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

Compactness and continuity of operators Measuring non-compactness of an operator T : X → Y . . . a bounded operator between Banach spaces. ◮ T is compact iff TB X is relatively compact iff χ ( TB X ) = 0. ◮ We set χ ( T ) = χ ( TB X ) , α ( T ) = α ( TB X ) etc. Compactness and continuity T is compact ⇔ T ∗ is compact ⇔ T ∗ | B Y ∗ is w ∗ -to-norm continuous Measuring discontinuity ◮ T ∗ | B Y ∗ is w ∗ -to-norm continuous iff ∀ ( y ∗ τ ) ⊂ B Y ∗ : ( 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 TB X is relatively compact iff χ ( TB X ) = 0. ◮ We set χ ( T ) = χ ( TB X ) , α ( T ) = α ( TB X ) etc. Compactness and continuity T is compact ⇔ T ∗ is compact ⇔ T ∗ | B Y ∗ is w ∗ -to-norm continuous Measuring discontinuity ◮ T ∗ | B Y ∗ is w ∗ -to-norm continuous iff ∀ ( y ∗ τ ) ⊂ B Y ∗ : ( 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 TB X is relatively compact iff χ ( TB X ) = 0. ◮ We set χ ( T ) = χ ( TB X ) , α ( T ) = α ( TB X ) etc. Compactness and continuity T is compact ⇔ T ∗ is compact ⇔ T ∗ | B Y ∗ is w ∗ -to-norm continuous Measuring discontinuity ◮ T ∗ | B Y ∗ is w ∗ -to-norm continuous iff ∀ ( y ∗ τ ) ⊂ B Y ∗ : ( y ∗ τ ) w ∗ -Cauchy ⇒ ( T ∗ y ∗ τ ) norm-Cauchy ◮ cont w ∗ →�·� ( T ∗ ) = sup { ca ( T ∗ y ∗ τ ) ; ( y ∗ τ ) ⊂ B Y ∗ 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 ∗ | B Y ∗ is w ∗ -to-norm continuous Measuring discontinuity ◮ T ∗ | B Y ∗ is w ∗ -to-norm continuous iff ∀ ( y ∗ τ ) ⊂ B Y ∗ : ( y ∗ τ ) w ∗ -Cauchy ⇒ ( T ∗ y ∗ τ ) norm-Cauchy ◮ cont w ∗ →�·� ( T ∗ ) = sup { ca ( T ∗ y ∗ τ ) ; ( y ∗ τ ) ⊂ B Y ∗ w ∗ -Cauchy } Compactness and continuity – quantitative relation 1 2 cont w ∗ →�·� ( T ∗ ) ≤ χ ( T ) ≤ cont 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 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. w ∗ ◮ [Banach-Alaoglu] A ⊂ X ◮ [Eberlein-Grothendieck] lim i lim j x ∗ i ( x j ) = lim j lim i x ∗ i ( x j ) whenever ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist. Smulyan] Any ( x n ) ⊂ A has a w-cluster point in X . ◮ [Eberlein- ˇ w . ◮ [James] Any x ∗ ∈ X ∗ attains its max on A 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. w ∗ ◮ [Banach-Alaoglu] A ⊂ X w ∗ wk ( A ) = � d ( A , X ) ◮ [Eberlein-Grothendieck] lim i lim j x ∗ i ( x j ) = lim j lim i x ∗ i ( x j ) whenever ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist. Smulyan] Any ( x n ) ⊂ A has a w-cluster point in X . ◮ [Eberlein- ˇ w . ◮ [James] Any x ∗ ∈ X ∗ attains its max on A 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. w ∗ ◮ [Banach-Alaoglu] A ⊂ X w ∗ wk ( A ) = � d ( A , X ) ◮ [Eberlein-Grothendieck] lim i lim j x ∗ i ( x j ) = lim j lim i x ∗ i ( x j ) whenever ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist. γ ( A ) = sup {| lim i lim j x ∗ i ( x j ) − lim j lim i x ∗ i ( x j ) | ; ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist } Smulyan] Any ( x n ) ⊂ A has a w-cluster point in X . ◮ [Eberlein- ˇ w . ◮ [James] Any x ∗ ∈ X ∗ attains its max on A 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. w ∗ ◮ [Banach-Alaoglu] A ⊂ X w ∗ wk ( A ) = � d ( A , X ) ◮ [Eberlein-Grothendieck] lim i lim j x ∗ i ( x j ) = lim j lim i x ∗ i ( x j ) whenever ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist. γ ( A ) = sup {| lim i lim j x ∗ i ( x j ) − lim j lim i x ∗ i ( x j ) | ; ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist } Smulyan] Any ( x n ) ⊂ A has a w-cluster point in X . ◮ [Eberlein- ˇ wck ( A ) = sup { dist ( clust w * (( x n )) , X ) : ( x n ) ⊂ A } w . ◮ [James] Any x ∗ ∈ X ∗ attains its max on A 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. w ∗ ◮ [Banach-Alaoglu] A ⊂ X w ∗ wk ( A ) = � d ( A , X ) ◮ [Eberlein-Grothendieck] lim i lim j x ∗ i ( x j ) = lim j lim i x ∗ i ( x j ) whenever ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist. γ ( A ) = sup {| lim i lim j x ∗ i ( x j ) − lim j lim i x ∗ i ( x j ) | ; ( x j ) ⊂ A , ( x ∗ i ) ⊂ B X ∗ and all limits exist } Smulyan] Any ( x n ) ⊂ A has a w-cluster point in X . ◮ [Eberlein- ˇ wck ( A ) = sup { dist ( clust w * (( x n )) , X ) : ( x n ) ⊂ A } w . ◮ [James] Any x ∗ ∈ X ∗ attains its max on A w ∗ Ja ( A ) = inf { r > 0 ; ∀ x ∗ ∈ E ∗ ∃ x ∗∗ ∈ A : 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 ) ≤ 2 Ja ( 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 ) ≤ 2 Ja ( 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 ) ≤ 2 Ja ( 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 ) ≤ 2 Ja ( 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 ) ≤ 2 Ja ( 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 ) ≤ 2 Ja ( 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 = c 0 (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 = c 0 (Γ) ◮ [KKS 2013] X = L 1 ( 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 = c 0 (Γ) ◮ [KKS 2013] X = L 1 ( µ ) � ( | f |− c χ E ) + d µ : c > 0 , µ ( E ) < + ∞} wk ( A ) = ω ( A ) = inf { sup f ∈ 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] ◮ wk ( A ) = ω ( A ) in the following spaces: ◮ [KKS 2013] X = c 0 (Γ) ◮ [KKS 2013] X = L 1 ( µ ) � ( | f |− c χ E ) + d µ : c > 0 , µ ( E ) < + ∞} wk ( A ) = ω ( A ) = inf { sup f ∈ A ◮ [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 = c 0 (Γ) ◮ [KKS 2013] X = L 1 ( µ ) � ( | f |− c χ E ) + d µ : c > 0 , µ ( E ) < + ∞} wk ( A ) = ω ( A ) = inf { sup f ∈ A ◮ [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 = c 0 (Γ) ◮ [KKS 2013] X = L 1 ( µ ) � ( | f |− c χ E ) + d µ : c > 0 , µ ( E ) < + ∞} wk ( A ) = ω ( A ) = inf { sup f ∈ A ◮ [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 ∃ ( A n ) ∞ n = 1 a cover of X ∀ n ∈ N : ω ( A n ) < 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 ∃ ( A n ) ∞ n = 1 a cover of X ∀ n ∈ N : ω ( A n ) < 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 ∃ ( A n ) ∞ n = 1 a cover of X ∀ n ∈ N : ω ( A n ) < ε. [An exercise] ◮ X is a subspace of WCG iff ∀ ε > 0 ∃ ( A n ) ∞ n = 1 a cover of X ∀ n ∈ N : wk ( A n ) < 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 ∃ ( A n ) ∞ n = 1 a cover of X ∀ n ∈ N : ω ( A n ) < ε. [An exercise] ◮ X is a subspace of WCG iff ∀ ε > 0 ∃ ( A n ) ∞ n = 1 a cover of X ∀ n ∈ N : wk ( A n ) < 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 ∗ | B X ∗ 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 ∗ | B X ∗ 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 on 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 ∗ | B X ∗ 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 on 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 B Y ∗ , 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 B Y ∗ , closed to finite unions σ . . . topology of uniform convergence of elements of A , i.e., the topology generated by the seminorms ρ A ( 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 B Y ∗ , closed to finite unions σ . . . topology of uniform convergence of elements of A , i.e., the topology generated by the seminorms ρ A ( x ) = sup {| x ∗ ( x ) | ; x ∗ ∈ A } , A ∈ A . Measuring σ -non-Cauchyness (oscillation) ( y ν ) ⊂ 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 B Y ∗ , closed to finite unions σ . . . topology of uniform convergence of elements of A , i.e., the topology generated by the seminorms ρ A ( x ) = sup {| x ∗ ( x ) | ; x ∗ ∈ A } , A ∈ A . Measuring σ -non-Cauchyness (oscillation) ( y ν ) ⊂ Y a bounded net ca σ ( y ν ) = sup { inf ν 0 ρ A - diam { y ν ; ν ≥ ν 0 } ; 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 B Y ∗ , closed to finite unions σ . . . topology of uniform convergence of elements of A , i.e., the topology generated by the seminorms ρ A ( x ) = sup {| x ∗ ( x ) | ; x ∗ ∈ A } , A ∈ A . Measuring σ -non-Cauchyness (oscillation) ( y ν ) ⊂ Y a bounded net ca σ ( y ν ) = sup { inf ν 0 ρ A - diam { y ν ; ν ≥ ν 0 } ; 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 B Y ∗ , closed to finite unions σ . . . topology of uniform convergence of elements of A , i.e., the topology generated by the seminorms ρ A ( x ) = sup {| x ∗ ( x ) | ; x ∗ ∈ A } , A ∈ A . Measuring σ -non-Cauchyness (oscillation) ( y ν ) ⊂ Y a bounded net ca σ ( y ν ) = sup { inf ν 0 ρ A - diam { y ν ; ν ≥ ν 0 } ; A ∈ A} Measuring discontinuity of linear operators T : X → Y bounded linear operator ◮ cont τ − σ ( T ) = sup { ca σ ( Tx ν ) ; ( x ν ) ⊂ B X 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 B Y ∗ , closed to finite unions σ . . . topology of uniform convergence of elements of A , i.e., the topology generated by the seminorms ρ A ( x ) = sup {| x ∗ ( x ) | ; x ∗ ∈ A } , A ∈ A . Measuring σ -non-Cauchyness (oscillation) ( y ν ) ⊂ Y a bounded net ca σ ( y ν ) = sup { inf ν 0 ρ A - diam { y ν ; ν ≥ ν 0 } ; A ∈ A} Measuring discontinuity of linear operators T : X → Y bounded linear operator ◮ cont τ − σ ( T ) = sup { ca σ ( Tx ν ) ; ( x ν ) ⊂ B X τ -Cauchy } ◮ cc τ − σ ( T ) = sup { ca σ ( Tx n ) ; ( x n ) ⊂ B X τ -Cauchy } Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view T : X → Y bounded linear operator 2 cont µ →�·� ( T ∗ ) ≤ ω ( T ) ≤ cont µ →�·� ( T ∗ ) ◮ [KKS 2013] 1 Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view T : X → Y bounded linear operator 2 cont µ →�·� ( T ∗ ) ≤ ω ( T ) ≤ cont µ →�·� ( T ∗ ) ◮ [KKS 2013] 1 2 cont ρ →�·� ( T ) ≤ ω ( T ∗ ) ≤ cont ρ →�·� ( T ) ◮ [KKS 2013] 1 Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view T : X → Y bounded linear operator 2 cont µ →�·� ( T ∗ ) ≤ ω ( T ) ≤ cont µ →�·� ( T ∗ ) ◮ [KKS 2013] 1 2 cont ρ →�·� ( T ) ≤ ω ( T ∗ ) ≤ cont ρ →�·� ( T ) ◮ [KKS 2013] 1 4 cont w ∗ → w ( T ∗ ) ≤ wk ( T ∗ ) ≤ cont w ∗ → w ( T ∗ ) 1 ◮ Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Weak compactness and continuity – quantitative view T : X → Y bounded linear operator 2 cont µ →�·� ( T ∗ ) ≤ ω ( T ) ≤ cont µ →�·� ( T ∗ ) ◮ [KKS 2013] 1 2 cont ρ →�·� ( T ) ≤ ω ( T ∗ ) ≤ cont ρ →�·� ( T ) ◮ [KKS 2013] 1 4 cont w ∗ → w ( T ∗ ) ≤ wk ( T ∗ ) ≤ cont w ∗ → w ( T ∗ ) 1 ◮ 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 2 cont µ →�·� ( T ∗ ) ≤ ω ( T ) ≤ cont µ →�·� ( T ∗ ) ◮ [KKS 2013] 1 2 cont ρ →�·� ( T ) ≤ ω ( T ∗ ) ≤ cont ρ →�·� ( T ) ◮ [KKS 2013] 1 4 cont w ∗ → w ( T ∗ ) ≤ wk ( T ∗ ) ≤ cont w ∗ → w ( T ∗ ) 1 ◮ Quantitative Gantmacher theorem ◮ ω ( T ) and ω ( T ∗ ) are incomparable. [K.Astala and H.-O.Tylli, 1990] 2 wk ( T ) ≤ wk ( T ∗ ) ≤ 2 wk ( T ) 1 ◮ [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 2 cont µ →�·� ( T ∗ ) ≤ ω ( T ) ≤ cont µ →�·� ( T ∗ ) ◮ [KKS 2013] 1 2 cont ρ →�·� ( T ) ≤ ω ( T ∗ ) ≤ cont ρ →�·� ( T ) ◮ [KKS 2013] 1 4 cont w ∗ → w ( T ∗ ) ≤ wk ( T ∗ ) ≤ cont w ∗ → w ( T ∗ ) 1 ◮ Quantitative Gantmacher theorem ◮ ω ( T ) and ω ( T ∗ ) are incomparable. [K.Astala and H.-O.Tylli, 1990] 2 wk ( T ) ≤ wk ( T ∗ ) ≤ 2 wk ( T ) 1 ◮ [C.Angosto and B.Cascales, 2009] Corollary 4 cont w ∗ → w ( T ∗ ) ≤ wk ( T ∗ ) ≤ 2 wk ( T ) ≤ 4 cont w ∗ → w ( T ∗ ) 1 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. χ m ( A ) = sup { χ 0 ( A | L , � · � ∞ ) : L ⊂ B X weakly compact } Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness Let A ⊂ X ∗ be bounded. χ m ( A ) = sup { χ 0 ( A | L , � · � ∞ ) : L ⊂ B X weakly compact } d ( A , K ) : K ⊂ X ∗ Mackey compact } ω m ( A ) = inf { � Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness Let A ⊂ X ∗ be bounded. χ m ( A ) = sup { χ 0 ( A | L , � · � ∞ ) : L ⊂ B X weakly compact } d ( A , K ) : K ⊂ X ∗ Mackey compact } ω m ( A ) = inf { � Let X = C 0 (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. χ m ( A ) = sup { χ 0 ( A | L , � · � ∞ ) : L ⊂ B X weakly compact } d ( A , K ) : K ⊂ X ∗ Mackey compact } ω m ( A ) = inf { � Let X = C 0 (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. χ m ( A ) = sup { χ 0 ( A | L , � · � ∞ ) : L ⊂ B X weakly compact } d ( A , K ) : K ⊂ X ∗ Mackey compact } ω m ( A ) = inf { � Let X = C 0 (Ω) ( Ω locally compact) and A ⊂ X ∗ be bounded ◮ A is weakly compact ⇔ A is Mackey compact. [A. Grothendieck, 1953] 2 χ m ( A ) ≤ ω m ( A ) = ω ( A ) = wk ( A ) ≤ πχ m ( A ) . [KS 2012] 1 ◮ Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness Let A ⊂ X ∗ be bounded. χ m ( A ) = sup { χ 0 ( A | L , � · � ∞ ) : L ⊂ B X weakly compact } d ( A , K ) : K ⊂ X ∗ Mackey compact } ω m ( A ) = inf { � Let X = C 0 (Ω) ( Ω locally compact) and A ⊂ X ∗ be bounded 2 χ m ( A ) ≤ ω m ( A ) = ω ( A ) = wk ( A ) ≤ πχ m ( A ) . [KS 2012] 1 ◮ Question Are the quantities χ m and ω m equivalent in any dual space? Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Measuring Mackey non-compactness Let A ⊂ X ∗ be bounded. χ m ( A ) = sup { χ 0 ( A | L , � · � ∞ ) : L ⊂ B X weakly compact } d ( A , K ) : K ⊂ X ∗ Mackey compact } ω m ( A ) = inf { � Let X = C 0 (Ω) ( Ω locally compact) and A ⊂ X ∗ be bounded 2 χ m ( A ) ≤ ω m ( A ) = ω ( A ) = wk ( A ) ≤ πχ m ( A ) . [KS 2012] 1 ◮ Question Are the quantities χ m and ω m equivalent in any dual space? Remark 2 χ m ( A ) ≤ ω m ( A ) holds always. 1 Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous operators Let T : X → Y be a bounded linear operator. TFAE: Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous operators Let T : X → Y be a bounded linear operator. TFAE: ◮ T is completely continuous, i.e., ( x n ) weakly convergent ⇒ ( Tx n ) norm convergent. Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous operators Let T : X → Y be a bounded linear operator. TFAE: ◮ T is completely continuous, i.e., ( x n ) weakly convergent ⇒ ( Tx n ) 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 operators Let T : X → Y be a bounded linear operator. TFAE: ◮ T is completely continuous, i.e., ( x n ) weakly convergent ⇒ ( Tx n ) 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 ∗ ( B Y ∗ ) is relatively Mackey compact. Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous operators Let T : X → Y be a bounded linear operator. TFAE: ◮ T is completely continuous, i.e., ( x n ) weakly convergent ⇒ ( Tx n ) 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 ∗ ( B Y ∗ ) is relatively Mackey compact. Quantitative version [KS 2012] ◮ dp ( T ) = sup { χ 0 ( TA ); A ⊂ B X weakly compact } Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity

Mackey compactness and completely continuous operators Let T : X → Y be a bounded linear operator. TFAE: ◮ T is completely continuous, i.e., ( x n ) weakly convergent ⇒ ( Tx n ) 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 ∗ ( B Y ∗ ) is relatively Mackey compact. Quantitative version [KS 2012] ◮ dp ( T ) = sup { χ 0 ( TA ); A ⊂ B X weakly compact } 2 χ m ( T ∗ ) ≤ dp ( T ) ≤ cc w →�·� ( T ) ≤ 4 χ m ( T ∗ ) 1 ◮ 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 : cc w →�·� ( T ) ≤ 2 ω ( T ∗ ) . [KKS 2013] 4. ∀ Y ∀ T : Y → X : cc w →�·� ( T ∗ ) ≤ 2 ω ( 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 : cc w →�·� ( T ) ≤ 2 ω ( T ∗ ) . [KKS 2013] 4. ∀ Y ∀ T : Y → X : cc w →�·� ( T ∗ ) ≤ 2 ω ( T ) . [KKS 2013] Sketch DPP cc w →�·� ( T ) cc ρ →�·� ( 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 : cc w →�·� ( T ) ≤ 2 ω ( T ∗ ) . [KKS 2013] 4. ∀ Y ∀ T : Y → X : cc w →�·� ( T ∗ ) ≤ 2 ω ( T ) . [KKS 2013] Sketch easy DPP cc w →�·� ( T ) cc ρ →�·� ( T ) cont ρ →�·� ( 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 : cc w →�·� ( T ) ≤ 2 ω ( T ∗ ) . [KKS 2013] 4. ∀ Y ∀ T : Y → X : cc w →�·� ( T ∗ ) ≤ 2 ω ( T ) . [KKS 2013] Sketch easy DPP above cc w →�·� ( T ) cc ρ →�·� ( T ) cont ρ →�·� ( T ) 2 ω ( 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 : cc w →�·� ( T ) ≤ 2 ω ( T ∗ ) . [KKS 2013] 4. ∀ Y ∀ T : Y → X : cc w →�·� ( T ∗ ) ≤ 2 ω ( T ) . [KKS 2013] Quantitative strengthening of DPP [KKS 2013] ◮ X has direct qDPP if ∃ C > 0 : ∀ Y ∀ T : X → Y : cc w →�·� ( T ) ≤ C wk ( T ∗ ) Ondˇ rej F.K. Kalenda Measuring noncompactness and discontinuity