Toeplitz and Asymptotic Toeplitz operators on H 2 ( D n ) Amit Maji (Joint work with Jaydeb Sarkar & Srijan Sarkar) Indian Statistical Institute, Bangalore Centre OTOA, December 13-22, 2016 Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 1 / 18
Aim To characterize Toeplitz operators on H 2 ( D n ). Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 2 / 18
Aim To characterize Toeplitz operators on H 2 ( D n ). To characterize asymptotically Toeplitz operators on H 2 ( D n ). Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 2 / 18
Aim To characterize Toeplitz operators on H 2 ( D n ). To characterize asymptotically Toeplitz operators on H 2 ( D n ). To generalize some of the recent results of Chalendar and Ross to vector-valued Hardy space H 2 E ( D ) and as well as quotient spaces of H 2 ( D n ). Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 2 / 18
Notation Open unit polydisc D n = { ( z 1 , . . . , z n ) ∈ C n : | z i | < 1 , i = 1 , . . . , n } . Distinguished boundary of D n T n = { ( z 1 , . . . , z n ) ∈ C n : | z i | = 1 , i = 1 , . . . , n } . Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 3 / 18
Notation Open unit polydisc D n = { ( z 1 , . . . , z n ) ∈ C n : | z i | < 1 , i = 1 , . . . , n } . Distinguished boundary of D n T n = { ( z 1 , . . . , z n ) ∈ C n : | z i | = 1 , i = 1 , . . . , n } . n =0 a n z n : � ∞ n =0 | a n | 2 < ∞} . Hardy space H 2 ( D ) = { f = � ∞ Vector-valued Hardy space n =0 a n z n : a n ∈ E and � ∞ E ( D ) = { f = � ∞ H 2 n =0 � a n � 2 E < ∞} . n =0 a n z n : sup H ∞ ( D ) = { f = � ∞ | a n | < ∞} . n ≥ 0 M z is the multiplication operator on H 2 ( D ) by the coordinate function z . Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 3 / 18
Notation Open unit polydisc D n = { ( z 1 , . . . , z n ) ∈ C n : | z i | < 1 , i = 1 , . . . , n } . Distinguished boundary of D n T n = { ( z 1 , . . . , z n ) ∈ C n : | z i | = 1 , i = 1 , . . . , n } . n =0 a n z n : � ∞ n =0 | a n | 2 < ∞} . Hardy space H 2 ( D ) = { f = � ∞ Vector-valued Hardy space n =0 a n z n : a n ∈ E and � ∞ E ( D ) = { f = � ∞ H 2 n =0 � a n � 2 E < ∞} . n =0 a n z n : sup H ∞ ( D ) = { f = � ∞ | a n | < ∞} . n ≥ 0 M z is the multiplication operator on H 2 ( D ) by the coordinate function z . � � a k z k : | a k | 2 < ∞ � � Hardy space over polydisc H 2 ( D n ) = f = , k ∈ N n k ∈ N n where k = ( k 1 , . . . , k n ) ∈ N n and z k = z k 1 1 · · · z k n n . For j = 1 , . . . , n , M z j are the multiplication operators on H 2 ( D n ) by the j th coordinate functions z j . Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 3 / 18
Multiplication operator For φ ∈ L ∞ ( T ), define M φ : L 2 ( T ) → L 2 ( T ) by M φ f = φ f for f ∈ L 2 ( T ). Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 4 / 18
Multiplication operator For φ ∈ L ∞ ( T ), define M φ : L 2 ( T ) → L 2 ( T ) by M φ f = φ f for f ∈ L 2 ( T ). The matrix of M φ with respect to the orthonormal basis { e in θ } ∞ n = −∞ of L 2 ( T ) = H 2 ( D ) ⊥ ⊕ H 2 ( D ) is ... ... ... ... φ 0 φ − 1 φ − 2 ... φ 1 φ 0 φ − 1 φ − 2 M φ = φ 2 φ 1 φ 0 φ − 1 φ − 2 ... φ 2 φ 1 φ 0 φ − 1 ... φ 2 φ 1 φ 0 ... ... ∞ φ n e in θ is a Fourier expansion of φ . � where φ = n = −∞ Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 4 / 18
Multiplication operator For φ ∈ L ∞ ( T ), define M φ : L 2 ( T ) → L 2 ( T ) by M φ f = φ f for f ∈ L 2 ( T ). The matrix of M φ with respect to the orthonormal basis { e in θ } ∞ n = −∞ of L 2 ( T ) = H 2 ( D ) ⊥ ⊕ H 2 ( D ) is ... ... ... ... φ 0 φ − 1 φ − 2 ... φ 1 φ 0 φ − 1 φ − 2 M φ = φ 2 φ 1 φ 0 φ − 1 φ − 2 ... φ 2 φ 1 φ 0 φ − 1 ... φ 2 φ 1 φ 0 ... ... ∞ φ n e in θ is a Fourier expansion of φ . � where φ = n = −∞ Toeplitz operator with symbol φ ∈ L ∞ ( T ) is the operator T φ defined by T φ f = P H 2 ( D ) ( φ f ) for f ∈ H 2 ( D ). Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 4 / 18
Toeplitz operator Toeplitz operators on the Hardy space (or, on the l 2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18
Toeplitz operator Toeplitz operators on the Hardy space (or, on the l 2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H 2 ( D ) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz operators J.Reine Angew. Math. 213:89–102, 1963/1964. Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18
Toeplitz operator Toeplitz operators on the Hardy space (or, on the l 2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H 2 ( D ) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz operators J.Reine Angew. Math. 213:89–102, 1963/1964. Brown-Halmos theorem characterize Toeplitz operators on H 2 ( D ) as follows: Let T be a bounded linear operator on H 2 ( D ). Then T is a Toeplitz operator if and only if M ∗ z TM z = T . Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18
Toeplitz operator Toeplitz operators on the Hardy space (or, on the l 2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H 2 ( D ) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz operators J.Reine Angew. Math. 213:89–102, 1963/1964. Brown-Halmos theorem characterize Toeplitz operators on H 2 ( D ) as follows: Let T be a bounded linear operator on H 2 ( D ). Then T is a Toeplitz operator if and only if M ∗ z TM z = T . The notion of Toeplitzness was extended to more general settings by Barr´ ıa and Halmos (1982) and Feintuch (1989). Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18
Toeplitz operator Toeplitz operators on the Hardy space (or, on the l 2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H 2 ( D ) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz operators J.Reine Angew. Math. 213:89–102, 1963/1964. Brown-Halmos theorem characterize Toeplitz operators on H 2 ( D ) as follows: Let T be a bounded linear operator on H 2 ( D ). Then T is a Toeplitz operator if and only if M ∗ z TM z = T . The notion of Toeplitzness was extended to more general settings by Barr´ ıa and Halmos (1982) and Feintuch (1989). A bounded linear operator T on H 2 ( D ) is (uniformly) asymptotically Toeplitz operator if { M ∗ m TM m z } m ≥ 1 converges in operator norm. z Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18
Toeplitz operator Toeplitz operators on the Hardy space (or, on the l 2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H 2 ( D ) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz operators J.Reine Angew. Math. 213:89–102, 1963/1964. Brown-Halmos theorem characterize Toeplitz operators on H 2 ( D ) as follows: Let T be a bounded linear operator on H 2 ( D ). Then T is a Toeplitz operator if and only if M ∗ z TM z = T . The notion of Toeplitzness was extended to more general settings by Barr´ ıa and Halmos (1982) and Feintuch (1989). A bounded linear operator T on H 2 ( D ) is (uniformly) asymptotically Toeplitz operator if { M ∗ m TM m z } m ≥ 1 converges in operator norm. z Feintuch (1989) gives a remarkable characterization of asymptotically Toeplitz operators: A bounded linear operator T on H 2 ( D ) is asymptotically Toeplitz if and only if T = compact + Toeplitz. Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18
Basic Definitions A closed subspace S of H is said to be invariant subspace of T ∈ B ( H ) if T ( S ) ⊆ S and S is said to be co-invariant subspace if T ∗ ( S ) ⊆ S . Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 6 / 18
Basic Definitions A closed subspace S of H is said to be invariant subspace of T ∈ B ( H ) if T ( S ) ⊆ S and S is said to be co-invariant subspace if T ∗ ( S ) ⊆ S . An operator T ∈ B ( H ) is said to be contraction if � T � ≤ 1. Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 6 / 18
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