Determinacy for the complex moment problem via positive definite - PowerPoint PPT Presentation

Determinacy for the complex moment problem via positive definite extensions Dariusz Cicho´ n December 2016, OTOA, Bangalore Joint work with J. Stochel and F.H. Szafraniec Dariusz Cicho´ n Determinacy via positive definite extensions

Introduction Notations: N = { ( m , n ) : m , n – integers such that m � 0 , n � 0 } , N + = { ( m , n ) : m , n – integers such that m + n � 0 } . Question: when a sequence γ = { γ m , n } m , n � 0 ⊂ C is a complex moment sequence? I.e. there exists a Borel measure µ on C such that � z m ¯ z n d µ ( z ) , c m , n = m , n � 0 . C An ‘iff’ criterion: PDE( γ ) is nonempty, where PDE( γ ) = { ˜ γ : ˜ γ is a positive definite extension of γ on N + } i.e. ˜ γ = { ˜ γ m , n } m + n � 0 ⊂ C satisfies ˜ γ | N = γ and � λ m , n ¯ λ p , q ˜ γ m + q , n + p � 0 m + n � 0 , p + q � 0 for every finitely supported { λ m , n } ∞ m + n � 0 ⊂ C . Dariusz Cicho´ n Determinacy via positive definite extensions

Introduction Notations: N = { ( m , n ) : m , n – integers such that m � 0 , n � 0 } , N + = { ( m , n ) : m , n – integers such that m + n � 0 } . Question: when a sequence γ = { γ m , n } m , n � 0 ⊂ C is a complex moment sequence? I.e. there exists a Borel measure µ on C such that � z m ¯ z n d µ ( z ) , c m , n = m , n � 0 . C An ‘iff’ criterion: PDE( γ ) is nonempty, where PDE( γ ) = { ˜ γ : ˜ γ is a positive definite extension of γ on N + } i.e. ˜ γ = { ˜ γ m , n } m + n � 0 ⊂ C satisfies ˜ γ | N = γ and � λ m , n ¯ λ p , q ˜ γ m + q , n + p � 0 m + n � 0 , p + q � 0 for every finitely supported { λ m , n } ∞ m + n � 0 ⊂ C . Dariusz Cicho´ n Determinacy via positive definite extensions

Introduction Notations: N = { ( m , n ) : m , n – integers such that m � 0 , n � 0 } , N + = { ( m , n ) : m , n – integers such that m + n � 0 } . Question: when a sequence γ = { γ m , n } m , n � 0 ⊂ C is a complex moment sequence? I.e. there exists a Borel measure µ on C such that � z m ¯ z n d µ ( z ) , c m , n = m , n � 0 . C An ‘iff’ criterion: PDE( γ ) is nonempty, where PDE( γ ) = { ˜ γ : ˜ γ is a positive definite extension of γ on N + } i.e. ˜ γ = { ˜ γ m , n } m + n � 0 ⊂ C satisfies ˜ γ | N = γ and � λ m , n ¯ λ p , q ˜ γ m + q , n + p � 0 m + n � 0 , p + q � 0 for every finitely supported { λ m , n } ∞ m + n � 0 ⊂ C . Dariusz Cicho´ n Determinacy via positive definite extensions

Introduction Notations: N = { ( m , n ) : m , n – integers such that m � 0 , n � 0 } , N + = { ( m , n ) : m , n – integers such that m + n � 0 } . Question: when a sequence γ = { γ m , n } m , n � 0 ⊂ C is a complex moment sequence? I.e. there exists a Borel measure µ on C such that � z m ¯ z n d µ ( z ) , c m , n = m , n � 0 . C An ‘iff’ criterion: PDE( γ ) is nonempty, where PDE( γ ) = { ˜ γ : ˜ γ is a positive definite extension of γ on N + } i.e. ˜ γ = { ˜ γ m , n } m + n � 0 ⊂ C satisfies ˜ γ | N = γ and � λ m , n ¯ λ p , q ˜ γ m + q , n + p � 0 m + n � 0 , p + q � 0 for every finitely supported { λ m , n } ∞ m + n � 0 ⊂ C . Dariusz Cicho´ n Determinacy via positive definite extensions

Why? The main reason for this to work: N + is semiperfect, i.e. every positive definite complex function on this semigroup can be represented via Borel measures. More specifically: if { ˜ γ m , n } m + n � 0 is positive definite on N + , then there are Borel measures µ 1 on C ∗ (without 0) and µ 2 on T (the unit circle) such that � � C ∗ z m ¯ z n d µ 1 ( z ) + z m ¯ z n d µ 2 ( z ) . γ m , n = ˜ δ m + n , 0 � �� � T the Dirac delta The pair ( µ 1 , µ 2 ) will be called representing for ˜ γ . Dariusz Cicho´ n Determinacy via positive definite extensions

Why? The main reason for this to work: N + is semiperfect, i.e. every positive definite complex function on this semigroup can be represented via Borel measures. More specifically: if { ˜ γ m , n } m + n � 0 is positive definite on N + , then there are Borel measures µ 1 on C ∗ (without 0) and µ 2 on T (the unit circle) such that � � C ∗ z m ¯ z n d µ 1 ( z ) + z m ¯ z n d µ 2 ( z ) . γ m , n = ˜ δ m + n , 0 � �� � T the Dirac delta The pair ( µ 1 , µ 2 ) will be called representing for ˜ γ . Dariusz Cicho´ n Determinacy via positive definite extensions

Why? The main reason for this to work: N + is semiperfect, i.e. every positive definite complex function on this semigroup can be represented via Borel measures. More specifically: if { ˜ γ m , n } m + n � 0 is positive definite on N + , then there are Borel measures µ 1 on C ∗ (without 0) and µ 2 on T (the unit circle) such that � � C ∗ z m ¯ z n d µ 1 ( z ) + z m ¯ z n d µ 2 ( z ) . γ m , n = ˜ δ m + n , 0 � �� � T the Dirac delta The pair ( µ 1 , µ 2 ) will be called representing for ˜ γ . Dariusz Cicho´ n Determinacy via positive definite extensions

Why? The main reason for this to work: N + is semiperfect, i.e. every positive definite complex function on this semigroup can be represented via Borel measures. More specifically: if { ˜ γ m , n } m + n � 0 is positive definite on N + , then there are Borel measures µ 1 on C ∗ (without 0) and µ 2 on T (the unit circle) such that � � C ∗ z m ¯ z n d µ 1 ( z ) + z m ¯ z n d µ 2 ( z ) . γ m , n = ˜ δ m + n , 0 � �� � T the Dirac delta The pair ( µ 1 , µ 2 ) will be called representing for ˜ γ . Dariusz Cicho´ n Determinacy via positive definite extensions

Why? The main reason for this to work: N + is semiperfect, i.e. every positive definite complex function on this semigroup can be represented via Borel measures. More specifically: if { ˜ γ m , n } m + n � 0 is positive definite on N + , then there are Borel measures µ 1 on C ∗ (without 0) and µ 2 on T (the unit circle) such that � � C ∗ z m ¯ z n d µ 1 ( z ) + z m ¯ z n d µ 2 ( z ) . γ m , n = ˜ δ m + n , 0 � �� � T the Dirac delta The pair ( µ 1 , µ 2 ) will be called representing for ˜ γ . Dariusz Cicho´ n Determinacy via positive definite extensions

Main question Is there any connection between the following two situations? 1 γ is a determinate complex moment sequence on N , i.e. the representing measure for γ is unique, 2 PDE( γ ) is a singleton, i.e. PDE( γ ) = { ˜ γ } . Remark: if ( µ 1 , µ 2 ) is representing for ˜ γ ∈ PDE( γ ), then the measure µ 1 + µ 2 ( T ) δ 0 is representing for γ . The natural condition appearing when dealing with determinacy of γ : p.d. extension ˜ γ ∈ PDE( γ ) is called semideterminate if for any two representing pairs of measures ( µ 1 , µ 2 ) and ( µ ′ 1 , µ ′ 2 ) for ˜ γ we have 1 and µ 2 ◦ ϕ − 1 = µ ′ 2 ◦ ϕ − 1 , µ 1 = µ ′ where ϕ : T ∋ z �→ z 2 ∈ T . This happens if γ is determinate. Dariusz Cicho´ n Determinacy via positive definite extensions

Main question Is there any connection between the following two situations? 1 γ is a determinate complex moment sequence on N , i.e. the representing measure for γ is unique, 2 PDE( γ ) is a singleton, i.e. PDE( γ ) = { ˜ γ } . Remark: if ( µ 1 , µ 2 ) is representing for ˜ γ ∈ PDE( γ ), then the measure µ 1 + µ 2 ( T ) δ 0 is representing for γ . The natural condition appearing when dealing with determinacy of γ : p.d. extension ˜ γ ∈ PDE( γ ) is called semideterminate if for any two representing pairs of measures ( µ 1 , µ 2 ) and ( µ ′ 1 , µ ′ 2 ) for ˜ γ we have 1 and µ 2 ◦ ϕ − 1 = µ ′ 2 ◦ ϕ − 1 , µ 1 = µ ′ where ϕ : T ∋ z �→ z 2 ∈ T . This happens if γ is determinate. Dariusz Cicho´ n Determinacy via positive definite extensions

Main question Is there any connection between the following two situations? 1 γ is a determinate complex moment sequence on N , i.e. the representing measure for γ is unique, 2 PDE( γ ) is a singleton, i.e. PDE( γ ) = { ˜ γ } . Remark: if ( µ 1 , µ 2 ) is representing for ˜ γ ∈ PDE( γ ), then the measure µ 1 + µ 2 ( T ) δ 0 is representing for γ . The natural condition appearing when dealing with determinacy of γ : p.d. extension ˜ γ ∈ PDE( γ ) is called semideterminate if for any two representing pairs of measures ( µ 1 , µ 2 ) and ( µ ′ 1 , µ ′ 2 ) for ˜ γ we have 1 and µ 2 ◦ ϕ − 1 = µ ′ 2 ◦ ϕ − 1 , µ 1 = µ ′ where ϕ : T ∋ z �→ z 2 ∈ T . This happens if γ is determinate. Dariusz Cicho´ n Determinacy via positive definite extensions

Main question Is there any connection between the following two situations? 1 γ is a determinate complex moment sequence on N , i.e. the representing measure for γ is unique, 2 PDE( γ ) is a singleton, i.e. PDE( γ ) = { ˜ γ } . Remark: if ( µ 1 , µ 2 ) is representing for ˜ γ ∈ PDE( γ ), then the measure µ 1 + µ 2 ( T ) δ 0 is representing for γ . The natural condition appearing when dealing with determinacy of γ : p.d. extension ˜ γ ∈ PDE( γ ) is called semideterminate if for any two representing pairs of measures ( µ 1 , µ 2 ) and ( µ ′ 1 , µ ′ 2 ) for ˜ γ we have 1 and µ 2 ◦ ϕ − 1 = µ ′ 2 ◦ ϕ − 1 , µ 1 = µ ′ where ϕ : T ∋ z �→ z 2 ∈ T . This happens if γ is determinate. Dariusz Cicho´ n Determinacy via positive definite extensions