A z k -invariant subspace without the wandering property Daniel - PowerPoint PPT Presentation

A z k -invariant subspace without the wandering property Daniel Seco Universidad Carlos III de Madrid and Instituto de Ciencias Matemáticas Workshop on Banach spaces and Banach lattices, ICMAT 12 th September 2019 Seco (UC3M/ICMAT) Wandering property ICMAT 1 / 13

Spaces over the disc Definition Dirichlet-type space , D α , is: ∞ | a k | 2 ( k + 1 ) α < ∞} � � a k z k , || f || 2 { f ∈ Hol ( D ) : f ( z ) = α = k ∈ N k = 0 Seco (UC3M/ICMAT) Wandering property ICMAT 2 / 13

Spaces over the disc Definition Dirichlet-type space , D α , is: ∞ | a k | 2 ( k + 1 ) α < ∞} � � a k z k , || f || 2 { f ∈ Hol ( D ) : f ( z ) = α = k ∈ N k = 0 Examples α = − 1, A 2 = Hol ( D ) ∩ L 2 ( D ) Seco (UC3M/ICMAT) Wandering property ICMAT 2 / 13

Spaces over the disc Definition Dirichlet-type space , D α , is: ∞ | a k | 2 ( k + 1 ) α < ∞} � � a k z k , || f || 2 { f ∈ Hol ( D ) : f ( z ) = α = k ∈ N k = 0 Examples α = − 1, A 2 = Hol ( D ) ∩ L 2 ( D ) α = 0, H 2 = Hol ( D ) ∩ L 2 ( T ) Seco (UC3M/ICMAT) Wandering property ICMAT 2 / 13

Spaces over the disc Definition Dirichlet-type space , D α , is: ∞ | a k | 2 ( k + 1 ) α < ∞} � � a k z k , || f || 2 { f ∈ Hol ( D ) : f ( z ) = α = k ∈ N k = 0 Examples α = − 1, A 2 = Hol ( D ) ∩ L 2 ( D ) α = 0, H 2 = Hol ( D ) ∩ L 2 ( T ) α = 1, D = Hol ( D ) ∩ { A ( f ( D )) < ∞} Seco (UC3M/ICMAT) Wandering property ICMAT 2 / 13

Spaces over the disc Definition Dirichlet-type space , D α , is: ∞ | a k | 2 ( k + 1 ) α < ∞} � � a k z k , || f || 2 { f ∈ Hol ( D ) : f ( z ) = α = k ∈ N k = 0 Examples α = − 1, A 2 = Hol ( D ) ∩ L 2 ( D ) α = 0, H 2 = Hol ( D ) ∩ L 2 ( T ) α = 1, D = Hol ( D ) ∩ { A ( f ( D )) < ∞} α > α ′ ⇒ D α ⊂ D α ′ Seco (UC3M/ICMAT) Wandering property ICMAT 2 / 13

Spaces over the disc Definition Dirichlet-type space , D α , is: ∞ | a k | 2 ( k + 1 ) α < ∞} � � a k z k , || f || 2 { f ∈ Hol ( D ) : f ( z ) = α = k ∈ N k = 0 Examples α = − 1, A 2 = Hol ( D ) ∩ L 2 ( D ) α = 0, H 2 = Hol ( D ) ∩ L 2 ( T ) α = 1, D = Hol ( D ) ∩ { A ( f ( D )) < ∞} α > α ′ ⇒ D α ⊂ D α ′ f ∈ D α ⇔ f ′ ∈ D α − 2 Seco (UC3M/ICMAT) Wandering property ICMAT 2 / 13

Spaces over the disc Definition Dirichlet-type space , D α , is: ∞ | a k | 2 ( k + 1 ) α < ∞} � � a k z k , || f || 2 { f ∈ Hol ( D ) : f ( z ) = α = k ∈ N k = 0 Examples α = − 1, A 2 = Hol ( D ) ∩ L 2 ( D ) α = 0, H 2 = Hol ( D ) ∩ L 2 ( T ) α = 1, D = Hol ( D ) ∩ { A ( f ( D )) < ∞} α > α ′ ⇒ D α ⊂ D α ′ f ∈ D α ⇔ f ′ ∈ D α − 2 Hilbert spaces with monomials as an orthogonal basis Seco (UC3M/ICMAT) Wandering property ICMAT 2 / 13

Invariant subspaces The (forward) shift operator is bdd: S : D α → D α : Sf ( z ) = zf ( z ) . A closed subspace V of D α is z k - invariant if S k V ⊂ V . Seco (UC3M/ICMAT) Wandering property ICMAT 3 / 13

Invariant subspaces The (forward) shift operator is bdd: S : D α → D α : Sf ( z ) = zf ( z ) . A closed subspace V of D α is z k - invariant if S k V ⊂ V . [ f ] z k (= [ f ]) = span { z tk f : t = 0 , 1 , 2 , . . . } . Seco (UC3M/ICMAT) Wandering property ICMAT 3 / 13

Invariant subspaces The (forward) shift operator is bdd: S : D α → D α : Sf ( z ) = zf ( z ) . A closed subspace V of D α is z k - invariant if S k V ⊂ V . [ f ] z k (= [ f ]) = span { z tk f : t = 0 , 1 , 2 , . . . } . Theorem (Beurling, ’49) For H 2 ( α = 0 ), M z-inv. subsp. ⇔ M = ϕ H 2 = [ ϕ ] with span ( ϕ ) = M ⊖ SM . Seco (UC3M/ICMAT) Wandering property ICMAT 3 / 13

Invariant subspaces The (forward) shift operator is bdd: S : D α → D α : Sf ( z ) = zf ( z ) . A closed subspace V of D α is z k - invariant if S k V ⊂ V . [ f ] z k (= [ f ]) = span { z tk f : t = 0 , 1 , 2 , . . . } . Theorem (Beurling, ’49) For H 2 ( α = 0 ), M z-inv. subsp. ⇔ M = ϕ H 2 = [ ϕ ] with span ( ϕ ) = M ⊖ SM . Theorem (Aleman, Richter, Sundberg, ’96) For A 2 ( α = − 1 ), M z-inv. ⇔ [ M ⊖ SM ] = M . Seco (UC3M/ICMAT) Wandering property ICMAT 3 / 13

Wandering subspaces Shimorin (’11) extended ARS’96 to α ∈ [ − 1 , 1 ] (different ideas for α < 0 and α > 0). Hedenmalm-Zhu (’92) and Nowak et al. (’17) showed that the analogous fails for α ≤ − 5 in some sense. Definition M has the z k wandering property if [ M ⊖ S k M ] z k = M . Seco (UC3M/ICMAT) Wandering property ICMAT 4 / 13

Wandering subspaces Shimorin (’11) extended ARS’96 to α ∈ [ − 1 , 1 ] (different ideas for α < 0 and α > 0). Hedenmalm-Zhu (’92) and Nowak et al. (’17) showed that the analogous fails for α ≤ − 5 in some sense. Definition M has the z k wandering property if [ M ⊖ S k M ] z k = M . If ∀ M z k -inv. subsp. of H , M has the wandering property, then the wandering subspace property (WSP) holds for z k in H . Seco (UC3M/ICMAT) Wandering property ICMAT 4 / 13

Wandering subspaces Shimorin (’11) extended ARS’96 to α ∈ [ − 1 , 1 ] (different ideas for α < 0 and α > 0). Hedenmalm-Zhu (’92) and Nowak et al. (’17) showed that the analogous fails for α ≤ − 5 in some sense. Definition M has the z k wandering property if [ M ⊖ S k M ] z k = M . If ∀ M z k -inv. subsp. of H , M has the wandering property, then the wandering subspace property (WSP) holds for z k in H . Norm dependent! Seco (UC3M/ICMAT) Wandering property ICMAT 4 / 13

Wandering subspaces Shimorin (’11) extended ARS’96 to α ∈ [ − 1 , 1 ] (different ideas for α < 0 and α > 0). Hedenmalm-Zhu (’92) and Nowak et al. (’17) showed that the analogous fails for α ≤ − 5 in some sense. Definition M has the z k wandering property if [ M ⊖ S k M ] z k = M . If ∀ M z k -inv. subsp. of H , M has the wandering property, then the wandering subspace property (WSP) holds for z k in H . Norm dependent! Really! Seco (UC3M/ICMAT) Wandering property ICMAT 4 / 13

The Problem More work on WSP for other multipliers (i.e. mult. by an inner function) by Carswell, Duren, Khavinson, Shapiro, Stessin, Sundberg, Weir... Seco (UC3M/ICMAT) Wandering property ICMAT 5 / 13

The Problem More work on WSP for other multipliers (i.e. mult. by an inner function) by Carswell, Duren, Khavinson, Shapiro, Stessin, Sundberg, Weir... CDS’02: Not known whether always WSP for mult. by a finite Blaschke product in A 2 . Seco (UC3M/ICMAT) Wandering property ICMAT 5 / 13

The Problem More work on WSP for other multipliers (i.e. mult. by an inner function) by Carswell, Duren, Khavinson, Shapiro, Stessin, Sundberg, Weir... CDS’02: Not known whether always WSP for mult. by a finite Blaschke product in A 2 . Conjecture k > 1. For z k in A 2 , the wandering prop. holds. Seco (UC3M/ICMAT) Wandering property ICMAT 5 / 13

The Problem More work on WSP for other multipliers (i.e. mult. by an inner function) by Carswell, Duren, Khavinson, Shapiro, Stessin, Sundberg, Weir... CDS’02: Not known whether always WSP for mult. by a finite Blaschke product in A 2 . Conjecture k > 1. For z k in A 2 , the wandering prop. holds. This is the problem we study (but do not solve) today. Seco (UC3M/ICMAT) Wandering property ICMAT 5 / 13

Results Theorem ∀ k ≥ 6 , ∀ α ∈ R , D α admits equiv. norm � · � : z k -WSP fails. Seco (UC3M/ICMAT) Wandering property ICMAT 6 / 13

Results Theorem ∀ k ≥ 6 , ∀ α ∈ R , D α admits equiv. norm � · � : z k -WSP fails. Theorem (Gallardo-Gutiérrez, Partington, Seco, ’19) ∀ k ≥ 1 , ∀ α ∈ [ − 1 , 1 ] , D α admits equiv. norm � · � : B-WSP holds (any B finite Blaschke product). Seco (UC3M/ICMAT) Wandering property ICMAT 6 / 13

Results Theorem ∀ k ≥ 6 , ∀ α ∈ R , D α admits equiv. norm � · � : z k -WSP fails. Theorem (Gallardo-Gutiérrez, Partington, Seco, ’19) ∀ k ≥ 1 , ∀ α ∈ [ − 1 , 1 ] , D α admits equiv. norm � · � : B-WSP holds (any B finite Blaschke product). What about the usual norms in D α ? Seco (UC3M/ICMAT) Wandering property ICMAT 6 / 13

Results Theorem ∀ k ≥ 6 , ∀ α ∈ R , D α admits equiv. norm � · � : z k -WSP fails. Theorem (Gallardo-Gutiérrez, Partington, Seco, ’19) ∀ k ≥ 1 , ∀ α ∈ [ − 1 , 1 ] , D α admits equiv. norm � · � : B-WSP holds (any B finite Blaschke product). What about the usual norms in D α ? ( k − 9 ) 2 ) , z k -WSP fails. 700 k ≥ 10, α < − ( 5 k + Seco (UC3M/ICMAT) Wandering property ICMAT 6 / 13

Results Theorem ∀ k ≥ 6 , ∀ α ∈ R , D α admits equiv. norm � · � : z k -WSP fails. Theorem (Gallardo-Gutiérrez, Partington, Seco, ’19) ∀ k ≥ 1 , ∀ α ∈ [ − 1 , 1 ] , D α admits equiv. norm � · � : B-WSP holds (any B finite Blaschke product). What about the usual norms in D α ? ( k − 9 ) 2 ) , z k -WSP fails. 700 k ≥ 10, α < − ( 5 k + log 2 log( k + 1 ) ≤ α ≤ 1, z k -WSP holds. − Seco (UC3M/ICMAT) Wandering property ICMAT 6 / 13