Toeplitz operators on the symmetrized bidisc (A joint work with T. - PowerPoint PPT Presentation

Toeplitz operators on the symmetrized bidisc (A joint work with T. Bhattacharyya and B. K. Das) Haripada Sau Indian Institute of Technology Bombay IWOTA – 2017 iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Brown-Halmos A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators , J. Reine Angew. Math. 213 (1963) 89-102. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

The symmetrized bidisc The symmetrized bidisc is G = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | < 1 , | z 2 | < 1 } . � �� � ���� s p iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

The symmetrized bidisc The symmetrized bidisc is G = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | < 1 , | z 2 | < 1 } . � �� � ���� s p This is the range of the symmetrization map π : D × D → C 2 defined by ( z 1 , z 2 ) �→ ( z 1 + z 2 , z 1 z 2 ) . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

The symmetrized bidisc The symmetrized bidisc is G = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | < 1 , | z 2 | < 1 } . � �� � ���� s p This is the range of the symmetrization map π : D × D → C 2 defined by ( z 1 , z 2 ) �→ ( z 1 + z 2 , z 1 z 2 ) . Γ := G = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | ≤ 1 , | z 2 | ≤ 1 } . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

People who have worked on this domain includes J. Agler, N. Young, P. Pflug, W. Zwonek, L. Kosinski, C. Costara, Z. Lykova, G. Bharali, O. Shalit, T. Bhattacharyya, J. Sarkar, S. Pal, S. Biswas, S. ShyamRoy, S. Lata. and B. K. Das. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

They Hardy space The Hardy space H 2 ( G ) of the symmetrized bidisc is the vector space of those holomorphic functions f on G which satisfy � | f ◦ π ( r e iθ 1 , r e iθ 2 ) | 2 | J ( r e iθ 1 , r e iθ 2 ) | 2 dθ 1 dθ 2 < ∞ sup 0 <r< 1 T × T where J is the complex Jacobian of the symmetrization map π and dθ i is the normalized Lebesgue measure on the unit circle T = { α : | α | = 1 } for all i = 1 , 2 . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

They Hardy space The Hardy space H 2 ( G ) of the symmetrized bidisc is the vector space of those holomorphic functions f on G which satisfy � | f ◦ π ( r e iθ 1 , r e iθ 2 ) | 2 | J ( r e iθ 1 , r e iθ 2 ) | 2 dθ 1 dθ 2 < ∞ sup 0 <r< 1 T × T where J is the complex Jacobian of the symmetrization map π and dθ i is the normalized Lebesgue measure on the unit circle T = { α : | α | = 1 } for all i = 1 , 2 . The norm of f ∈ H 2 ( G ) is defined to be � 1 � � 1 / 2 | f ◦ π ( r e iθ 1 , r e iθ 2 ) | 2 | J ( r e iθ 1 , r e iθ 2 ) | 2 dθ 1 dθ 2 sup 0 <r< 1 , � J � T × T where � J � 2 = � T × T | J ( e iθ 1 , e iθ 2 ) | 2 dθ 1 dθ 2 . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

The L 2 space The distinguished boundary of Γ is b Γ = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | = 1 = | z 2 |} = π ( T × T ) . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

The L 2 space The distinguished boundary of Γ is b Γ = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | = 1 = | z 2 |} = π ( T × T ) . The Hilbert space L 2 ( b Γ) consists of the following functions: � | f ◦ π ( e iθ 1 , e iθ 2 ) | 2 | J ( e iθ 1 , e iθ 2 ) | 2 dθ 1 dθ 2 < ∞} . { f : b Γ → C : T × T iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

The L 2 space The distinguished boundary of Γ is b Γ = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | = 1 = | z 2 |} = π ( T × T ) . The Hilbert space L 2 ( b Γ) consists of the following functions: � | f ◦ π ( e iθ 1 , e iθ 2 ) | 2 | J ( e iθ 1 , e iθ 2 ) | 2 dθ 1 dθ 2 < ∞} . { f : b Γ → C : T × T Theorem The space H 2 ( G ) sits isometrically inside the space L 2 ( b Γ) . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Toeplitz operators Let L ∞ ( b Γ) be the vectors space consisting of { ϕ : b Γ → C : ∃ M > 0 , such that | ϕ ( s, p ) | ≤ M a.e. in b Γ } . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Toeplitz operators Let L ∞ ( b Γ) be the vectors space consisting of { ϕ : b Γ → C : ∃ M > 0 , such that | ϕ ( s, p ) | ≤ M a.e. in b Γ } . For a function ϕ in L ∞ ( b Γ) , let M ϕ be the operator on L 2 ( b Γ) defined by M ϕ f ( s, p ) = ϕ ( s, p ) f ( s, p ) , for all f in L 2 ( b Γ) . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Toeplitz operators Let L ∞ ( b Γ) be the vectors space consisting of { ϕ : b Γ → C : ∃ M > 0 , such that | ϕ ( s, p ) | ≤ M a.e. in b Γ } . For a function ϕ in L ∞ ( b Γ) , let M ϕ be the operator on L 2 ( b Γ) defined by M ϕ f ( s, p ) = ϕ ( s, p ) f ( s, p ) , for all f in L 2 ( b Γ) . Definition For a function ϕ in L ∞ ( b Γ) , the Toeplitz operator with symbol ϕ , denoted by T ϕ , is defined by T ϕ f = PrM ϕ f , for all f in H 2 ( G ) . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Brown-Halmos relations on disc, polydisc and ball A bounded operator T on H 2 ( D ) is a Toeplitz operator if and only if T ∗ z TT z = T. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Brown-Halmos relations on disc, polydisc and ball A bounded operator T on H 2 ( D ) is a Toeplitz operator if and only if T ∗ z TT z = T. A bounded operator T on H 2 ( D n ) is a Toeplitz operator if and only if T ∗ z i TT z i = T for every 1 ≤ i ≤ n. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Brown-Halmos relations on disc, polydisc and ball A bounded operator T on H 2 ( D ) is a Toeplitz operator if and only if T ∗ z TT z = T. A bounded operator T on H 2 ( D n ) is a Toeplitz operator if and only if T ∗ z i TT z i = T for every 1 ≤ i ≤ n. A bounded operator T on H 2 ( B n ) is a Toeplitz operator if and only if T ∗ z 1 TT z 1 + T ∗ z 2 TT z 2 + · · · + T ∗ z n TT z n = T. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Brown-Halmos relations for the symmetrized bidisc In the symmetrized bidisc, the pair ( T s , T p ) satisfies T ∗ s T p = T s and T ∗ p T p = I. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Brown-Halmos relations for the symmetrized bidisc In the symmetrized bidisc, the pair ( T s , T p ) satisfies T ∗ s T p = T s and T ∗ p T p = I. Theorem A bounded operator T on H 2 ( G ) is a Toeplitz operator if and only if T ∗ s TT p = TT s and T ∗ p TT p = T. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Brown-Halmos relations for the symmetrized bidisc In the symmetrized bidisc, the pair ( T s , T p ) satisfies T ∗ s T p = T s and T ∗ p T p = I. Theorem A bounded operator T on H 2 ( G ) is a Toeplitz operator if and only if T ∗ s TT p = TT s and T ∗ p TT p = T. Corollary If T commutes with both T s and T p , then T is a Toeplitz operator. iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Analytic Toeplitz operators and their characterizations Definition A Toeplitz operator with symbol ϕ is called an analytic Toeplitz operator if ϕ is in H ∞ ( G ) . iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Analytic Toeplitz operators and their characterizations Definition A Toeplitz operator with symbol ϕ is called an analytic Toeplitz operator if ϕ is in H ∞ ( G ) . Theorem For a Toeplitz operator with symbol ϕ the following are equivalent: iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Analytic Toeplitz operators and their characterizations Definition A Toeplitz operator with symbol ϕ is called an analytic Toeplitz operator if ϕ is in H ∞ ( G ) . Theorem For a Toeplitz operator with symbol ϕ the following are equivalent: (i) T ϕ is an analytic Toeplitz operator; iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc

Analytic Toeplitz operators and their characterizations Definition A Toeplitz operator with symbol ϕ is called an analytic Toeplitz operator if ϕ is in H ∞ ( G ) . Theorem For a Toeplitz operator with symbol ϕ the following are equivalent: (i) T ϕ is an analytic Toeplitz operator; (ii) T ϕ commutes with T p ; iisclogo Haripada Sau Indian Institute of Technology Bombay Toeplitz operators – Symmetrized bidisc