Ando dilation and its applications Bata Krishna Das Indian - PowerPoint PPT Presentation

Ando dilation and its applications Bata Krishna Das Indian Institute of Technology Bombay OTOA - 2016 ISI Bangalore, December 20 ( joint work with J. Sarkar and S. Sarkar ) B. K. Das Ando dilation and its applications

Introduction D : Open unit disc. H 2 E ( D ) : E -valued Hardy space over the unit disc. The shift operator on H 2 E ( D ) is denoted by M z . For a contraction T , D T := ( I − TT ∗ ) 1 / 2 is the defect operator and D T := RanD T is the defect space of T . A contraction T on H is pure if T ∗ n → 0 in S.O.T. B. K. Das Ando dilation and its applications

Introduction D : Open unit disc. H 2 E ( D ) : E -valued Hardy space over the unit disc. The shift operator on H 2 E ( D ) is denoted by M z . For a contraction T , D T := ( I − TT ∗ ) 1 / 2 is the defect operator and D T := RanD T is the defect space of T . A contraction T on H is pure if T ∗ n → 0 in S.O.T. Theorem (Nagy-Foias) Let T be a contraction on a Hilbert space H . Then T has a unique minimal unitary dilation. B. K. Das Ando dilation and its applications

Introduction D : Open unit disc. H 2 E ( D ) : E -valued Hardy space over the unit disc. The shift operator on H 2 E ( D ) is denoted by M z . For a contraction T , D T := ( I − TT ∗ ) 1 / 2 is the defect operator and D T := RanD T is the defect space of T . A contraction T on H is pure if T ∗ n → 0 in S.O.T. Theorem (Nagy-Foias) Let T be a contraction on a Hilbert space H . Then T has a unique minimal unitary dilation. • von Neumann inequality: For any polynomial p ∈ C [ z ], � p ( T ) � ≤ sup | p ( z ) | . z ∈ D B. K. Das Ando dilation and its applications

Ando dilation Theorem (T. Ando) Let ( T 1 , T 2 ) be a pair of commuting contractions on H . Then ( T 1 , T 2 ) dilates to a pair of commuting unitaries ( U 1 , U 2 ) . B. K. Das Ando dilation and its applications

Ando dilation Theorem (T. Ando) Let ( T 1 , T 2 ) be a pair of commuting contractions on H . Then ( T 1 , T 2 ) dilates to a pair of commuting unitaries ( U 1 , U 2 ) . • von Neumann inequality: For any polynomial p ∈ C [ z 1 , z 2 ], � p ( T 1 , T 2 ) � ≤ ( z 1 , z 2 ) ∈ D 2 | p ( z 1 , z 2 ) | . sup B. K. Das Ando dilation and its applications

Ando dilation Theorem (T. Ando) Let ( T 1 , T 2 ) be a pair of commuting contractions on H . Then ( T 1 , T 2 ) dilates to a pair of commuting unitaries ( U 1 , U 2 ) . • von Neumann inequality: For any polynomial p ∈ C [ z 1 , z 2 ], � p ( T 1 , T 2 ) � ≤ ( z 1 , z 2 ) ∈ D 2 | p ( z 1 , z 2 ) | . sup Definition A variety V = { ( z 1 , z 2 ) ∈ D 2 : p ( z 1 , z 2 ) = 0 } is a distinguished variety of the bidisc if V ∩ ∂ D 2 = V ∩ ( ∂ D ) 2 . B. K. Das Ando dilation and its applications

Distinguished variety of the bidisc Theorem (Agler and McCarthy) V is a distinguished variety of the bidisc if and only if there is a matrix valued inner function Φ such that V = { ( z 1 , z 2 ) ∈ D 2 : det ( z 1 I − Φ( z 2 )) = 0 } . B. K. Das Ando dilation and its applications

Distinguished variety of the bidisc Theorem (Agler and McCarthy) V is a distinguished variety of the bidisc if and only if there is a matrix valued inner function Φ such that V = { ( z 1 , z 2 ) ∈ D 2 : det ( z 1 I − Φ( z 2 )) = 0 } . A distinguished variety V of the bidisc � ( M z , M Φ ) on some H 2 C m ( D ) with Φ is a matrix valued inner function. B. K. Das Ando dilation and its applications

Distinguished variety of the bidisc Theorem (Agler and McCarthy) V is a distinguished variety of the bidisc if and only if there is a matrix valued inner function Φ such that V = { ( z 1 , z 2 ) ∈ D 2 : det ( z 1 I − Φ( z 2 )) = 0 } . A distinguished variety V of the bidisc � ( M z , M Φ ) on some H 2 C m ( D ) with Φ is a matrix valued inner function. Theorem (Agler and McCarthy) Let ( T 1 , T 2 ) be a pair of commuting strict matrices. Then there is a distinguished variety V of the bidisc such that � p ( T 1 , T 2 ) � ≤ sup | p ( z 1 , z 2 ) | ( p ∈ C [ z 1 , z 2 ]) . ( z 1 , z 2 ) ∈ V B. K. Das Ando dilation and its applications

Question and realization formula Question: What are the commuting pair of contractions ( T 1 , T 2 ) which dilates to a pair of commuting isometries ( M z , M Φ ) on H 2 E ( D ) for some finite dimensional Hilbert space E ? B. K. Das Ando dilation and its applications

Question and realization formula Question: What are the commuting pair of contractions ( T 1 , T 2 ) which dilates to a pair of commuting isometries ( M z , M Φ ) on H 2 E ( D ) for some finite dimensional Hilbert space E ? T 1 has to be a pure contraction with dim D T 1 < ∞ . B. K. Das Ando dilation and its applications

Question and realization formula Question: What are the commuting pair of contractions ( T 1 , T 2 ) which dilates to a pair of commuting isometries ( M z , M Φ ) on H 2 E ( D ) for some finite dimensional Hilbert space E ? T 1 has to be a pure contraction with dim D T 1 < ∞ . Is that all we need? B. K. Das Ando dilation and its applications

Question and realization formula Question: What are the commuting pair of contractions ( T 1 , T 2 ) which dilates to a pair of commuting isometries ( M z , M Φ ) on H 2 E ( D ) for some finite dimensional Hilbert space E ? T 1 has to be a pure contraction with dim D T 1 < ∞ . Is that all we need? Φ is an operator valued multiplier of H 2 E ( D ) if and only if � A � B there exist a Hilbert space H and an isometry U = in C D B ( E ⊕ H ) such that Φ( z ) = A + zB ( I − zD ) − 1 C for all z ∈ D . B. K. Das Ando dilation and its applications

Ando type dilation Let ( T 1 , T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D T i < ∞ for i = 1 , 2. B. K. Das Ando dilation and its applications

Ando type dilation Let ( T 1 , T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D T i < ∞ for i = 1 , 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T 1 ( D ). B. K. Das Ando dilation and its applications

Ando type dilation Let ( T 1 , T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D T i < ∞ for i = 1 , 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T 1 ( D ). Consider the operator equality ( I − T 1 T ∗ 1 )+ T 1 ( I − T 2 T ∗ 2 ) T ∗ 1 = T 2 ( I − T 1 T ∗ 1 ) T ∗ 2 +( I − T 2 T ∗ 2 ). B. K. Das Ando dilation and its applications

Ando type dilation Let ( T 1 , T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D T i < ∞ for i = 1 , 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T 1 ( D ). Consider the operator equality ( I − T 1 T ∗ 1 )+ T 1 ( I − T 2 T ∗ 2 ) T ∗ 1 = T 2 ( I − T 1 T ∗ 1 ) T ∗ 2 +( I − T 2 T ∗ 2 ). U : { (D T 1 h , D T 2 h ) : h ∈ H } → { (D T 1 T ∗ 2 h , D T 2 h ) : h ∈ H } defines an isometry defined by (D T 1 h , D T 2 T ∗ 1 h ) �→ (D T 1 T ∗ 2 h , D T 2 h ) ( h ∈ H ) . B. K. Das Ando dilation and its applications

Ando type dilation Let ( T 1 , T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D T i < ∞ for i = 1 , 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T 1 ( D ). Consider the operator equality ( I − T 1 T ∗ 1 )+ T 1 ( I − T 2 T ∗ 2 ) T ∗ 1 = T 2 ( I − T 1 T ∗ 1 ) T ∗ 2 +( I − T 2 T ∗ 2 ). U : { (D T 1 h , D T 2 h ) : h ∈ H } → { (D T 1 T ∗ 2 h , D T 2 h ) : h ∈ H } defines an isometry defined by (D T 1 h , D T 2 T ∗ 1 h ) �→ (D T 1 T ∗ 2 h , D T 2 h ) ( h ∈ H ) . Extend U to a unitary in B ( D T 1 ⊕ D T 2 ). B. K. Das Ando dilation and its applications

Ando type dilation Let ( T 1 , T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D T i < ∞ for i = 1 , 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T 1 ( D ). Consider the operator equality ( I − T 1 T ∗ 1 )+ T 1 ( I − T 2 T ∗ 2 ) T ∗ 1 = T 2 ( I − T 1 T ∗ 1 ) T ∗ 2 +( I − T 2 T ∗ 2 ). U : { (D T 1 h , D T 2 h ) : h ∈ H } → { (D T 1 T ∗ 2 h , D T 2 h ) : h ∈ H } defines an isometry defined by (D T 1 h , D T 2 T ∗ 1 h ) �→ (D T 1 T ∗ 2 h , D T 2 h ) ( h ∈ H ) . Extend U to a unitary in B ( D T 1 ⊕ D T 2 ). Let Φ ∈ H ∞ B ( D T 1 ) ( D ) be the matrix valued inner function corresponding to U ∗ . Then M ∗ Φ is the co-isometric extension of T ∗ 2 . B. K. Das Ando dilation and its applications

Dilation and sharp von Neumann inequality Theorem Let ( T 1 , T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D T i < ∞ , i = 1 , 2 . Then ( T 1 , T 2 ) dilates to ( M z , M Φ ) on H 2 D T 1 ( D ) . Therefore, there exists a variety V ⊂ D 2 such that � p ( T 1 , T 2 ) � ≤ sup | p ( z 1 , z 2 ) | ( p ∈ C [ z 1 , z 2 ]) . ( z 1 , z 2 ) ∈ V If, in addition, T 2 is pure then V can be taken to be a distinguished variety of the bidisc. B. K. Das Ando dilation and its applications

Berger-Coburn-Lebow representation A pure pair of commuting isometries is a pair of commuting isometries ( V 1 , V 1 ) with V 1 V 2 is pure. B. K. Das Ando dilation and its applications