Ando dilation and its applications

Transcription

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

2 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.

3 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.

4 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

5 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 ).

6 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

7 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.

8 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}.

9 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.

10 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

11 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?

12 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 <.

13 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?

14 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

15 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.

16 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).

17 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 ).

18 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).

19 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 ).

20 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.

21 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.

22 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.

23 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).

24 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.

25 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.

26 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 ).

27 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.

28 References J. Agler and J. E. McCarthy, Distinguished Varieties, Acta Math. 194 (2005), 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), 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: