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)
Introduction D : Open unit disc. HE 2 (D) : E-valued Hardy space over the unit disc. The shift operator on HE 2(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.
Introduction D : Open unit disc. HE 2 (D) : E-valued Hardy space over the unit disc. The shift operator on HE 2(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.
Introduction D : Open unit disc. HE 2 (D) : E-valued Hardy space over the unit disc. The shift operator on HE 2(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
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 ).
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 ) sup p(z 1, z 2 ). (z 1,z 2 ) D 2
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 ) sup p(z 1, z 2 ). (z 1,z 2 ) D 2 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.
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}.
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.
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
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 HE 2 (D) for some finite dimensional Hilbert space E?
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 HE 2 (D) for some finite dimensional Hilbert space E? T 1 has to be a pure contraction with dim D T1 <.
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 HE 2 (D) for some finite dimensional Hilbert space E? T 1 has to be a pure contraction with dim D T1 <. Is that all we need?
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 HE 2 (D) for some finite dimensional Hilbert space E? T 1 has to be a pure contraction with dim D T1 <. Is that all we need? Φ is an operator valued multiplier of HE 2 (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 for all z D. Φ(z) = A + zb(i zd) 1 C
Ando type dilation Let (T 1, T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D Ti < for i = 1, 2.
Ando type dilation Let (T 1, T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D Ti < for i = 1, 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T1 (D).
Ando type dilation Let (T 1, T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D Ti < for i = 1, 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T1 (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 2T 2 ).
Ando type dilation Let (T 1, T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D Ti < for i = 1, 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T1 (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 2T 2 ). U : {(D T1 h, D T2 h) : h H} {(D T1 T 2 h, D T 2 h) : h H} defines an isometry defined by (D T1 h, D T2 T 1 h) (D T1 T 2 h, D T2 h) (h H).
Ando type dilation Let (T 1, T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D Ti < for i = 1, 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T1 (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 2T 2 ). U : {(D T1 h, D T2 h) : h H} {(D T1 T 2 h, D T 2 h) : h H} defines an isometry defined by (D T1 h, D T2 T 1 h) (D T1 T 2 h, D T2 h) (h H). Extend U to a unitary in B(D T1 D T2 ).
Ando type dilation Let (T 1, T 2 ) be a pair of commuting contractions on H with T 1 is pure and dim D Ti < for i = 1, 2. Let M z be the minimal isometric dilation of T 1 on H 2 D T1 (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 2T 2 ). U : {(D T1 h, D T2 h) : h H} {(D T1 T 2 h, D T 2 h) : h H} defines an isometry defined by (D T1 h, D T2 T 1 h) (D T1 T 2 h, D T2 h) (h H). Extend U to a unitary in B(D T1 D T2 ). Let Φ HB(D T1 )(D) be the matrix valued inner function corresponding to U. Then MΦ is the co-isometric extension of T2.
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 Ti <, i = 1, 2. Then (T 1, T 2 ) dilates to (M z, M Φ ) on HD 2 T1 (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.
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.
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. Theorem (B-C-L) A pure pair of commuting isometries (V 1, V 2 ) is unitary equivalent to a commuting pair of isometries (M Φ, M Ψ ) on HE 2 for some Hilbert space E with Φ(z) = (zp + P)U and Ψ(z) = U(zP + P ) where U is a unitary in B(E) and P is a projection in B(E).
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. Theorem (B-C-L) A pure pair of commuting isometries (V 1, V 2 ) is unitary equivalent to a commuting pair of isometries (M Φ, M Ψ ) on HE 2 for some Hilbert space E with Φ(z) = (zp + P)U and Ψ(z) = U(zP + P ) where U is a unitary in B(E) and P is a projection in B(E). H 2 E (D) is the model space of the pure isometry V 1V 2. Pure pair of commuting isometry (V 1, V 2 ) (E, U, P). M z = M Φ M Ψ.
Dilation and factorization of pure pair of contractions A pure pair of commuting contractions is a pair commuting contractions (T 1, T 2 ) with T 1 T 2 is pure.
Dilation and factorization of pure pair of contractions A pure pair of commuting contractions is a pair commuting contractions (T 1, T 2 ) with T 1 T 2 is pure. Theorem A pure pair of commuting contractions (T 1, T 2 ) dilates to a pure pair of commuting isometries corresponding to a triple (D T1 D T2, U, P) where U is a unitary and P is a projection in B(D T1 D T2 ).
Dilation and factorization of pure pair of contractions A pure pair of commuting contractions is a pair commuting contractions (T 1, T 2 ) with T 1 T 2 is pure. Theorem A pure pair of commuting contractions (T 1, T 2 ) dilates to a pure pair of commuting isometries corresponding to a triple (D T1 D T2, U, P) where U is a unitary and P is a projection in B(D T1 D T2 ). Theorem Let T be a pure contraction on H and let T = P Q M z Q be the Sz.-Nagy and Foias representation of T. TFAE (i) T = T 1 T 2 for some commuting contractions T 1 and T 2 on H. (ii) There exist B(D T )-valued polynomial φ and ψ of degree 1 such that Q is a joint (Mφ, M ψ )- invariant subspace, P Q M z Q = P Q M φψ Q = P Q M ψφ Q.
References J. Agler and J. E. McCarthy, Distinguished Varieties, Acta Math. 194 (2005), 133-153. C.A. Berger, L.A. Coburn and A. Lebow, Representation and index theory for C -algebras generated by commuting isometries, J. Funct. Anal. 27 (1978), 51-99. B. K. Das and J. Sarkar, Ando dilations, von Neumann inequality, and distinguished varieties, J. Funct. Anal. (to appear). B.K. Das, J. Sarkar and S. Sarkar, Factorizations of contractions, arxiv:1607.05815.