Dynamics of weighted composition operators on function spaces defined by local properties Thomas Kalmes Faculty of Mathematics Chemnitz Technical University Paweł Domański Memorial Conference Będlewo July 1 - 7, 2018 T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 1 / 16
Continuous linear operator T on a lcs E is called (topologically) transitive : ⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U ) ∩ V � = ∅ ( E separable Fréchet, equivalent to T hypercyclic , i.e. there is x ∈ E s.th. { T m x ; m ∈ N 0 } is dense in E .) T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16
Continuous linear operator T on a lcs E is called (topologically) transitive : ⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U ) ∩ V � = ∅ weakly mixing : ⇔ T ⊕ T transitive on E ⊕ E , i.e. ∀ U j , V j ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U j ) ∩ V j � = ∅ ( j = 1 , 2) T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16
Continuous linear operator T on a lcs E is called (topologically) transitive : ⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U ) ∩ V � = ∅ weakly mixing : ⇔ T ⊕ T transitive on E ⊕ E , i.e. ∀ U j , V j ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U j ) ∩ V j � = ∅ ( j = 1 , 2) power bounded : ⇔ { T m ; m ∈ N 0 } is equicontinuous, i.e. ∀ p ∈ cs ( E ) ∃ q ∈ cs ( E ) ∀ m ∈ N 0 , x ∈ E : p ( T m x ) ≤ q ( x ) Albanese, Bonet, Ricker ’09: E Fréchet-Montel, T power bounded � n − 1 ⇒ T uniformly mean ergodic , i.e. ∀ x ∈ E ∃ lim n →∞ 1 m =0 T m x n and convergence is uniform on bounded subsets of E . T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16
Continuous linear operator T on a lcs E is called (topologically) transitive : ⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U ) ∩ V � = ∅ weakly mixing : ⇔ T ⊕ T transitive on E ⊕ E , i.e. ∀ U j , V j ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U j ) ∩ V j � = ∅ ( j = 1 , 2) power bounded : ⇔ { T m ; m ∈ N 0 } is equicontinuous, i.e. ∀ p ∈ cs ( E ) ∃ q ∈ cs ( E ) ∀ m ∈ N 0 , x ∈ E : p ( T m x ) ≤ q ( x ) Albanese, Bonet, Ricker ’09: E Fréchet-Montel, T power bounded � n − 1 ⇒ T uniformly mean ergodic , i.e. ∀ x ∈ E ∃ lim n →∞ 1 m =0 T m x n and convergence is uniform on bounded subsets of E . Folklore (see e.g. Yoshida, 1980): E sequentially complete lcs, T continuous, linear, power bounded ⇒ (exp( sT )) s ≥ 0 C 0 -semigroup, where exp( sT ) x = � ∞ s m m ! T m x m =0 T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16
Continuous linear operator T on a lcs E is called (topologically) transitive : ⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U ) ∩ V � = ∅ weakly mixing : ⇔ T ⊕ T transitive on E ⊕ E , i.e. ∀ U j , V j ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U j ) ∩ V j � = ∅ ( j = 1 , 2) power bounded : ⇔ { T m ; m ∈ N 0 } is equicontinuous, i.e. ∀ p ∈ cs ( E ) ∃ q ∈ cs ( E ) ∀ m ∈ N 0 , x ∈ E : p ( T m x ) ≤ q ( x ) Several authors investigated these properties for weighted composition operators C w,ψ ( f ) = w · ( f ◦ ψ ) on various function spaces, e.g. Große-Erdmann, Mortini ’09; Zając ’16; Bonet, Domański ’12; Przestacki ’17; Beltrán-Meneu, Gómez-Callado, Jordá, Jornet ’16;... T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16
Continuous linear operator T on a lcs E is called (topologically) transitive : ⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U ) ∩ V � = ∅ weakly mixing : ⇔ T ⊕ T transitive on E ⊕ E , i.e. ∀ U j , V j ⊆ E open, non-empty ∃ m ∈ N 0 : T m ( U j ) ∩ V j � = ∅ ( j = 1 , 2) power bounded : ⇔ { T m ; m ∈ N 0 } is equicontinuous, i.e. ∀ p ∈ cs ( E ) ∃ q ∈ cs ( E ) ∀ m ∈ N 0 , x ∈ E : p ( T m x ) ≤ q ( x ) Several authors investigated these properties for weighted composition operators C w,ψ ( f ) = w · ( f ◦ ψ ) on various function spaces, e.g. Große-Erdmann, Mortini ’09; Zając ’16; Bonet, Domański ’12; Przestacki ’17; Beltrán-Meneu, Gómez-Callado, Jordá, Jornet ’16;... Objective: study these dynamical properties for weighted composition operators C w,ψ ( f ) = w · ( f ◦ ψ ) on function spaces "in a general framework". T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16
Ω locally compact, σ -compact, non-compact Hausdorff space, F a sheaf of K -valued functions on Ω , i.e. T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16
Ω locally compact, σ -compact, non-compact Hausdorff space, F a sheaf of K -valued functions on Ω , i.e. ∀ X ⊆ Ω open: F ( X ) is a K -vector space of K -valued functions on X s.th. ∀ Y ⊆ X ⊆ Ω open: r Y X : F ( X ) → F ( Y ) , f �→ f | Y well-defined (Gluing) ∀ open cover ( X ι ) ι ∈ I of an open set X ⊆ Ω ∀ ( f ι ) ι ∈ I ∈ � ι ∈ I F ( X ι ) with f ι | X ι ∩ X κ = f κ | X ι ∩ X κ ( ι, κ ∈ I ) there is f ∈ F ( X ) with f | X ι = f ι ( ι ∈ I ) . T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16
Ω locally compact, σ -compact, non-compact Hausdorff space, F a sheaf of K -valued functions on Ω , i.e. ∀ X ⊆ Ω open: F ( X ) is a K -vector space of K -valued functions on X s.th. ∀ Y ⊆ X ⊆ Ω open: r Y X : F ( X ) → F ( Y ) , f �→ f | Y well-defined (Gluing) ∀ open cover ( X ι ) ι ∈ I of an open set X ⊆ Ω ∀ ( f ι ) ι ∈ I ∈ � ι ∈ I F ( X ι ) with f ι | X ι ∩ X κ = f κ | X ι ∩ X κ ( ι, κ ∈ I ) there is f ∈ F ( X ) with f | X ι = f ι ( ι ∈ I ) . ⇒ ∀ X ⊆ Ω open ∀ ( X n ) n ∈ N 0 open, relatively compact exhaustion of X : ∼ proj ( F ( X n +1 ) , r X n F ( X ) X n +1 ) n ∈ N 0 = � { ( f n ) n ∈ N 0 ∈ F ( X n ); ∀ n ∈ N : f n | X n − 1 = f n − 1 } = n �→ via f ( f | X n ) n ∈ N 0 T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16
Ω locally compact, σ -compact, non-compact Hausdorff space, F a sheaf of K -valued functions on Ω , i.e. ∀ X ⊆ Ω open: F ( X ) is a K -vector space of K -valued functions on X s.th. ∀ Y ⊆ X ⊆ Ω open: r Y X : F ( X ) → F ( Y ) , f �→ f | Y well-defined (Gluing) ∀ open cover ( X ι ) ι ∈ I of an open set X ⊆ Ω ∀ ( f ι ) ι ∈ I ∈ � ι ∈ I F ( X ι ) with f ι | X ι ∩ X κ = f κ | X ι ∩ X κ ( ι, κ ∈ I ) there is f ∈ F ( X ) with f | X ι = f ι ( ι ∈ I ) . ⇒ ∀ X ⊆ Ω open ∀ ( X n ) n ∈ N 0 open, relatively compact exhaustion of X : ∼ proj ( F ( X n +1 ) , r X n F ( X ) X n +1 ) n ∈ N 0 = � { ( f n ) n ∈ N 0 ∈ F ( X n ); ∀ n ∈ N : f n | X n − 1 = f n − 1 } = n �→ via f ( f | X n ) n ∈ N 0 Examples: Ω = R d , F ( X ) = C ∞ ( X ) , F ( X ) = C ( X ) , F ( X ) = A ( X ) , or for Ω = C , F ( X ) = H ( X ) . L p ( X ) is not a sheaf. T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16
We define the following properties for a sheaf of functions F on Ω : ( F 1 ) ∀ X ⊆ Ω : F ( X ) is a webbed, ultrabornological Hausdorff lcs, F ( X ) ⊆ C ( X ) with ∀ x ∈ X : δ x ∈ F ( X ) ′ , T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16
We define the following properties for a sheaf of functions F on Ω : ( F 1 ) ∀ X ⊆ Ω : F ( X ) is a webbed, ultrabornological Hausdorff lcs, F ( X ) ⊆ C ( X ) with ∀ x ∈ X : δ x ∈ F ( X ) ′ , and ∀ open, rel. comp. exh. ( X n ) n ∈ N 0 of X : F ( X ) ∼ = proj ( F ( X n +1 ) , r X n X n +1 ) n ∈ N 0 topologically T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16
We define the following properties for a sheaf of functions F on Ω : ( F 1 ) ∀ X ⊆ Ω : F ( X ) is a webbed, ultrabornological Hausdorff lcs, F ( X ) ⊆ C ( X ) with ∀ x ∈ X : δ x ∈ F ( X ) ′ , and ∀ open, rel. comp. exh. ( X n ) n ∈ N 0 of X : F ( X ) ∼ = proj ( F ( X n +1 ) , r X n X n +1 ) n ∈ N 0 topologically ( F 2 ) ∀ K ⋐ Ω ∃ f K ∈ F (Ω) ∀ x ∈ K : f K ( x ) � = 0 T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16
We define the following properties for a sheaf of functions F on Ω : ( F 1 ) ∀ X ⊆ Ω : F ( X ) is a webbed, ultrabornological Hausdorff lcs, F ( X ) ⊆ C ( X ) with ∀ x ∈ X : δ x ∈ F ( X ) ′ , and ∀ open, rel. comp. exh. ( X n ) n ∈ N 0 of X : F ( X ) ∼ = proj ( F ( X n +1 ) , r X n X n +1 ) n ∈ N 0 topologically ( F 2 ) ∀ K ⋐ Ω ∃ f K ∈ F (Ω) ∀ x ∈ K : f K ( x ) � = 0 ( F 3 ) ∀ x, y ∈ Ω , x � = y ∃ f ∈ F (Ω) : f ( x ) = 0 , f ( y ) = 1 T. Kalmes (TU Chemnitz) Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16
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