Characterization of invariant subspaces in the polydisc

Transcription

1 Characterization of invariant subspaces in the polydisc Amit Maji IIIT Guwahati (Joint work with Aneesh M., Jaydeb Sarkar & Sankar T. R.) Recent Advances in Operator Theory and Operator Algebras Indian Statistical Institute, Bangalore December 13-19, 2018 Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 1 / 25

2 Characterization of invariant subspaces... Aim To give a complete characterization of (joint) invariant subspaces for (M z1,..., M zn ) on the Hardy space H 2 (D n ) over the unit polydisc D n in C n, n > 1. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 2 / 25

3 Characterization of invariant subspaces... Aim To give a complete characterization of (joint) invariant subspaces for (M z1,..., M zn ) on the Hardy space H 2 (D n ) over the unit polydisc D n in C n, n > 1. To discuss about a complete set of unitary invariants for invariant subspaces as well as unitarily equivalent invariant subspaces. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 2 / 25

4 Characterization of invariant subspaces... Aim To give a complete characterization of (joint) invariant subspaces for (M z1,..., M zn ) on the Hardy space H 2 (D n ) over the unit polydisc D n in C n, n > 1. To discuss about a complete set of unitary invariants for invariant subspaces as well as unitarily equivalent invariant subspaces. To classify a large class of n-tuples of commuting isometries. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 2 / 25

5 Characterization of invariant subspaces... Notation Unit polydisc D n = {(z 1,..., z n ) C n : z i < 1, i = 1,..., n}. Hardy space H 2 (D) = {f = n=0 a nz n : n=0 a n 2 < }. Vector-valued Hardy space HE(D) 2 = {f = a n z n : a n E and n=0 a n 2 E < }, n=0 where E is some Hilbert space. M z denote the multiplication operator on HE 2 (D) defined by (M z f )(w) = wf (w) (f H 2 E(D), w D). HB(E) (D) denote the space of bounded B(E)-valued holomorphic functions on D. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 3 / 25

6 Invariant subspaces: Motivation A closed subspace M of a Hilbert space H is said to be invariant subspace under T B(H) if T (M) M. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 4 / 25

7 Invariant subspaces: Motivation A closed subspace M of a Hilbert space H is said to be invariant subspace under T B(H) if T (M) M. Given an invariant subspace M of T B(H), one can view T as an operator matrix with respect to the decomposition H = M M [ ] T M. 0 Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 4 / 25

8 Invariant subspaces: Motivation A closed subspace M of a Hilbert space H is said to be invariant subspace under T B(H) if T (M) M. Given an invariant subspace M of T B(H), one can view T as an operator matrix with respect to the decomposition H = M M [ ] T M. 0 One of the most famous open problems in operator theory and function theory is the so-called invariant subspace problem: Does every bounded linear operator have a non-trivial closed invariant subspace? Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 4 / 25

9 Invariant subspaces: Motivation The celebrated Beurling theorem (1949) says that a non-zero closed subspace S of H 2 (D) is invariant for M z if and only if there exists an inner function θ H (D) such that S = θh 2 (D). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 5 / 25

10 Invariant subspaces: Motivation The celebrated Beurling theorem (1949) says that a non-zero closed subspace S of H 2 (D) is invariant for M z if and only if there exists an inner function θ H (D) such that S = θh 2 (D). One may now ask whether an analogous characterization holds for (joint) invariant subspaces for (M z1,..., M zn ) on H 2 (D n ), n > 1, i.e., Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 5 / 25

11 Invariant subspaces: Motivation The celebrated Beurling theorem (1949) says that a non-zero closed subspace S of H 2 (D) is invariant for M z if and only if there exists an inner function θ H (D) such that S = θh 2 (D). One may now ask whether an analogous characterization holds for (joint) invariant subspaces for (M z1,..., M zn ) on H 2 (D n ), n > 1, i.e., if S is an invariant subspace for (M z1,..., M zn ) on H 2 (D n ), then S = ψh 2 (D n ), where ψ H (D n ) is an inner function. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 5 / 25

12 Invariant subspaces: Motivation The celebrated Beurling theorem (1949) says that a non-zero closed subspace S of H 2 (D) is invariant for M z if and only if there exists an inner function θ H (D) such that S = θh 2 (D). One may now ask whether an analogous characterization holds for (joint) invariant subspaces for (M z1,..., M zn ) on H 2 (D n ), n > 1, i.e., if S is an invariant subspace for (M z1,..., M zn ) on H 2 (D n ), then S = ψh 2 (D n ), where ψ H (D n ) is an inner function. Rudin s pathological examples (Rudin (1969)): There exist invariant subspaces S 1 and S 2 for (M z1, M z2 ) on H 2 (D 2 ) such that (1) S 1 is not finitely generated, and (2) S 2 H (D 2 ) = {0}. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 5 / 25

13 Invariant subspaces: Motivation To understand the structure of a large class of operators we need to understand the structure of shift invariant subspaces for the vector-valued Hardy space over the unit disc. Theorem (Beurling-Lax-Halmos) A non-zero closed subspace S of HE 2(D) is invariant for M z if and only if there exists a closed subspace F E and an inner function Θ HB(F,E) (D) such that S = ΘH 2 F(D). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 6 / 25

14 Invariant subspaces: Motivation To understand the structure of a large class of operators we need to understand the structure of shift invariant subspaces for the vector-valued Hardy space over the unit disc. Theorem (Beurling-Lax-Halmos) A non-zero closed subspace S of HE 2(D) is invariant for M z if and only if there exists a closed subspace F E and an inner function Θ HB(F,E) (D) such that Idea S = ΘH 2 F(D). Identify Hardy space over polydisc H 2 (D n+1 ) to the H 2 (D n )-valued Hardy space over disc H 2 H 2 (D n ) (D). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 6 / 25

15 Invariant subspaces: Motivation To understand the structure of a large class of operators we need to understand the structure of shift invariant subspaces for the vector-valued Hardy space over the unit disc. Theorem (Beurling-Lax-Halmos) A non-zero closed subspace S of HE 2(D) is invariant for M z if and only if there exists a closed subspace F E and an inner function Θ HB(F,E) (D) such that Idea S = ΘH 2 F(D). Identify Hardy space over polydisc H 2 (D n+1 ) to the H 2 (D n )-valued Hardy space over disc H 2 H 2 (D n ) (D). Represent (M z1, M z2,..., M zn+1 ) on H 2 (D n+1 ) to (M z, M κ1,..., M κn ) on HH 2 2 (D n ) (D), where κ i HB(H 2 (D n ))(D), i = 1,..., n, is a constant as well as simple and explicit B(H 2 (D n ))-valued analytic function. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 6 / 25

16 Basic definitions A closed subspace S H 2 E (Dn ) is called a (joint) invariant subspace for (M z1,..., M zn ) on H 2 E (Dn ) if z i S S, for all i = 1,..., n. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 7 / 25

17 Basic definitions A closed subspace S H 2 E (Dn ) is called a (joint) invariant subspace for (M z1,..., M zn ) on H 2 E (Dn ) if z i S S, for all i = 1,..., n. An isometry V on H is called a pure isometry (or shift) if V m 0 in SOT. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 7 / 25

18 Basic definitions A closed subspace S H 2 E (Dn ) is called a (joint) invariant subspace for (M z1,..., M zn ) on H 2 E (Dn ) if z i S S, for all i = 1,..., n. An isometry V on H is called a pure isometry (or shift) if V m 0 in SOT. Let V be a pure isometry on H. Then where W = ker V = H V H. H = V m W, m=0 Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 7 / 25

19 Basic definitions A closed subspace S H 2 E (Dn ) is called a (joint) invariant subspace for (M z1,..., M zn ) on H 2 E (Dn ) if z i S S, for all i = 1,..., n. An isometry V on H is called a pure isometry (or shift) if V m 0 in SOT. Let V be a pure isometry on H. Then H = V m W, m=0 where W = ker V = H V H. The natural map Π V : H HW 2 (D) defined by Π V (V m η) = z m η, for all m 0 and η W, is a unitary operator and Π V V = M z Π V. We call Π V the Wold-von Neumann decomposition of the shift V. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 7 / 25

20 Commutator of shift Theorem 1 (Maji, Sarkar, & Sankar 18) Let V be a pure isometry on H, and let C be a bounded operator on H. Let Pi V be the Wold-von Neumann decomposition of V. Let W = Ker(V ) and assume that M = Π V CΠ V. Then CV = VC, if and only if where M = M Θ, Θ(z) = P W (I H zv ) 1 C W (z D). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 8 / 25

21 Commutator of shift Theorem 1 (Maji, Sarkar, & Sankar 18) Let V be a pure isometry on H, and let C be a bounded operator on H. Let Pi V be the Wold-von Neumann decomposition of V. Let W = Ker(V ) and assume that M = Π V CΠ V. Then CV = VC, if and only if where M = M Θ, Θ(z) = P W (I H zv ) 1 C W (z D). Remark In addition if CV = V C, then Θ(z) = C W = Θ(0) (z D), as C(I VV ) = (I VV )C and V m W = 0 for all m 1. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 8 / 25

22 Preparation for main result For the sake of simplicity we discuss firstly for n = 2, i.e., vector-valued Hardy space over the bidisc H 2 E (D2 ). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 9 / 25

23 Preparation for main result For the sake of simplicity we discuss firstly for n = 2, i.e., vector-valued Hardy space over the bidisc H 2 E (D2 ). We now identify HE 2(D2 ) with H 2 HE 2 (D)(D) by the canonical unitaries H 2 E(D 2 ) Û H 2 (D) H 2 E(D) Ũ H 2 H 2 E (D)(D) where and Û(z k1 1 zk2 2 η) = zk1 (z k2 1 η), ( k 1, k 2 0, η E ) Ũ(z k ζ) = z k ζ, ( k 0, ζ H 2 E(D) ). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 9 / 25

24 Preparation for main result For the sake of simplicity we discuss firstly for n = 2, i.e., vector-valued Hardy space over the bidisc H 2 E (D2 ). We now identify HE 2(D2 ) with H 2 HE 2 (D)(D) by the canonical unitaries H 2 E(D 2 ) Û H 2 (D) H 2 E(D) Ũ H 2 H 2 E (D)(D) where and Û(z k1 1 zk2 2 η) = zk1 (z k2 1 η), ( k 1, k 2 0, η E ) Ũ(z k ζ) = z k ζ, ( k 0, ζ H 2 E(D) ). Set U = ŨÛ. Then it follows that U : H2 E (D2 ) H 2 HE 2 (D)(D) is a unitary operator. Since ÛM z1 = (M z I H 2 E (D) ))Û and ÛM z2 = (I H 2 (D) M z1 )Û, we have UM z1 = M z U, and UM z2 = M κ1 U, where κ 1 (w) = M z1 for w D. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 9 / 25

25 Preparation for main result Let E be a Hilbert space and let E n = H 2 (D n ) E for n 1. Let κ i H B(E n) (D) denote the B(E n )-valued constant function on D defined by κ i (w) = M zi B(E n ), for all w D, and let M κi denote the multiplication operator on H 2 E n (D) defined by M κi f = κ i f, for all f H 2 E n (D) and i = 1,..., n. Then Theorem 2 (Maji, Aneesh, Sarkar, & Sankar 18) (i) (M z1, M z2..., M zn+1 ) on H 2 E (Dn+1 ) and (M z, M κ1,..., M κn ) on H 2 E n (D) are unitarily equivalent. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 10 / 25

26 H 2 H 2 E (D)(D). Set V = M z S and V 1 = M κ1 S. Preparation for main result Let S H 2 H 2 E (D)(D) be a closed invariant subspace for (M z, M κ1 ) on Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 11 / 25

27 H 2 H 2 E (D)(D). Set V = M z S and V 1 = M κ1 S. Preparation for main result Let S H 2 H 2 E (D)(D) be a closed invariant subspace for (M z, M κ1 ) on Let Π V : S HW 2 (D) be the Wold-von Neumann decomposition of V on S. Then Π V V Π V = M z and Π V V 1 Π V = M Φ1, where for all w D, Φ 1 H B(W) (D). Φ 1 (w) = P W (I S wv ) 1 V 1 W, Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 11 / 25

28 H 2 H 2 E (D)(D). Set V = M z S and V 1 = M κ1 S. Preparation for main result Let S H 2 H 2 E (D)(D) be a closed invariant subspace for (M z, M κ1 ) on Let Π V : S HW 2 (D) be the Wold-von Neumann decomposition of V on S. Then Π V V Π V = M z and Π V V 1 Π V = M Φ1, where for all w D, Φ 1 H B(W) (D). Φ 1 (w) = P W (I S wv ) 1 V 1 W, Let i S denote the inclusion map i S : S H 2 E n (D). Then HW(D) 2 Π V S i S H 2 HE 2 (D)(D) i.e., Π S = i S Π V : H2 W (D) H2 HE 2 (D)(D) is an isometry and ran Π S = S. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 11 / 25

29 Main result We have invariant subspace result on vector-valued Hardy space over polydisc setting: Theorem 2 (Maji, Aneesh, Sarkar, & Sankar 18) (ii) Let E be a Hilbert space, S HE 2 n (D) be a closed subspace, and let W = S zs. Then S is invariant for (M z, M κ1,..., M κn ) if and only if (M Φ1,..., M Φn ) is an n-tuple of commuting shifts on HW 2 (D) and there exists an inner function Θ HB(W,E n) (D) such that and where for all w D and i = 1,..., n. S = ΘH 2 W(D), κ i Θ = ΘΦ i, Φ i (w) = P W (I S wp S M z ) 1 M κi W, Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 12 / 25

30 Remarks One obvious necessary condition for a closed subspace S of H 2 E n (D) to be (joint) invariant for (M z, M κ1,..., M κn ) is that S is invariant for M z, and, consequently S = ΘH 2 W(D), where W = S zs and Θ HB(W,E n) (D) is the Beurling, Lax and Halmos inner function. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 13 / 25

31 Remarks One obvious necessary condition for a closed subspace S of H 2 E n (D) to be (joint) invariant for (M z, M κ1,..., M κn ) is that S is invariant for M z, and, consequently S = ΘH 2 W(D), where W = S zs and Θ HB(W,E n) (D) is the Beurling, Lax and Halmos inner function. Again κ i S S, = κ i Θ = ΘΓ i, for some Γ i B(HW 2 (D)), i = 1,..., n (by Douglas s range inclusion theorem ). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 13 / 25

32 Remarks One obvious necessary condition for a closed subspace S of H 2 E n (D) to be (joint) invariant for (M z, M κ1,..., M κn ) is that S is invariant for M z, and, consequently S = ΘH 2 W(D), where W = S zs and Θ HB(W,E n) (D) is the Beurling, Lax and Halmos inner function. Again κ i S S, = κ i Θ = ΘΓ i, for some Γ i B(HW 2 (D)), i = 1,..., n (by Douglas s range inclusion theorem ). In the above theorem, we prove that Γ i is explicit, that is Γ i = Φ i H B(W) (D), for all i = 1,..., n, and (Γ 1,..., Γ n ) is an n-tuple of commuting shifts on H 2 W (D). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 13 / 25

33 Remarks Uniqueness Let S be an invariant subspace for (M z, M κ1,..., M κn ) on H 2 E n (D). Then S = ΘH 2 W (D) and κ i Θ = ΘΦ i (i = 1,..., n), from the above Theorem. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 14 / 25

34 Remarks Uniqueness Let S be an invariant subspace for (M z, M κ1,..., M κn ) on H 2 E n (D). Then S = ΘH 2 W (D) and κ i Θ = ΘΦ i (i = 1,..., n), from the above Theorem. Now suppose S = ΘH 2 W (D) and κ i Θ = Θ Φ i for some Hilbert space W, inner function Θ H B( W) (D) and shift M Φ i on H (D), i = 1,..., n. 2 W Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 14 / 25

35 Remarks Uniqueness Let S be an invariant subspace for (M z, M κ1,..., M κn ) on H 2 E n (D). Then S = ΘH 2 W (D) and κ i Θ = ΘΦ i (i = 1,..., n), from the above Theorem. Now suppose S = ΘH 2 W (D) and κ i Θ = Θ Φ i for some Hilbert space W, inner function Θ H B( W) (D) and shift M Φ i on H (D), i = 1,..., n. 2 W Then there exists a unitary operator τ : W W such that Θ = Θτ, and τφ i = Φ i τ, for all i = 1,..., n. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 14 / 25

36 Nested invariant subspaces Theorem (Maji, Aneesh, Sarkar, & Sankar 18) Let E be a Hilbert space, and let S 1 = Θ 1 H 2 W 1 (D) and S 2 = Θ 2 H 2 W 2 (D) be two invariant subspaces for (M z, M κ1,..., M κn ) on H 2 E n (D) and W j = S j zs j for j = 1, 2. Let Φ j,i (w) = P Wj (I Sj wp Sj M z ) 1 M κi Wj, for all w D, j = 1, 2, and i = 1,..., n. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 15 / 25

37 Nested invariant subspaces Theorem (Maji, Aneesh, Sarkar, & Sankar 18) Let E be a Hilbert space, and let S 1 = Θ 1 H 2 W 1 (D) and S 2 = Θ 2 H 2 W 2 (D) be two invariant subspaces for (M z, M κ1,..., M κn ) on H 2 E n (D) and W j = S j zs j for j = 1, 2. Let Φ j,i (w) = P Wj (I Sj wp Sj M z ) 1 M κi Wj, for all w D, j = 1, 2, and i = 1,..., n. Then S 1 S 2 if and only if there exists an inner multiplier Ψ H B(W 1,W 2) (D) such that Θ 1 = Θ 2 Ψ and ΨΦ 1,i = Φ 2,i Ψ for all i = 1,..., n. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 15 / 25

38 Unitarily equivalent invariant subspaces Definition Let S and S be invariant subspaces for the (n + 1)-tuples of multiplication operators (M z, M κ1,..., M κn ) on H 2 E n (D) and H 2 Ẽ n (D), respectively. We say that S and S are unitarily equivalent, and write S = S, if there is a unitary map U : S S such that for all i = 1,..., n. UM z S = M z S U and UM κ i S = M κi S U, Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 16 / 25

39 Unitarily equivalent invariant subspaces Definition Let S and S be invariant subspaces for the (n + 1)-tuples of multiplication operators (M z, M κ1,..., M κn ) on H 2 E n (D) and H 2 Ẽ n (D), respectively. We say that S and S are unitarily equivalent, and write S = S, if there is a unitary map U : S S such that for all i = 1,..., n. Identification UM z S = M z S U and UM κ i S = M κi S U, There exists a unitary operator U E : H 2 E (Dn+1 ) H 2 E n (D) such that and U E M z1 = M z U E, U E M zi+1 = M κi U E, for all i = 1,..., n. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 16 / 25

40 Intertwining maps Let F be another Hilbert space, and let X : H 2 E (Dn+1 ) H 2 F (Dn+1 ) be a bounded linear operator such that XM zi = M zi X, (0.1) for all i = 1,..., n + 1. Set X n = U F XU E. Then X n : H 2 E n (D) H 2 F n (D) is bounded and for all i = 1,..., n. Definition X n M z = M z X n and X n M κi = M κi X n, (0.2) Any map satisfying (0.2) will be referred to module maps. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 17 / 25

41 Unitarily equivalent invariant subspaces Theorem (Maji, Aneesh, Sarkar, & Sankar 18) Let S H 2 F n (D) be a closed invariant subspace for (M z, M κ1,..., M κn ) on H 2 F n (D). Then S = H 2 E n (D) if and only if there exists an isometric module map X n : H 2 E n (D) H 2 F n (D) such that S = X n H 2 E n (D). Moreover, in this case dim E dim F. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 18 / 25

42 Unitarily equivalent invariant subspaces Theorem (Maji, Aneesh, Sarkar, & Sankar 18) Let S H 2 F n (D) be a closed invariant subspace for (M z, M κ1,..., M κn ) on H 2 F n (D). Then S = H 2 E n (D) if and only if there exists an isometric module map X n : H 2 E n (D) H 2 F n (D) such that S = X n H 2 E n (D). Moreover, in this case dim E dim F. Corollary Let S H 2 H n (D) be a closed invariant subspace for (M z, M κ1,..., M κn ) on H 2 H n (D). Then S = H 2 H n (D) if and only if there exists an isometric module map X n : H 2 H n (D) H 2 H n (D) such that S = X n (H 2 H n (D)). The above corollary was first observed by Agrawal, Clark and Douglas (1986). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 18 / 25

43 A complete set of unitary invariants Definition Let E and Ẽ be Hilbert spaces, and let {Ψ 1,..., Ψ n } HB(E) (D) and { Ψ 1,..., Ψ n } H (D). We say that {Ψ B(Ẽ) 1,..., Ψ n } and { Ψ 1,..., Ψ n } coincide if there exists a unitary operator τ : E Ẽ such that for all z D and i = 1,..., n. τψ i (z) = Ψ i (z)τ, Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 19 / 25

44 A complete set of unitary invariants Definition Let E and Ẽ be Hilbert spaces, and let {Ψ 1,..., Ψ n } HB(E) (D) and { Ψ 1,..., Ψ n } H (D). We say that {Ψ B(Ẽ) 1,..., Ψ n } and { Ψ 1,..., Ψ n } coincide if there exists a unitary operator τ : E Ẽ such that for all z D and i = 1,..., n. τψ i (z) = Ψ i (z)τ, Theorem (Maji, Aneesh, Sarkar, & Sankar 18) Let E and Ẽ be Hilbert spaces. Let S HE 2 n (D) and S H 2 (D) be invariant Ẽ n subspaces for (M z, M κ1,..., M κn ) on HE 2 n (D) and H 2 (D), respectively. Then Ẽ n S = S if and only if {Φ 1,..., Φ n } and { Φ 1,..., Φ n } coincide. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 19 / 25

45 Representations of model isometries Question Given a Hilbert space E, characterize (n + 1)-tuples of commuting shifts on Hilbert spaces that are unitarily equivalent to (M z, M κ1,..., M κn ) on H 2 E n (D). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 20 / 25

46 Representations of model isometries Question Given a Hilbert space E, characterize (n + 1)-tuples of commuting shifts on Hilbert spaces that are unitarily equivalent to (M z, M κ1,..., M κn ) on H 2 E n (D). Answer Answer to this question is related to (numerical invariant) the rank of an operator associated with the Szegö kernel on D n+1. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 20 / 25

47 Representations of model isometries Definition The defect operator corresponding to an m-tuple of commuting contractions (T 1,..., T m ) on a Hilbert space H is defined (see, Guo & Yang (2004)) as S 1 m (T 1,..., T m ) = 0 k m ( 1) k T k1 1 T km where k = k 1 + k k m, 0 k i 1, i = 1,..., m. m T k1 m, 1 T km Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 21 / 25

48 Representations of model isometries Definition The defect operator corresponding to an m-tuple of commuting contractions (T 1,..., T m ) on a Hilbert space H is defined (see, Guo & Yang (2004)) as S 1 m (T 1,..., T m ) = 0 k m ( 1) k T k1 1 T km where k = k 1 + k k m, 0 k i 1, i = 1,..., m. m T k1 m, 1 T km Definition We say that (T 1,..., T m ) is of rank p (p N { }) if rank [S 1 m (T 1,..., T m )] = p, and we write rank (T 1,..., T m ) = p. Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 21 / 25

49 Representations of model isometries Let (V, V 1..., V n ) be an (n + 1)-tuple of doubly commuting shifts on H. Then Sarkar (2014) proved that (V, V 1,..., V n ) on H and (M z1,..., M zn+1 ) on HD 2 (Dn+1 ) are unitarily equivalent, where ( n ) D = ran S 1 n+1 (V, V 1,..., V n ) = ker Vi ker V. i=1 Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 22 / 25

50 Representations of model isometries Let (V, V 1..., V n ) be an (n + 1)-tuple of doubly commuting shifts on H. Then Sarkar (2014) proved that (V, V 1,..., V n ) on H and (M z1,..., M zn+1 ) on HD 2 (Dn+1 ) are unitarily equivalent, where ( n ) D = ran S 1 n+1 (V, V 1,..., V n ) = ker Vi ker V. i=1 Theorem (Maji, Aneesh, Sarkar, & Sankar 18) Let (V, V 1,..., V n ) be an (n + 1)-tuple of doubly commuting shifts on some Hilbert space H. Let W = H V H, and let Ψ i (z) = V i W (i = 1,..., n), for all z D. Then (V, V 1,..., V n ) on H, (M z, M Ψ1,..., M Ψn ) on HW 2 (D), and (M z, M κ1,..., M κn ) on HE 2 n (D) are unitarily equivalent, where E is a Hilbert space and dim E = rank (V, V 1,..., V n ). Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 22 / 25

51 References O. P. Agrawal, D. N. Clark and R. G. Douglas, Invariant subspaces in the polydisk, Pacific J. Math., 121 (1986), H. Bercovici, R.G. Douglas and C. Foias, Canonical models for bi-isometries, A panorama of modern operator theory and related topics, , Oper. Theory Adv. Appl., 218, Birkhäuser/Springer Basel AG, Basel, H. Bercovici, R.G. Douglas and C. Foias, On the classification of multi-isometries, Acta Sci. Math. (Szeged), 72 (2006), 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. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space. North-Holland, Amsterdam-London, K. Guo and R. Yang, The core function of submodules over the bidisk, Indiana Univ. Math. J. 53 (2004), Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 23 / 25

52 References A. Maji, A. Mundayadan, J. Sarkar, Sankar T. R, Characterization of invariant subspaces in the polydisc, Journal of Operator Theory, (To appear). A. Maji, J. Sarkar, Sankar T. R, Pairs of commuting isometries - I, Studia Mathematica, (To appear). D. Popovici, A Wold-type decomposition for commuting isometric pairs, Proc. Amer. Math. Soc. 132 (2004) Y. Qin and R. Yang, A note on Rudin s pathological submodule in H 2 (D 2 ), Integral Equations Operator Theory 88 (2017), W. Rudin, Function Theory in Polydiscs, Benjamin, New York, J. Sarkar, Wold decomposition for doubly commuting isometries, Linear Algebra Appl. 445 (2014), H. Wold, A study in the analysis of stationary time series, Stockholm, Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 24 / 25

53 Thank You!