Frames generated from exponential of operators Akram Aldroubi Vanderbilt University ICERM 2018 Supported by NSF/DMS – Typeset by Foil T EX –
Operator induced frames ICERM 18 Outline 1. Statement of the problem – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Outline 1. Statement of the problem 2. Frames induced by powers of an operator – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Outline 1. Statement of the problem 2. Frames induced by powers of an operator 3. Semi-continuous frames induced by powers of an operator – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Problem Statement General Statement: Let H be a Hilbert space, A a bounded operator on H , and G a countable subset of H . Find conditions on A , and G , such that the system { A n g } g ∈G ,n =0 , 1 , 2 ,...,L is a frame of H for some L ∈ N ∪ {∞} . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Problem Statement General Statement: Let H be a Hilbert space, A a bounded operator on H , and G a countable subset of H . Find conditions on A , and G , such that the system { A n g } g ∈G ,n =0 , 1 , 2 ,...,L is a frame of H for some L ∈ N ∪ {∞} . Alternatively { A t g } g ∈G ,t ∈ [0 ,L ] is a semi-continuous frame. – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Problem Statement General Statement: Let H be a Hilbert space, A a bounded operator on H , and G a countable subset of H . Find conditions on A , and G , such that the system { A n g } g ∈G ,n =0 , 1 , 2 ,...,L is a frame of H for some L ∈ N ∪ {∞} . Alternatively { A t g } g ∈G ,t ∈ [0 ,L ] is a semi-continuous frame. Motivation : Sampling and reconstruction of functions that are evolving in time under the action of an evolution operator. – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Definitions Let A ∈ B ( H ) , G ⊂ H , τ ⊂ R , and E = { A t g : g ∈ G , t ∈ T } ⊂ H . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Definitions Let A ∈ B ( H ) , G ⊂ H , τ ⊂ R , and E = { A t g : g ∈ G , t ∈ T } ⊂ H . |� f, A t g �| 2 dµ ( t ) ≤ C 2 � f � 2 , for all f ∈ H � 1. E is Bessel if � g ∈G T – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Definitions Let A ∈ B ( H ) , G ⊂ H , τ ⊂ R , and E = { A t g : g ∈ G , t ∈ T } ⊂ H . |� f, A t g �| 2 dµ ( t ) ≤ C 2 � f � 2 , for all f ∈ H � 1. E is Bessel if � g ∈G T 2. E is a (semi-continuous) frame for H if there exists C 1 , C 2 > 0 such that � C 1 � f � 2 ≤ � |� f, A t g �| 2 dµ ( t ) ≤ C 2 � f � 2 . g ∈G T – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Armenak Petrosian– 2017] If for ∈ A B ( H ) there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N ( L = ∞ ) is a frame for H , then for every f ∈ H , ( A ∗ ) n f → 0 as n → ∞ . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Armenak Petrosian– 2017] If for ∈ A B ( H ) there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N ( L = ∞ ) is a frame for H , then for every f ∈ H , ( A ∗ ) n f → 0 as n → ∞ . Thus the spectral radius ρ ( A ) , for dim( H ) = ∞ is s.t. ρ ( A ∗ ) ⊂ D . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Armenak Petrosian– 2017] If for ∈ A B ( H ) there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N ( L = ∞ ) is a frame for H , then for every f ∈ H , ( A ∗ ) n f → 0 as n → ∞ . Thus the spectral radius ρ ( A ) , for dim( H ) = ∞ is s.t. ρ ( A ∗ ) ⊂ D . Example : H = ℓ 2 ( N ) , Ax = (0 , x 1 , x 2 , . . . ) , G = { e 1 = (1 , 0 , . . . ) } . Then, A n e 1 = A n (1 , 0 , . . . ) = e n . Thus, { A n g } g ∈G ,n ∈ N is a frame for H (orthonormal basis). ( A ∗ ) n x = ( x n , x n +1 , . . . ) → 0 . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [ A., Armenak Petrosian– 2017] If for ∈ A B ( H ) there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N ( L = ∞ ) is a frame for H , then for every f ∈ H , ( A ∗ ) n f → 0 as n → ∞ . Corollary. [A., Armenak Petrosian– 2017] If A ∈ B ( H ) is unitary, then no G is such that { A n g } g ∈G ,n ∈ N is a frame for H . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [ A., Armenak Petrosian– 2017] If for ∈ A B ( H ) there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N ( L = ∞ ) is a frame for H , then for every f ∈ H , ( A ∗ ) n f → 0 as n → ∞ . Corollary. [A., Armenak Petrosian– 2017] If A ∈ B ( H ) is unitary, then no G is such that { A n g } g ∈G ,n ∈ N is a frame for H . Example : H = ℓ 2 ( Z ) , Ae n = e n +1 is the right shift operator, � ( A ∗ ) n f � = � f � . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [ A., Armenak Petrosian– 2017] If for ∈ A B ( H ) there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N ( L = ∞ ) is a frame for H , then for every f ∈ H , ( A ∗ ) n f → 0 as n → ∞ . Corollary. [A., Armenak Petrosian– 2017] If A ∈ B ( H ) is unitary, then no G is such that { A n g } g ∈G ,n ∈ N is a frame for H . Example : H = ℓ 2 ( Z ) , Ae n = e n +1 is the right shift operator, � ( A ∗ ) n f � = � f � . More recent: Christensen, Hasannasabjaldehbakhani and Philipp– ICCHA7–2018): Also A n f → 0 as n → ∞ . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Armenak Petrosian– 2017] If � A � ≤ 1 and ( A ∗ ) n f → 0 as n → ∞ , then there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N is a (tight) frame for H . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Armenak Petrosian– 2017] If � A � ≤ 1 and ( A ∗ ) n f → 0 as n → ∞ , then there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N is a (tight) frame for H . Characterization (Cabrelli, Molter, Paternostro, Philipp – 2018) (1) ( A ∗ ) n f → 0 , f ∈ H ; (2) G is Bessel; and (3) There exists an invertible S such that ASA ∗ = S − G where Gf = � g � f, g � g . Then { A n g } g ∈G ,n ∈ N is a frame for H . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Armenak Petrosian– 2017] If � A � ≤ 1 and ( A ∗ ) n f → 0 as n → ∞ , then there exists G ⊂ H such that { A n g } g ∈G ,n ∈ N is a (tight) frame for H . Characterization (Cabrelli, Molter, Paternostro, Philipp – 2018) (1) ( A ∗ ) n f → 0 , f ∈ H ; (2) G is Bessel; and (3) There exists an invertible S such that ASA ∗ = S − G where Gf = � g � f, g � g . Then { A n g } g ∈G ,n ∈ N is a frame for H . � f, A n g � A n g S is necessarily the frame oparator Sf = � g ∈ G,n ≥ 0 – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Finite set G Theorem. [A., Armenak Petrosian– 2017] If dim H = ∞ , |G| < ∞ , and { A n g } g ∈G ,n ∈ N is a frame for H , then � A � ≥ 1 . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Cabrelli, Cakmak, Molter, Pertrosyan–JFA 17] Then, { A n g } n ≥ 0 is a Let A ∈ B ( H ) be normal, dim( H ) = ∞ . frame for H if and only if 1. A = � λ j P j , where P j are rank one ortho-projections. j ∈ N 2. | λ k | < 1 for all k , and | λ k | → 1 and { λ k } satisfies Carleson | λ n − λ k | � condition inf n λ n λ k | ≥ δ for some δ > 0 . | 1 − ¯ k � = n � P j g � √ 3. 0 < C 1 ≤ 1 −| λ j | 2 ≤ C 2 < ∞ for some constants C 1 , C 2 . – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Cabrelli, Cakmak, Molter, Pertrosyan–JFA 17] Then, { A n g } n ≥ 0 is a Let A ∈ B ( H ) be normal, dim( H ) = ∞ . frame for H if and only if 1. A = � λ j P j , where P j are rank one ortho-projections. j ∈ N 2. | λ k | < 1 for all k , and | λ k | → 1 and { λ k } satisfies Carleson | λ n − λ k | � condition inf n λ n λ k | ≥ δ for some δ > 0 . | 1 − ¯ k � = n � P j g � √ 3. 0 < C 1 ≤ 1 −| λ j | 2 ≤ C 2 < ∞ for some constants C 1 , C 2 . Generalization by Cabrelli,Molter, Paternostro, Philipp |G| < ∞ –2018 – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Theorem. [A., Cabrelli, Cakmak, Molter, Pertrosyan–JFA 17] Then, { A n g } n ≥ 0 is a Let A ∈ B ( H ) be normal, dim( H ) = ∞ . frame for H if and only if 1. A = � λ j P j , where P j are rank one ortho-projections. j ∈ N 2. | λ k | < 1 for all k , and | λ k | → 1 and { λ k } satisfies Carleson | λ n − λ k | � condition inf n λ n λ k | ≥ δ for some δ > 0 . | 1 − ¯ k � = n � P j g � √ 3. 0 < C 1 ≤ 1 −| λ j | 2 ≤ C 2 < ∞ for some constants C 1 , C 2 . Generalization by Cabrelli,Molter, Paternostro, Philipp |G| < ∞ –2018 – Typeset by Foil T EX – Akram Aldroubi
Operator induced frames ICERM 18 Conjecture : If A is a normal operator on H , then the system � A n g � � A n g � � g ∈G ,n ≥ 0 is not a frame for H . – Typeset by Foil T EX – Akram Aldroubi
Recommend
More recommend
