The Pick–Nevanlinna problem: from metric geometry to matrix positivity Gautam Bharali Indian Institute of Science bharali@iisc.ac.in Eigenfunction 2019 (with Apoorva Khare) Indian Institute of Science April 12, 2019 Gautam Bharali The Pick–Nevanlinna problem
1 The central problem The Pick–Nevanlinna Interpolation Problem in general: Gautam Bharali The Pick–Nevanlinna problem
1 The central problem The Pick–Nevanlinna Interpolation Problem in general: ( ∗ ) Ω k ⊂ C n k are domains, k = 1 , 2 . Given M distinct points z 1 , . . . , z M ∈ Ω 1 , and points w 1 , . . . , w M in Ω 2 , find nec. & suff. conditions on ( z 1 , w 1 ) , ( z 2 , w 2 ) , . . . , ( z M , w M ) , such that there exists a holomorphic map F : Ω 1 → Ω 2 satisfying F ( z j ) = w j , 1 ≤ j ≤ M . Gautam Bharali The Pick–Nevanlinna problem
1 The central problem The Pick–Nevanlinna Interpolation Problem in general: ( ∗ ) Ω k ⊂ C n k are domains, k = 1 , 2 . Given M distinct points z 1 , . . . , z M ∈ Ω 1 , and points w 1 , . . . , w M in Ω 2 , find nec. & suff. conditions on ( z 1 , w 1 ) , ( z 2 , w 2 ) , . . . , ( z M , w M ) , such that there exists a holomorphic map F : Ω 1 → Ω 2 satisfying F ( z j ) = w j , 1 ≤ j ≤ M . ( ∗ ) derives its name from the solution to this problem for Ω 1 = Ω 2 = D given by G. Pick & rediscovered by R. Nevanlinna. Gautam Bharali The Pick–Nevanlinna problem
1 The central problem The Pick–Nevanlinna Interpolation Problem in general: ( ∗ ) Ω k ⊂ C n k are domains, k = 1 , 2 . Given M distinct points z 1 , . . . , z M ∈ Ω 1 , and points w 1 , . . . , w M in Ω 2 , find nec. & suff. conditions on ( z 1 , w 1 ) , ( z 2 , w 2 ) , . . . , ( z M , w M ) , such that there exists a holomorphic map F : Ω 1 → Ω 2 satisfying F ( z j ) = w j , 1 ≤ j ≤ M . ( ∗ ) derives its name from the solution to this problem for Ω 1 = Ω 2 = D given by G. Pick & rediscovered by R. Nevanlinna. Theorem (G. Pick, R. Nevanlinna) . Let z 1 , . . . , z M be distinct points in D and w 1 , . . . , w M ∈ D . There exists F ∈ Hol( D ; D ) satisfying F ( z j ) = w j , 1 ≤ j ≤ M , iff the matrix � 1 − w k w j � M 1 − z k z j j,k =1 is positive semi-definite. Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Given a non-empty set S , a Hilbert function space on S is a complex Hilbert space H with the following properties: Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Given a non-empty set S , a Hilbert function space on S is a complex Hilbert space H with the following properties: ◮ elements of H are C -valued functions on S ; Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Given a non-empty set S , a Hilbert function space on S is a complex Hilbert space H with the following properties: ◮ elements of H are C -valued functions on S ; ◮ eval x is a bounded linear functional ∀ x ∈ S . Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Given a non-empty set S , a Hilbert function space on S is a complex Hilbert space H with the following properties: ◮ elements of H are C -valued functions on S ; ◮ eval x is a bounded linear functional ∀ x ∈ S . Equip C n with a complex inner product. Define the vector space Mult( H , H ⊗ C n ) := { φ : S → C n | hφ ∈ H ⊗ C n ∀ h ∈ H} viewed as a subspace of B ( H , H ⊗ C n ) . Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Given a non-empty set S , a Hilbert function space on S is a complex Hilbert space H with the following properties: ◮ elements of H are C -valued functions on S ; ◮ eval x is a bounded linear functional ∀ x ∈ S . Equip C n with a complex inner product. Define the vector space Mult( H , H ⊗ C n ) := { φ : S → C n | hφ ∈ H ⊗ C n ∀ h ∈ H} viewed as a subspace of B ( H , H ⊗ C n ) . If we write M φ ( h ) := h ⊗ φ ( = hφ ) , h ∈ H , then it’s easy to show: ⇒ ( I − M φ M ∗ � M φ � op ≤ 1 ⇐ φ ) is positive semi-definite. Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Given a non-empty set S , a Hilbert function space on S is a complex Hilbert space H with the following properties: ◮ elements of H are C -valued functions on S ; ◮ eval x is a bounded linear functional ∀ x ∈ S . Equip C n with a complex inner product. Define the vector space Mult( H , H ⊗ C n ) := { φ : S → C n | hφ ∈ H ⊗ C n ∀ h ∈ H} viewed as a subspace of B ( H , H ⊗ C n ) . If we write M φ ( h ) := h ⊗ φ ( = hφ ) , h ∈ H , then it’s easy to show: ⇒ ( I − M φ M ∗ � M φ � op ≤ 1 ⇐ φ ) is positive semi-definite. Denote by K ( · , x ) ∈ H the Riesz representative of eval x . Gautam Bharali The Pick–Nevanlinna problem
2 Towards a necessary condition How does one even guess such a condition? Sarason gave the following argument — which he says is implicit in Nevanlinna’s approach: Given a non-empty set S , a Hilbert function space on S is a complex Hilbert space H with the following properties: ◮ elements of H are C -valued functions on S ; ◮ eval x is a bounded linear functional ∀ x ∈ S . Equip C n with a complex inner product. Define the vector space Mult( H , H ⊗ C n ) := { φ : S → C n | hφ ∈ H ⊗ C n ∀ h ∈ H} viewed as a subspace of B ( H , H ⊗ C n ) . If we write M φ ( h ) := h ⊗ φ ( = hφ ) , h ∈ H , then it’s easy to show: ⇒ ( I − M φ M ∗ � M φ � op ≤ 1 ⇐ φ ) is positive semi-definite. Denote by K ( · , x ) ∈ H the Riesz representative of eval x . With these constructs, we discover . . . Gautam Bharali The Pick–Nevanlinna problem
3 Towards a necessary condition, cont’d. Proposition (Sarason) . Let S be a non-empty set and H a Hilbert function space on it. Fix � · , · � on C n . Let x 1 , . . . , x M be distinct points in S and w 1 , . . . , w M ∈ C n s.t. � w j � ≤ 1 , 1 ≤ j ≤ M . Gautam Bharali The Pick–Nevanlinna problem
3 Towards a necessary condition, cont’d. Proposition (Sarason) . Let S be a non-empty set and H a Hilbert function space on it. Fix � · , · � on C n . Let x 1 , . . . , x M be distinct points in S and w 1 , . . . , w M ∈ C n s.t. � w j � ≤ 1 , 1 ≤ j ≤ M . It there exists a φ ∈ Mult( H , H ⊗ C n ) with � M φ � op ≤ 1 satisfying φ ( x j ) = w j , 1 ≤ j ≤ M , then the matrix �� I − ( · w j )( · w k ) ∗ � � M K ( x j , x k ) j,k =1 is positive semi-definite. Gautam Bharali The Pick–Nevanlinna problem
3 Towards a necessary condition, cont’d. Proposition (Sarason) . Let S be a non-empty set and H a Hilbert function space on it. Fix � · , · � on C n . Let x 1 , . . . , x M be distinct points in S and w 1 , . . . , w M ∈ C n s.t. � w j � ≤ 1 , 1 ≤ j ≤ M . It there exists a φ ∈ Mult( H , H ⊗ C n ) with � M φ � op ≤ 1 satisfying φ ( x j ) = w j , 1 ≤ j ≤ M , then the matrix �� I − ( · w j )( · w k ) ∗ � � M K ( x j , x k ) j,k =1 is positive semi-definite. Since computing adjoints of operators on H ⊗ C n takes time, we’ll consider the case n = 1 . Gautam Bharali The Pick–Nevanlinna problem
3 Towards a necessary condition, cont’d. Proposition (Sarason) . Let S be a non-empty set and H a Hilbert function space on it. Fix � · , · � on C n . Let x 1 , . . . , x M be distinct points in S and w 1 , . . . , w M ∈ C n s.t. � w j � ≤ 1 , 1 ≤ j ≤ M . It there exists a φ ∈ Mult( H , H ⊗ C n ) with � M φ � op ≤ 1 satisfying φ ( x j ) = w j , 1 ≤ j ≤ M , then the matrix �� I − ( · w j )( · w k ) ∗ � � M K ( x j , x k ) j,k =1 is positive semi-definite. Since computing adjoints of operators on H ⊗ C n takes time, we’ll consider the case n = 1 . In this case � g, M ∗ φ K ( · , x ) � = � φg, K ( · , x ) � = φ ( x ) g ( x ) = φ ( x ) � g, K ( · , x ) � ∀ g ∈ H ⇒ M ∗ φ K ( · , x ) = φ ( x ) K ( · , x ) . Gautam Bharali The Pick–Nevanlinna problem
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