactness Comparison of the two approaches Weak compactness and continuity Mackey non-compactness Measuring Mackey non-compactness Mackey compactness and continuity Application: Dunford-Pettis property

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

References

[CKS 2012] B.Cascales, O.Kalenda and J.Spurn´ y: A quantitative version of James’ compactness theorem, Proc. Edinburgh

• Math. Soc., II. Ser. 55 (2012), no. 2, 369-386.

[KKS 2013] M.Kaˇ cena, O.Kalenda and J.Spurn´ y: Quantitative Dunford-Pettis property, Advances in Math. 234 (2013), 488-527. [KS 2012] O.Kalenda and J.Spurn´ y: Quantification of the reciprocal Dunford-Pettis property, Studia Math. 210 (2012), no. 3, 261-278. . . . and some recent observations

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators Weak non-compactness Two approaches to weak noncompactness Comparison of the two approaches Weak compactness and continuity Mackey non-compactness Measuring Mackey non-compactness Mackey compactness and continuity Application: Dunford-Pettis property

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Theorem

Let (X, d) be a complete metric space and A ⊂ X. TFAE:

◮ A is relatively compact. ◮ A is totally bounded. ◮ Any sequence in A has a subsequence converging in X.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Theorem

Let (X, d) be a ✘✘✘✘

✘ ❳❳❳❳ ❳

complete metric space and A ⊂ X. TFAE:

◮ A is totally bounded. ◮ Any sequence in A has a Cauchy subsequence.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Theorem

Let (X, d) be a ✘✘✘✘

✘ ❳❳❳❳ ❳

complete metric space and A ⊂ X. TFAE:

◮ A is totally bounded. ◮ Any sequence in A has a Cauchy subsequence.

Measures of noncompactness

Let (X, d) be a metric space and A ⊂ X.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Theorem

Let (X, d) be a ✘✘✘✘

✘ ❳❳❳❳ ❳

complete metric space and A ⊂ X. TFAE:

◮ A is totally bounded. ◮ Any sequence in A has a Cauchy subsequence.

Measures of noncompactness

Let (X, d) be a metric space and A ⊂ X.

◮ α(A) = inf{ε > 0; A = n i=1 Ai, diam Ai < ε}

[Kuratowski 1930]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Theorem

Let (X, d) be a ✘✘✘✘

✘ ❳❳❳❳ ❳

complete metric space and A ⊂ X. TFAE:

◮ A is totally bounded. ◮ Any sequence in A has a Cauchy subsequence.

Measures of noncompactness

Let (X, d) be a metric space and A ⊂ X.

◮ α(A) = inf{ε > 0; A = n i=1 Ai, diam Ai < ε}

[Kuratowski 1930]

◮ χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

[Gohberg, Goldenˇ stein and Marcus 1957]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Theorem

Let (X, d) be a ✘✘✘✘

✘ ❳❳❳❳ ❳

complete metric space and A ⊂ X. TFAE:

◮ A is totally bounded. ◮ Any sequence in A has a Cauchy subsequence.

Measures of noncompactness

Let (X, d) be a metric space and A ⊂ X.

◮ α(A) = inf{ε > 0; A = n i=1 Ai, diam Ai < ε}

[Kuratowski 1930]

◮ χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

[Gohberg, Goldenˇ stein and Marcus 1957]

◮ χ0(A) = inf{ε > 0; ∃F ⊂ A finite : A ⊂ U(F, ε)}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Theorem

Let (X, d) be a ✘✘✘✘

✘ ❳❳❳❳ ❳

complete metric space and A ⊂ X. TFAE:

◮ A is totally bounded. ◮ Any sequence in A has a Cauchy subsequence.

Measures of noncompactness

Let (X, d) be a metric space and A ⊂ X.

◮ α(A) = inf{ε > 0; A = n i=1 Ai, diam Ai < ε}

[Kuratowski 1930]

◮ χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

[Gohberg, Goldenˇ stein and Marcus 1957]

◮ χ0(A) = inf{ε > 0; ∃F ⊂ A finite : A ⊂ U(F, ε)} ◮ β(A) = sup

• {inf ca (xkn) ; kn ր ∞}; (xk) ⊂ A
• ca (xk) (= osc(xk)) = infn∈N diam{xk; k ≥ n}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Measures of noncompactness

Let (X, d) be a metric space and A ⊂ X.

◮ α(A) = inf{ε > 0; A = n i=1 Ai, diam Ai < ε}

[Kuratowski 1930]

◮ χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

[Gohberg, Goldenˇ stein and Marcus 1957]

◮ χ0(A) = inf{ε > 0; ∃F ⊂ A finite : A ⊂ U(F, ε)} ◮ β(A) = sup

• {inf ca (xkn) ; kn ր ∞}; (xk) ⊂ A
• ca (xk) (= osc(xk)) = infn∈N diam{xk; k ≥ n}

Proposition

◮ χ(A) = 0 ⇔ A is totally bounded

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring non-compactness in a metric space

Measures of noncompactness

Let (X, d) be a metric space and A ⊂ X.

◮ α(A) = inf{ε > 0; A = n i=1 Ai, diam Ai < ε}

[Kuratowski 1930]

◮ χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

[Gohberg, Goldenˇ stein and Marcus 1957]

◮ χ0(A) = inf{ε > 0; ∃F ⊂ A finite : A ⊂ U(F, ε)} ◮ β(A) = sup

• {inf ca (xkn) ; kn ր ∞}; (xk) ⊂ A
• ca (xk) (= osc(xk)) = infn∈N diam{xk; k ≥ n}

Proposition

◮ χ(A) = 0 ⇔ A is totally bounded ◮ χ(A) ≤ χ0(A) ≤ β(A) ≤ α(A) ≤ 2χ(A)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact iff χ(TBX) = 0.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact iff χ(TBX) = 0. ◮ We set χ(T) = χ(TBX), α(T) = α(TBX) etc.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact iff χ(TBX) = 0. ◮ We set χ(T) = χ(TBX), α(T) = α(TBX) etc.

Compactness and continuity

T is compact ⇔ T ∗ is compact

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact iff χ(TBX) = 0. ◮ We set χ(T) = χ(TBX), α(T) = α(TBX) etc.

Compactness and continuity

T is compact ⇔ T ∗ is compact ⇔ T ∗|BY∗ is w∗-to-norm continuous

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact iff χ(TBX) = 0. ◮ We set χ(T) = χ(TBX), α(T) = α(TBX) etc.

Compactness and continuity

T is compact ⇔ T ∗ is compact ⇔ T ∗|BY∗ is w∗-to-norm continuous

Measuring discontinuity

◮ T ∗|BY∗ is w∗-to-norm continuous iff ∀(y∗ τ ) ⊂ BY ∗ :

(y∗

τ ) w∗-convergent ⇒ (T ∗y∗ τ ) norm-convergent

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact iff χ(TBX) = 0. ◮ We set χ(T) = χ(TBX), α(T) = α(TBX) etc.

Compactness and continuity

T is compact ⇔ T ∗ is compact ⇔ T ∗|BY∗ is w∗-to-norm continuous

Measuring discontinuity

◮ T ∗|BY∗ is w∗-to-norm continuous iff ∀(y∗ τ ) ⊂ BY ∗ :

(y∗

τ ) w∗-Cauchy ⇒ (T ∗y∗ τ ) norm-Cauchy

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Measuring non-compactness of an operator

T : X → Y . . . a bounded operator between Banach spaces.

◮ T is compact iff TBX is relatively compact iff χ(TBX) = 0. ◮ We set χ(T) = χ(TBX), α(T) = α(TBX) etc.

Compactness and continuity

T is compact ⇔ T ∗ is compact ⇔ T ∗|BY∗ is w∗-to-norm continuous

Measuring discontinuity

◮ T ∗|BY∗ is w∗-to-norm continuous iff ∀(y∗ τ ) ⊂ BY ∗ :

(y∗

τ ) w∗-Cauchy ⇒ (T ∗y∗ τ ) norm-Cauchy ◮ contw∗→· (T ∗) = sup{ca (T ∗y∗ τ ) ; (y∗ τ ) ⊂ BY ∗ w∗-Cauchy}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Compactness and continuity of operators

Compactness and continuity

T is compact ⇔ T ∗ is compact ⇔ T ∗|BY∗ is w∗-to-norm continuous

Measuring discontinuity

◮ T ∗|BY∗ is w∗-to-norm continuous iff ∀(y∗ τ ) ⊂ BY ∗ :

(y∗

τ ) w∗-Cauchy ⇒ (T ∗y∗ τ ) norm-Cauchy ◮ contw∗→· (T ∗) = sup{ca (T ∗y∗ τ ) ; (y∗ τ ) ⊂ BY ∗ w∗-Cauchy}

Compactness and continuity – quantitative relation

1 2 contw∗→· (T ∗) ≤ χ(T) ≤ contw∗→· (T ∗)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators Weak non-compactness Two approaches to weak noncompactness Comparison of the two approaches Weak compactness and continuity Mackey non-compactness Measuring Mackey non-compactness Mackey compactness and continuity Application: Dunford-Pettis property

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring weak non-compactness - de Blasi approach

X . . . a Banach space A . . . a bounded subset of X.

Recall:

χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring weak non-compactness - de Blasi approach

X . . . a Banach space A . . . a bounded subset of X.

Recall:

χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

Measuring how much a set sticks out of another one

• d(A, B) = sup{dist(a, B); a ∈ A}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring weak non-compactness - de Blasi approach

X . . . a Banach space A . . . a bounded subset of X.

Recall:

χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

Measuring how much a set sticks out of another one

• d(A, B) = sup{dist(a, B); a ∈ A}

Hausdorff measure of noncompactness reformulated

χ(A) = inf{ d(A, F); F ⊂ X finite}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring weak non-compactness - de Blasi approach

X . . . a Banach space A . . . a bounded subset of X.

Recall:

χ(A) = inf{ε > 0; ∃F ⊂ X finite : A ⊂ U(F, ε)}

Measuring how much a set sticks out of another one

• d(A, B) = sup{dist(a, B); a ∈ A}

Hausdorff measure of noncompactness reformulated

χ(A) = inf{ d(A, F); F ⊂ X finite} = inf{ d(A, K); K ⊂ X compact}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring weak non-compactness - de Blasi approach

X . . . a Banach space A . . . a bounded subset of X.

Measuring how much a set sticks out of another one

• d(A, B) = sup{dist(a, B); a ∈ A}

Hausdorff measure of noncompactness reformulated

χ(A) = inf{ d(A, F); F ⊂ X finite} = inf{ d(A, K); K ⊂ X compact}

De Blasi measure of weak noncompactness

◮ ω(A) = inf{

d(A, K); K ⊂ X weakly compact}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring weak non-compactness - de Blasi approach

X . . . a Banach space A . . . a bounded subset of X.

Measuring how much a set sticks out of another one

• d(A, B) = sup{dist(a, B); a ∈ A}

Hausdorff measure of noncompactness reformulated

χ(A) = inf{ d(A, F); F ⊂ X finite} = inf{ d(A, K); K ⊂ X compact}

De Blasi measure of weak noncompactness

◮ ω(A) = inf{

d(A, K); K ⊂ X weakly compact}

◮ [de Blasi 1977] ω(A) = 0 ⇔ A is relatively weakly compact

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Other measures of weak noncompactness

Let X be a Banach space and A ⊂ X a bounded set. TFAE

◮ A is relatively weakly compact. ◮ [Banach-Alaoglu] A w∗

⊂ X

◮ [Eberlein-Grothendieck] limi limj x∗ i (xj) = limj limi x∗ i (xj)

whenever (xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist. ◮ [Eberlein- ˇ

Smulyan] Any (xn) ⊂ A has a w-cluster point in X.

◮ [James] Any x∗ ∈ X ∗ attains its max on A w.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Other measures of weak noncompactness

Let X be a Banach space and A ⊂ X a bounded set. TFAE

◮ A is relatively weakly compact. ◮ [Banach-Alaoglu] A w∗

⊂ X wk(A) = d(A

w∗

, X)

◮ [Eberlein-Grothendieck] limi limj x∗ i (xj) = limj limi x∗ i (xj)

whenever (xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist. ◮ [Eberlein- ˇ

Smulyan] Any (xn) ⊂ A has a w-cluster point in X.

◮ [James] Any x∗ ∈ X ∗ attains its max on A w.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Other measures of weak noncompactness

Let X be a Banach space and A ⊂ X a bounded set. TFAE

◮ A is relatively weakly compact. ◮ [Banach-Alaoglu] A w∗

⊂ X wk(A) = d(A

w∗

, X)

◮ [Eberlein-Grothendieck] limi limj x∗ i (xj) = limj limi x∗ i (xj)

whenever (xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist.

γ(A) = sup{| limi limj x∗

i (xj) − limj limi x∗ i (xj)|;

(xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist} ◮ [Eberlein- ˇ

Smulyan] Any (xn) ⊂ A has a w-cluster point in X.

◮ [James] Any x∗ ∈ X ∗ attains its max on A w.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Other measures of weak noncompactness

Let X be a Banach space and A ⊂ X a bounded set. TFAE

◮ A is relatively weakly compact. ◮ [Banach-Alaoglu] A w∗

⊂ X wk(A) = d(A

w∗

, X)

◮ [Eberlein-Grothendieck] limi limj x∗ i (xj) = limj limi x∗ i (xj)

whenever (xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist.

γ(A) = sup{| limi limj x∗

i (xj) − limj limi x∗ i (xj)|;

(xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist} ◮ [Eberlein- ˇ

Smulyan] Any (xn) ⊂ A has a w-cluster point in X. wck(A) = sup{dist(clustw*((xn)), X) : (xn) ⊂ A}

◮ [James] Any x∗ ∈ X ∗ attains its max on A w.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Other measures of weak noncompactness

Let X be a Banach space and A ⊂ X a bounded set. TFAE

◮ A is relatively weakly compact. ◮ [Banach-Alaoglu] A w∗

⊂ X wk(A) = d(A

w∗

, X)

◮ [Eberlein-Grothendieck] limi limj x∗ i (xj) = limj limi x∗ i (xj)

whenever (xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist.

γ(A) = sup{| limi limj x∗

i (xj) − limj limi x∗ i (xj)|;

(xj) ⊂ A, (x∗

i ) ⊂ BX ∗ and all limits exist} ◮ [Eberlein- ˇ

Smulyan] Any (xn) ⊂ A has a w-cluster point in X. wck(A) = sup{dist(clustw*((xn)), X) : (xn) ⊂ A}

◮ [James] Any x∗ ∈ X ∗ attains its max on A w.

Ja(A) = inf{r > 0; ∀x∗ ∈ E∗ ∃x∗∗ ∈ A

w∗

: x∗∗(x∗) = sup x∗(A) & dist(x∗∗, X) ≤ r}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Quantitative characterizations of weak compactness

Theorem

wk(A) ≤ γ(A) ≤ 2Ja(A) ≤ 2 wck(A) ≤ 2 wk(A)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Quantitative characterizations of weak compactness

Theorem

wk(A) ≤ γ(A) ≤ 2Ja(A) ≤ 2 wck(A) ≤ 2 wk(A)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Quantitative characterizations of weak compactness

Theorem

wk(A) ≤ γ(A) ≤ 2Ja(A) ≤ 2 wck(A) ≤ 2 wk(A) Quantitative versions of

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Quantitative characterizations of weak compactness

Theorem

wk(A) ≤ γ(A) ≤ 2Ja(A) ≤ 2 wck(A) ≤ 2 wk(A) Quantitative versions of

◮ Eberlein-Grothendieck theorem

[M.Fabian, P .H´ ajek, V.Montesinos and V.Zizler, 2005]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Quantitative characterizations of weak compactness

Theorem

wk(A) ≤ γ(A) ≤ 2Ja(A) ≤ 2 wck(A) ≤ 2 wk(A) Quantitative versions of

◮ Eberlein-Grothendieck theorem

[M.Fabian, P .H´ ajek, V.Montesinos and V.Zizler, 2005]

◮ Eberlein- ˇ

Smulyan theorem [C.Angosto and B.Cascales, 2008]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Quantitative characterizations of weak compactness

Theorem

wk(A) ≤ γ(A) ≤ 2Ja(A) ≤ 2 wck(A) ≤ 2 wk(A) Quantitative versions of

◮ Eberlein-Grothendieck theorem

[M.Fabian, P .H´ ajek, V.Montesinos and V.Zizler, 2005]

◮ Eberlein- ˇ

Smulyan theorem [C.Angosto and B.Cascales, 2008]

◮ James theorem [CKS 2012]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

◮ wk(A) = ω(A) in the following spaces:

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

◮ wk(A) = ω(A) in the following spaces:

◮ [KKS 2013] X = c0(Γ) Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

◮ wk(A) = ω(A) in the following spaces:

◮ [KKS 2013] X = c0(Γ) ◮ [KKS 2013] X = L1(µ) Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

◮ wk(A) = ω(A) in the following spaces:

◮ [KKS 2013] X = c0(Γ) ◮ [KKS 2013] X = L1(µ)

wk(A) = ω(A) = inf{sup

f∈A

• (|f|−cχE)+ dµ : c > 0, µ(E) < +∞}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

◮ wk(A) = ω(A) in the following spaces:

◮ [KKS 2013] X = c0(Γ) ◮ [KKS 2013] X = L1(µ)

wk(A) = ω(A) = inf{sup

f∈A

• (|f|−cχE)+ dµ : c > 0, µ(E) < +∞}

◮ [in preparation] X = N(H) or X = K(H) Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

◮ wk(A) = ω(A) in the following spaces:

◮ [KKS 2013] X = c0(Γ) ◮ [KKS 2013] X = L1(µ)

wk(A) = ω(A) = inf{sup

f∈A

• (|f|−cχE)+ dµ : c > 0, µ(E) < +∞}

◮ [in preparation] X = N(H) or X = K(H)

Question

Let X = C(K). Are ω(A) and wk(A) equivalent for bounded subsets of X?

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness

◮ Easy: wk(A) ≤ ω(A) ≤ χ(A) ◮ In general: wk(A) and ω(A) are not equivalent.

[K.Astala and H.-O.Tylli, 1990]

◮ wk(A) = ω(A) in the following spaces:

◮ [KKS 2013] X = c0(Γ) ◮ [KKS 2013] X = L1(µ)

wk(A) = ω(A) = inf{sup

f∈A

• (|f|−cχE)+ dµ : c > 0, µ(E) < +∞}

◮ [in preparation] X = N(H) or X = K(H)

Question

Let X = C(K). Are ω(A) and wk(A) equivalent for bounded subsets of X? Is it true at least for K = [0, 1]?

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness II

Theorem

Let X be a Banach space.

◮ X is WCG iff

∀ ε > 0 ∃ (An)∞

n=1 a cover of X ∀n ∈ N : ω(An) < ε.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness II

Theorem

Let X be a Banach space.

◮ X is WCG iff

∀ ε > 0 ∃ (An)∞

n=1 a cover of X ∀n ∈ N : ω(An) < ε.

[An exercise]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness II

Theorem

Let X be a Banach space.

◮ X is WCG iff

∀ ε > 0 ∃ (An)∞

n=1 a cover of X ∀n ∈ N : ω(An) < ε.

[An exercise]

◮ X is a subspace of WCG iff

∀ ε > 0 ∃ (An)∞

n=1 a cover of X ∀n ∈ N : wk(An) < ε.

[M.Fabian, V.Montesinos and V.Zizler, 2004]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Comparison of measures of weak non-compactness II

Theorem

Let X be a Banach space.

◮ X is WCG iff

∀ ε > 0 ∃ (An)∞

n=1 a cover of X ∀n ∈ N : ω(An) < ε.

[An exercise]

◮ X is a subspace of WCG iff

∀ ε > 0 ∃ (An)∞

n=1 a cover of X ∀n ∈ N : wk(An) < ε.

[M.Fabian, V.Montesinos and V.Zizler, 2004]

Remark

If ω and wk are equivalent in C(K) spaces, it easily follows that Eberlein compact spaces are preserved by continuous images.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity

Let T : X → Y be a bounded linear operator.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity

Let T : X → Y be a bounded linear operator. T weakly compact ⇔ T ∗ weakly compact [Gantmacher 1940]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity

Let T : X → Y be a bounded linear operator. T ∗|BX∗ w∗-to-w continuous

• T weakly compact

⇔ T ∗ weakly compact [Gantmacher 1940]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity

Let T : X → Y be a bounded linear operator. T ∗|BX∗ w∗-to-w continuous

• T weakly compact

⇔ T ∗ weakly compact

• T ∗ Mackey-to-norm continuous

[Gantmacher 1940] [Grothendieck 1953] µ(X ∗, X) = topology of uniform convergence

• n weakly compact subsets of X

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity

Let T : X → Y be a bounded linear operator. T ∗|BX∗ w∗-to-w continuous

• T weakly compact

⇔ T ∗ weakly compact

• T ∗ Mackey-to-norm continuous

T Right-to-norm continuous [Gantmacher 1940] [Grothendieck 1953] µ(X ∗, X) = topology of uniform convergence

• n weakly compact subsets of X

[Peralta, Villanueva, Maitland Wright and Ylinen 2007] ρ(X, X ∗) = µ(X ∗∗, X ∗)|X

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X A . . . family of subsets of BY ∗, closed to finite unions

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X A . . . family of subsets of BY ∗, closed to finite unions σ . . . topology of uniform convergence of elements of A, i.e., the topology generated by the seminorms ρA(x) = sup{|x∗(x)|; x∗ ∈ A}, A ∈ A.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X A . . . family of subsets of BY ∗, closed to finite unions σ . . . topology of uniform convergence of elements of A, i.e., the topology generated by the seminorms ρA(x) = sup{|x∗(x)|; x∗ ∈ A}, A ∈ A.

Measuring σ-non-Cauchyness (oscillation)

(yν) ⊂ Y a bounded net

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X A . . . family of subsets of BY ∗, closed to finite unions σ . . . topology of uniform convergence of elements of A, i.e., the topology generated by the seminorms ρA(x) = sup{|x∗(x)|; x∗ ∈ A}, A ∈ A.

Measuring σ-non-Cauchyness (oscillation)

(yν) ⊂ Y a bounded net caσ (yν) = sup{inf

ν0 ρA- diam{yν; ν ≥ ν0}; A ∈ A}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X A . . . family of subsets of BY ∗, closed to finite unions σ . . . topology of uniform convergence of elements of A, i.e., the topology generated by the seminorms ρA(x) = sup{|x∗(x)|; x∗ ∈ A}, A ∈ A.

Measuring σ-non-Cauchyness (oscillation)

(yν) ⊂ Y a bounded net caσ (yν) = sup{inf

ν0 ρA- diam{yν; ν ≥ ν0}; A ∈ A}

Measuring discontinuity of linear operators

T : X → Y bounded linear operator

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X A . . . family of subsets of BY ∗, closed to finite unions σ . . . topology of uniform convergence of elements of A, i.e., the topology generated by the seminorms ρA(x) = sup{|x∗(x)|; x∗ ∈ A}, A ∈ A.

Measuring σ-non-Cauchyness (oscillation)

(yν) ⊂ Y a bounded net caσ (yν) = sup{inf

ν0 ρA- diam{yν; ν ≥ ν0}; A ∈ A}

Measuring discontinuity of linear operators

T : X → Y bounded linear operator

◮ contτ−σ (T) = sup{caσ (Txν) ; (xν) ⊂ BX τ-Cauchy}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring discontinuity

X, Y . . . Banach spaces τ . . . a locally convex topology on X A . . . family of subsets of BY ∗, closed to finite unions σ . . . topology of uniform convergence of elements of A, i.e., the topology generated by the seminorms ρA(x) = sup{|x∗(x)|; x∗ ∈ A}, A ∈ A.

Measuring σ-non-Cauchyness (oscillation)

(yν) ⊂ Y a bounded net caσ (yν) = sup{inf

ν0 ρA- diam{yν; ν ≥ ν0}; A ∈ A}

Measuring discontinuity of linear operators

T : X → Y bounded linear operator

◮ contτ−σ (T) = sup{caσ (Txν) ; (xν) ⊂ BX τ-Cauchy} ◮ ccτ−σ (T) = sup{caσ (Txn) ; (xn) ⊂ BX τ-Cauchy}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view

T : X → Y bounded linear operator

◮ [KKS 2013] 1 2 contµ→· (T ∗) ≤ ω(T) ≤ contµ→· (T ∗)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view

T : X → Y bounded linear operator

◮ [KKS 2013] 1 2 contµ→· (T ∗) ≤ ω(T) ≤ contµ→· (T ∗) ◮ [KKS 2013] 1 2 contρ→· (T) ≤ ω(T ∗) ≤ contρ→· (T)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view

T : X → Y bounded linear operator

◮ [KKS 2013] 1 2 contµ→· (T ∗) ≤ ω(T) ≤ contµ→· (T ∗) ◮ [KKS 2013] 1 2 contρ→· (T) ≤ ω(T ∗) ≤ contρ→· (T) ◮ 1 4 contw∗→w (T ∗) ≤ wk(T ∗) ≤ contw∗→w (T ∗)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view

T : X → Y bounded linear operator

◮ [KKS 2013] 1 2 contµ→· (T ∗) ≤ ω(T) ≤ contµ→· (T ∗) ◮ [KKS 2013] 1 2 contρ→· (T) ≤ ω(T ∗) ≤ contρ→· (T) ◮ 1 4 contw∗→w (T ∗) ≤ wk(T ∗) ≤ contw∗→w (T ∗)

Quantitative Gantmacher theorem

◮ ω(T) and ω(T ∗) are incomparable.

[K.Astala and H.-O.Tylli, 1990]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view

T : X → Y bounded linear operator

◮ [KKS 2013] 1 2 contµ→· (T ∗) ≤ ω(T) ≤ contµ→· (T ∗) ◮ [KKS 2013] 1 2 contρ→· (T) ≤ ω(T ∗) ≤ contρ→· (T) ◮ 1 4 contw∗→w (T ∗) ≤ wk(T ∗) ≤ contw∗→w (T ∗)

Quantitative Gantmacher theorem

◮ ω(T) and ω(T ∗) are incomparable.

[K.Astala and H.-O.Tylli, 1990]

◮ 1 2 wk(T) ≤ wk(T ∗) ≤ 2 wk(T)

[C.Angosto and B.Cascales, 2009]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view

T : X → Y bounded linear operator

◮ [KKS 2013] 1 2 contµ→· (T ∗) ≤ ω(T) ≤ contµ→· (T ∗) ◮ [KKS 2013] 1 2 contρ→· (T) ≤ ω(T ∗) ≤ contρ→· (T) ◮ 1 4 contw∗→w (T ∗) ≤ wk(T ∗) ≤ contw∗→w (T ∗)

Quantitative Gantmacher theorem

◮ ω(T) and ω(T ∗) are incomparable.

[K.Astala and H.-O.Tylli, 1990]

◮ 1 2 wk(T) ≤ wk(T ∗) ≤ 2 wk(T)

[C.Angosto and B.Cascales, 2009]

Corollary

1 4 contw∗→w (T ∗) ≤ wk(T ∗) ≤ 2 wk(T) ≤ 4 contw∗→w (T ∗)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators Weak non-compactness Two approaches to weak noncompactness Comparison of the two approaches Weak compactness and continuity Mackey non-compactness Measuring Mackey non-compactness Mackey compactness and continuity Application: Dunford-Pettis property

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded. χm(A) = sup{χ0(A|L, · ∞) : L ⊂ BX weakly compact}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded. χm(A) = sup{χ0(A|L, · ∞) : L ⊂ BX weakly compact} ωm(A) = inf{ d(A, K) : K ⊂ X ∗ Mackey compact}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded. χm(A) = sup{χ0(A|L, · ∞) : L ⊂ BX weakly compact} ωm(A) = inf{ d(A, K) : K ⊂ X ∗ Mackey compact} Let X = C0(Ω) (Ω locally compact) and A ⊂ X ∗ be bounded

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded. χm(A) = sup{χ0(A|L, · ∞) : L ⊂ BX weakly compact} ωm(A) = inf{ d(A, K) : K ⊂ X ∗ Mackey compact} Let X = C0(Ω) (Ω locally compact) and A ⊂ X ∗ be bounded

◮ A is weakly compact ⇔ A is Mackey compact.

[A. Grothendieck, 1953]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded. χm(A) = sup{χ0(A|L, · ∞) : L ⊂ BX weakly compact} ωm(A) = inf{ d(A, K) : K ⊂ X ∗ Mackey compact} Let X = C0(Ω) (Ω locally compact) and A ⊂ X ∗ be bounded

◮ A is weakly compact ⇔ A is Mackey compact.

[A. Grothendieck, 1953]

◮ 1 2χm(A) ≤ ωm(A) = ω(A) = wk(A) ≤ πχm(A). [KS 2012]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded. χm(A) = sup{χ0(A|L, · ∞) : L ⊂ BX weakly compact} ωm(A) = inf{ d(A, K) : K ⊂ X ∗ Mackey compact} Let X = C0(Ω) (Ω locally compact) and A ⊂ X ∗ be bounded

◮ 1 2χm(A) ≤ ωm(A) = ω(A) = wk(A) ≤ πχm(A). [KS 2012]

Question

Are the quantities χm and ωm equivalent in any dual space?

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness

Let A ⊂ X ∗ be bounded. χm(A) = sup{χ0(A|L, · ∞) : L ⊂ BX weakly compact} ωm(A) = inf{ d(A, K) : K ⊂ X ∗ Mackey compact} Let X = C0(Ω) (Ω locally compact) and A ⊂ X ∗ be bounded

◮ 1 2χm(A) ≤ ωm(A) = ω(A) = wk(A) ≤ πχm(A). [KS 2012]

Question

Are the quantities χm and ωm equivalent in any dual space?

Remark

1 2χm(A) ≤ ωm(A) holds always.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous

• perators

Let T : X → Y be a bounded linear operator. TFAE:

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous

• perators

Let T : X → Y be a bounded linear operator. TFAE:

◮ T is completely continuous, i.e.,

(xn) weakly convergent ⇒ (Txn) norm convergent.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous

• perators

Let T : X → Y be a bounded linear operator. TFAE:

◮ T is completely continuous, i.e.,

(xn) weakly convergent ⇒ (Txn) norm convergent.

◮ T is Dunford-Pettis, i.e.,

T(A) is norm-compact for each A ⊂ X weakly compact.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous

• perators

Let T : X → Y be a bounded linear operator. TFAE:

◮ T is completely continuous, i.e.,

(xn) weakly convergent ⇒ (Txn) norm convergent.

◮ T is Dunford-Pettis, i.e.,

T(A) is norm-compact for each A ⊂ X weakly compact.

◮ [A.Grothendieck, 1953] T ∗ is Mackey compact, i.e.,

T ∗(BY ∗) is relatively Mackey compact.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous

• perators

Let T : X → Y be a bounded linear operator. TFAE:

◮ T is completely continuous, i.e.,

(xn) weakly convergent ⇒ (Txn) norm convergent.

◮ T is Dunford-Pettis, i.e.,

T(A) is norm-compact for each A ⊂ X weakly compact.

◮ [A.Grothendieck, 1953] T ∗ is Mackey compact, i.e.,

T ∗(BY ∗) is relatively Mackey compact.

Quantitative version [KS 2012]

◮ dp(T) = sup{χ0(TA); A ⊂ BX weakly compact}

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous

• perators

Let T : X → Y be a bounded linear operator. TFAE:

◮ T is completely continuous, i.e.,

(xn) weakly convergent ⇒ (Txn) norm convergent.

◮ T is Dunford-Pettis, i.e.,

T(A) is norm-compact for each A ⊂ X weakly compact.

◮ [A.Grothendieck, 1953] T ∗ is Mackey compact, i.e.,

T ∗(BY ∗) is relatively Mackey compact.

Quantitative version [KS 2012]

◮ dp(T) = sup{χ0(TA); A ⊂ BX weakly compact} ◮ 1 2χm(T ∗) ≤ dp(T) ≤ ccw→· (T) ≤ 4χm(T ∗)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Plan

Compactness in metric spaces, norm-compactness Measuring non-compactness in a metric space Norm-compactness and continuity of operators Weak non-compactness Two approaches to weak noncompactness Comparison of the two approaches Weak compactness and continuity Mackey non-compactness Measuring Mackey non-compactness Mackey compactness and continuity Application: Dunford-Pettis property

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T ∗ is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T ∗ is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

• 3. ∀Y ∀T : X → Y : ccw→· (T) ≤ 2ω(T ∗). [KKS 2013]
• 4. ∀Y ∀T : Y → X : ccw→· (T ∗) ≤ 2ω(T). [KKS 2013]

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T ∗ is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

• 3. ∀Y ∀T : X → Y : ccw→· (T) ≤ 2ω(T ∗). [KKS 2013]
• 4. ∀Y ∀T : Y → X : ccw→· (T ∗) ≤ 2ω(T). [KKS 2013]

Sketch

ccw→· (T)

DPP

≤ ccρ→· (T)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T ∗ is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

• 3. ∀Y ∀T : X → Y : ccw→· (T) ≤ 2ω(T ∗). [KKS 2013]
• 4. ∀Y ∀T : Y → X : ccw→· (T ∗) ≤ 2ω(T). [KKS 2013]

Sketch

ccw→· (T)

DPP

≤ ccρ→· (T)

easy

≤ contρ→· (T)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T ∗ is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

• 3. ∀Y ∀T : X → Y : ccw→· (T) ≤ 2ω(T ∗). [KKS 2013]
• 4. ∀Y ∀T : Y → X : ccw→· (T ∗) ≤ 2ω(T). [KKS 2013]

Sketch

ccw→· (T)

DPP

≤ ccρ→· (T)

easy

≤ contρ→· (T)

above

≤ 2ω(T ∗)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T ∗ is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

• 3. ∀Y ∀T : X → Y : ccw→· (T) ≤ 2ω(T ∗). [KKS 2013]
• 4. ∀Y ∀T : Y → X : ccw→· (T ∗) ≤ 2ω(T). [KKS 2013]

Quantitative strengthening of DPP [KKS 2013]

◮ X has direct qDPP if

∃C > 0 : ∀Y∀T : X → Y : ccw→· (T) ≤ C wk(T ∗)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Application: Dunford-Pettis property

Let X be a Banach space. The following assertions are equivalent to the Dunford-Pettis property of X.

• 1. ∀Y ∀T : X → Y:

T ∗ is weakly compact ⇒ T is completely continuous.

• 2. ∀Y ∀T : Y → X:

T is weakly compact ⇒ T ∗ is completely continuous.

• 3. ∀Y ∀T : X → Y : ccw→· (T) ≤ 2ω(T ∗). [KKS 2013]
• 4. ∀Y ∀T : Y → X : ccw→· (T ∗) ≤ 2ω(T). [KKS 2013]

Quantitative strengthening of DPP [KKS 2013]

◮ X has direct qDPP if

∃C > 0 : ∀Y∀T : X → Y : ccw→· (T) ≤ C wk(T ∗)

◮ X has dual qDPP if

∃C > 0 : ∀Y∀T : Y → X : ccw→· (T ∗) ≤ C wk(T)

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Thank you for your attention.

Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity