( f(rz) 2 E. E E (D n ) := sup f(z) L(E,E ) <.

Transcription

1 DOUBLY COMMUTING SUBMODULES OF THE HARDY MODULE OVER POLYDISCS JAYDEB SARKAR, AMOL SASANE, AND BRETT D. WICK Abstract. In this note we establish a vector-valued version of Beurling s Theorem (the Lax-Halmos Theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the weak completion problem in H (D n ). 1. Introduction and Statement of Main Results In [B], Beurling described all the invariant subspaces for the operator M z of multiplication by z on the Hilbert space H 2 (D) of the disc. In [L], Peter Lax extended Beurling s result to the (finite-dimensional) vector-valued case (while also considering the Hardy space of the half plane). Lax s vectorial case proof was further extended to infinite-dimensional vector spaces by Halmos, see [NF]. The characterization of M z -invariant subspaces obtained is the following famous result. Theorem 1.1 (Beurling-Lax-Halmos). Let S be a closed nonzero subspace of H 2 E (D). Then S is invariant under multiplication by z if and only if there exists a Hilbert space E and an inner function Θ H E E (D) such that S = ΘH 2 E (D). For n N and E a Hilbert space, HE 2 (D n ) is the set of all E -valued holomorphic functions in the polydisc D n, where D := {z C : z < 1} (with boundary T) such that f H 2 E (Dn ) := sup 0<r<1 ( ) 1/2 f(rz) 2 E dz < +. T n On the other hand, if L(E, E ) denotes the set of all continuous linear transformations from E to E, then HE E (D n ) denotes the set of all L(E, E )- valued holomorphic functions with f H E E (D n ) := sup f(z) L(E,E ) <. z D n 2010 Mathematics Subject Classification. Primary 46J15; Secondary 47A15, 30H05, 47A56. Key words and phrases. Invariant subspace, shift operator, Doubly commuting, Hardy algebra on the polydisc, Completion Problem. Research supported in part by National Science Foundation DMS grants # and #

2 2 J. SARKAR, A. SASANE, AND B. D. WICK An operator-valued Θ H E E (D n ) inner if the pointwise a.e. boundary values are isometries: (Θ(ζ)) Θ(ζ) = I E for almost all ζ T n. A natural question is then to ask what happens in the case of several variables, for example when one considers the Hardy space H 2 E (D n ) of the polydisc D n. It is known that in general, a Beurling-Lax-Halmos type characterization of subspaces of the Hardy Hilbert space is not possible [R]. It is however, easy to see that the Hardy space on the polydisc H 2 E (D n ), when n > 1, satisfies the doubly commuting property, that is, for all 1 i < j n M z i M zj = M zj M z i. We impose this additional assumption to the submodules of H 2 E (D n ) and call that class of submodules as doubly commuting submodules. More precisely: Definition 1.2. A commuting family of bounded linear operators {T 1,..., T n } on some Hilbert space H is said to be doubly commuting if T i T j = T j T i, for all 1 i, j n and i j. A closed subspace S of H 2 E (Dn ) which is invariant under M z1,, M zn is said to be a doubly commuting submodule if S is a submodule, that is, M zi S S for all i and the family of module multiplication operators {R z1,..., R zn } where R zi := M zi S, for all 1 i n, is doubly commuting, that is, for all i j in {1,..., n}. R zi R z j = R z j R zi, In this note we completely characterize the doubly commuting submodules of the vector-valued Hardy module H 2 E (D n ) over the polydisc, and this is the content of our main theorem. This result is an analogue of the classical Beurling-Lax-Halmos Theorem on the Hardy space over the unit disc. Theorem 1.3. Let S be a closed nonzero subspace of H 2 E (D n ). Then S is a doubly commuting submodule if and only if there exists a Hilbert space E with E E, where the inclusion is up to unitary equivalence, and an inner function Θ H E E (D n ) such that S = M Θ H 2 E(D n ).

3 DOUBLY COMMUTING SUB-HARDY-MODULES 3 In the special scalar case E = C and when n = 2 (the bidisc), this characterization was obtained by Mandrekar in [M], and the proof given there relies on the Wold decomposition for two variables [S]. Our proof is based on the more natural language of Hilbert modules and a generalization of Wold decomposition for doubly commuting isometries [Sa]. As an application of this theorem, we can establish a version of the Weak Completion Property for the algebra H (D n ). Suppose that E E c. Recall that the Completion Problem for H (D n ) is the problem of characterizing the functions f HE E c (D n ) such that there exists an invertible function F HE c E c (D n ) with F E = f. In the case of H (D), the Completion Problem was settled by Tolokonnikov in [To]. In that paper, it was pointed out that there is a close connection between the Completion Problem and the characterization of invariant subspaces of H 2 (D). Using Theorem 1.3 we then have the following analogue of the results in [To]. Theorem 1.4 (Tolokonnikov s Lemma for the Polydisc). Let f H E E c (D n ) with E E c and dim E, dim E c <. Then the following statements are equivalent: (i) There exists a function g H E c E (Dn ) such that gf I in D n and the operators M z1,..., M zn doubly commute on the subspace ker M g. (ii) There exists a function F H E c E c (D n ) such that F E = f, F Ec E is inner, and F 1 H E c E c (D n ). Remark 1.5. Theorem 1.4 for the polydisc is different from Tolokonnikov s lemma in the disc in which one does not demand that the completion F has the property that F Ec E is inner. But, from the proof of Tolokonnikov s lemma in the case of the disc (see [N]), one can see that the following statements are equivalent for f H E E c (D) with E E c and dim E < : (i) There exists a function g HE c E (D) such that gf I in D. (ii) There exists a function F HE c E c (D) such that F E = f, and F 1 HE c E c (D). (ii ) There exists a function F HE c E c (D) such that F E = f, F Ec E is inner, and F 1 HE c E c (D). In the polydisc case it is unclear how the conditions (II) There exists a function F H E c E c (D n ) such that F E = f, and F 1 H E c E c (D n ). (II ) There exists a function F H E c E c (D n ) such that F E = f, F Ec E is inner, and F 1 H E c E c (D n ).

4 4 J. SARKAR, A. SASANE, AND B. D. WICK are related. We refer to the Completion Problem in (II) as the Strong Completion Problem, while the one in (II ) as the Weak Completion Problem. Whether the two are equivalent is an open problem. We also remark that in the disc case, Tolokonnikov s Lemma was proved by Sergei Treil [T] without any assumptions about the finite dimensionality of E, E c. However, our proof of Theorem 1.4 relies on Lemma 3.1, whose validity we do not know without the assumption on the finite dimensionality of E and E c. Example 1.6. As a simple illustration of Theorem 1.4, take n = 3, dim E = 1, dim E c = 3 and f := ez 1 e z 2 (H (D 3 )) 3 1. e z 3 With g := [ e z ] (H (D 2 )) 1 3, we see that gf = 1. We have ker M g = φ 1 φ 2 (H 2 (D 3 )) 3 1 : e z 1 φ 1 = 0 φ 3 = φ 1 φ 2 (H 2 (D 3 )) 3 1 : φ 1 = 0 = Θ(H2 (D 2 )) 2 1, φ 3 where Θ is the inner function Θ := (H (D 3 )) As Θ is inner, it follows from Theorem 1.3 that M z1, M z2, M z3 doubly commute on the submodule Θ(H 2 (D 3 )) 2 1 = ker M g. Hence f can be completed to an invertible matrix. In fact, with F := [ f Θ ] = ez e z 2 1 0, e z one can easily see that F is invertible as an element of (H (D 3 )) 3 3. In Section 2 we give a proof of Theorem 1.3, and subsequently, in Section 3, we use this theorem to study the Weak Completion Problem for H (D n ), providing a proof of Theorem Beurling-Lax-Halmos Theorem for the Polydisc In this section we present a complete characterization of reducing submodules and a proof of the Beurling-Lax-Halmos theorem for doubly commuting submodules of H 2 E (Dn ).

5 DOUBLY COMMUTING SUB-HARDY-MODULES 5 Recall that a closed subspace S HE 2 (Dn ) is said to be a reducing submodule of HE 2 (Dn ) if M zi S, Mz i S S for all i = 1,..., n. We start by reviewing some definitions and some well-known facts about the vector-valued Hardy space over polydisc. For more details about reproducing kernel Hilbert spaces over domains in C n, we refer the reader to [DMS]. Let n S(z, w) = (1 w j z j ) 1. ((z, w) D n D n ) j=1 be the Cauchy kernel on the polydisc D n. Then for some Hilbert space E, the kernel function S E of H 2 E (Dn ) is given by S E (z, w) = S(z, w)i E. ((z, w) D n D n ) In particular, {S(, w)η : w D n, η E} is a total subset for H 2 E (Dn ), that is, span{s(, w)η : w D n, η E} = H 2 E(D n ), where S(, w) H 2 (D n ) and (S(, w))(z) = S(z, w), for all z, w D n. Moreover, for all f H 2 E (Dn ), w D n and η E we have f, S(, w)η H 2 E (D n ) = f(w), η E. Note also that for the multiplication operator M zi on H 2 E (Dn ) M z i (S(, w)η) = w i (S(, w)η), where w D n, η E and 1 i n. We also have S 1 (z, w) = ( 1) l z i1 z il w i1 w il, for all z, w D n. For H 2 E (Dn ) we set S 1 E (M z, M z ) := 0 i 1 <...<i l n 0 i 1 <...<i l n ( 1) l M zi1 M zil M z i1 M z il. The following Lemma is well-known in the study of reproducing kernel Hilbert spaces. Lemma 2.1. Let E be a Hilbert space. Then S 1 E (M z, M z ) = P E, where P E is the orthogonal projection of H 2 E (Dn ) onto the space of all constant functions.

6 6 J. SARKAR, A. SASANE, AND B. D. WICK Proof. for all z, w D n and η, ζ E we have S 1 E (M z, M z ) (S(, z)η), (S(, w)ζ) H 2 E (D n ) = ( 1) l M zi1 M zil Mz i1 Mz il (S(, z)η), (S(, w)ζ) = = 0 i 1 <...<i l n 0 i 1 <...<i l n 0 i 1 <...<i l n ( 1) M l z i1 Mz il (S(, z)η), Mz i1 Mz il (S(, w)ζ) = S 1 (w, z)s(w, z) η, ζ E = η, ζ E ( 1) l z i1 z il w i1 w il S(, z), S(, w) H 2 (D n ) η, ζ E = P E S(, z)η, S(, w)ζ H 2 E (D n ) Since {S(, z)η : z D n, η E} is a total subset of H 2 E (Dn ), we have that This completes the proof. S 1 E (M z, M z ) = P E. In the following proposition we characterize the reducing submodules of H 2 E (Dn ). Proposition 2.2. Let S be a closed subspace of H 2 E (Dn ). Then S is a reducing submodule of H 2 E (Dn ) if and only if for some closed subspace E of E. S = H 2 E (D n ), Proof. Let S be a reducing submodule of H 2 E (Dn ), that is, for all 1 i n we have M zi P S = P S M zi. By Lemma 2.1 P E P S = S 1 E (M z, M z )P S = P S S 1 E (M z, M z ) = P S P E. In particular, that P S P E is an orthogonal projection and where E := E S. Hence, for any P S P E = P E P S = P E, f = k N n a k z k S, where a k E for all k N n, we have f = P S f = P S ( k N n M k z a k ) = k N n M k z P S a k. H 2 E (Dn ) H 2 E (Dn )

7 DOUBLY COMMUTING SUB-HARDY-MODULES 7 But P S a k = P S P E a k E. Consequently, M k z P S a k H 2 E (D n ) for all k N n and hence f H 2 E (D n ). That is, S H 2 E (D n ). For the reverse inclusion, it is enough to observe that E S and that S is a reducing submodule. The converse part is immediate. Hence the lemma follows. Let S be a doubly commuting submodule of H 2 E (Dn ). Then R zi R z i = M zi P S M z i P S = M zi P S M z i, implies that R zi Rz i is an orthogonal projection of S onto z i S and hence I S R zi Rz i is an orthogonal projection of S onto S z i S, that is, for all i = 1,..., n. Define for all i = 1,..., n, and I S R zi R z i = P S zi S, W i = ran(i S R zi R z i ) = S z i S, W = n W i. Now let S be a doubly commuting submodule of H 2 E (Dn ). By doubly commutativity of S it follows that (also see [Sa]) (I S R zi R z i )(I S R zj R z j ) = (I S R zj R z j )(I S R zi R z i ), for all i j. Therefore {(I S R zi Rz i )} n is a family of commuting orthogonal projections and hence (2.1) n n n n W = W i = (S z i S) = ran(i S R zi Rz i )) = ran( (I S R zi Rz i )). Now we present a wandering subspace theorem concerning doubly commuting submodules of H 2 E (Dn ). The result is a consequence of a several variables analogue of the classical Wold decomposition theorem as obtained by Gaspar and Suciu [GS]. We provide a direct proof (also see Corollary 3.2 in [Sa]). Theorem 2.3. Let S be a doubly commuting submodule of H 2 E (Dn ). Then S = k N n z k W. Proof. First, note that if M is a submodule of HE 2 (Dn ) then Rz k i M Mz k i HE(D 2 n ) = {0}, k N k N

8 8 J. SARKAR, A. SASANE, AND B. D. WICK for each i = 1,..., n. Therefore, R zi is a shift, that is, the unitary part k N R k z i M in the Wold decomposition (cf. [NF], [Sa]) of R zi on M is trivial for all i = 1,..., n. Moreover, if S is doubly commuting then R zi (I S R zj R z j ) = (I S R zj R z j )R zi, for all i j. Therefore W j is a R zi -reducing subspace for all i j. Note also that for all 1 m < n, m+1 and hence m+1 W i = ran( (I S R zi R z i )) m m = ran( (I S R zi Rz i ) R zm+1 Rz m+1 (I S R zi Rz i )) m m = ran( (I S R zi Rz i ) R zm+1 (I S R zi Rz i )Rz m+1 ) = (W 1... W m ) z m+1 (W 1 W m ), (W 1... W m ) z m+1 (W 1 W m ) = We use mathematical induction to prove that for all 2 m n, we have S = k N m z k (W 1... W m ). First, by Wold decomposition theorem for the shift R z1 S = R k 1 z 1 W 1 = z k 1 1 W 1. k 1 N k 1 N m+1 W i. on S we have Again by applying Wold decomposition for R z2 W1 L(W 1 ) we have and hence W 1 = R k 2 z 2 (W 1 z 2 W 1 ) = z k 2 2 (W 1 W 2 ), k 2 N k 2 N S = k 1 N z k 1 1 Finally, let ( k 2 N ) z k 2 2 (W 1 W 2 ) = k 1,k 2 N z k 1 1 z k 2 2 (W 1 W 2 ). S = z k (W 1... W m ), k N m for some m < n. Then we again apply the Wold decomposition on the isometry R zm+1 W1... W m L(W 1... W m )

9 to obtain W 1... W m = DOUBLY COMMUTING SUB-HARDY-MODULES 9 k m+1 N = k m+1 N ) z k m+1 m+1 ((W 1... W m ) z m+1 W 1... W m z k m+1 m+1 (W 1... W m W m+1 ), which yields S = z k (W 1... W m+1 ). k N m+1 This completes the proof. We now turn to the proof of Theorem 1.3. Proof of Theorem 1.3. By Theorem 2.3 we have (2.2) S = n z k ( W i ). k N n Now define the Hilbert space E by n E = W i, and the linear operator V : HE 2 (Dn ) HE 2 (D n ) by ( ) V = Mz k a k, k N n where k N n a k z k k N n a k z k H 2 E(D n ) and a k E for all k N n. Observe that k N n M k z a k 2 H 2 E (Dn ) = k N n z k a k 2 H 2 E (Dn ) = k N n z k a k 2 H 2 E (Dn ), where the last equality follows from the orthogonal decomposition of S in (2.2). Therefore, k N n M k z a k 2 H 2 E (Dn ) = k N n z k a k 2 H 2 E (Dn ) = k N n a k 2 H 2 E (Dn ) = k N n a k 2 E = k N n z k a k 2 H 2 E (Dn ), and hence V is an isometry. Moreover, for all k N n and η E we have V M zi (z k η) = V (z k+e i η) = M k+e i z η = M zi (Mz k η) = M zi V (z k η), that is, V M zi Therefore, = M zi V for all i = 1,..., n. Hence V is a module map. V = M Θ,

10 10 J. SARKAR, A. SASANE, AND B. D. WICK for some bounded holomorphic function Θ H E E (D n ) (cf. page 655 in [BLTT]). Moreover, since V is an isometry, we have M ΘM Θ = I H 2 E (D n ), that is, that Θ is an inner function. Also since M zi E S for all i = 1,..., n we have that ranv S. Also by (2.2) that S ranv. Hence it follows that that is, Finally, for all i = 1,..., n, we have ranv = ranm Θ = S, S = ΘH 2 E(D n ). S z i S = ΘH 2 E(D n ) z i ΘH 2 E(D n ) = {Θf : f H 2 E(D n ), M z i Θf = 0}, and hence by (2.1) E = n W i = n (S z i S) = {Θf : Mz i Θf = 0, f HE(D 2 n ), i = 1,..., n} {g H 2 E (D n ) : M z i g = 0, i = 1,..., n} = E, that is, E E. To prove the converse part, let S = M Θ H 2 E (Dn ) be a submodule of H 2 E (D n ) for some inner function Θ H E E (D n ). Then and hence for all i j, P S = M Θ M Θ, M zi P S M z j = M zi M Θ M ΘM z j = M Θ M zi M z j M Θ = M Θ M z j M zi M Θ This implies = M Θ M z j M ΘM Θ M zi M Θ = M Θ M ΘM z j M zi M Θ M Θ = P S M z j M zi P S. R z j R zi = P S M z j P S M zi S = P S M z j M zi S = M zi P S M z j = R zi R z j, that is, S is a doubly commuting submodule. This completes the proof.

11 DOUBLY COMMUTING SUB-HARDY-MODULES Tolokonnikov s Lemma for the Polydisc We will need the following lemma, which is a polydisc version of a similar result proved in the case of the disc in Nikolski s book [N]*p Here we use the notation M g for the multiplication operator on HE 2 induced by g HE E. Lemma 3.1 (Lemma on Local Rank). Let E, E c be Hilbert spaces, with dim E, dim E c <. Let g H E c E (Dn ) be such that ker M g = {h H 2 E c (D n ) : g(z)h(z) 0} = ΘH 2 E a (D n ), where E a is a Hilbert space and Θ is a L(E a, E c )-valued inner function. Then where rank g := max rank g(ζ). ζ Dn dim E c = dim E a + rank g, Proof. We have ker M g = {h H 2 E c (D n ) : gh 0}. If ζ D n, then let [ker M g ](ζ) := {h(ζ) : h ker M g }. We claim that [ker M g ](ζ) = Θ(ζ)E a. Indeed, let v [ker M g ](ζ). Then v = h(ζ) for some element h ker M g = ΘH 2 E a (D n ). So h = Θφ, for some φ H 2 E a (D n ). In particular, v = h(ζ) = Θ(ζ)φ(ζ), where φ(ζ) E a. So (3.1) [ker M g ](ζ) Θ(ζ)E a. On the other hand, if w Θ(ζ)E a, then w = Θ(ζ)x, where x E a. Consider the constant function x mapping D z x x E a. Clearly x H 2 E a (D n ). So h := Θx ΘH 2 E a (D n ) = ker M g. Hence w = Θ(ζ)x = (Θx)(ζ) = h(ζ), and so w [ker M g ](ζ). So we also have that (3.2) Θ(ζ)E a [ker M g ](ζ). Our claim that [ker M g ](ζ) = Θ(ζ)E a follows from (3.1) and (3.2). Suppose that for a ζ D n, v [ker M g ](ζ). Then v = h(ζ) for some h ker M g. Thus gh 0 in D n, and in particular, g(ζ)v = g(ζ)h(ζ) = 0. Thus v ker g(ζ). So we have that [ker M g ](ζ) ker g(ζ). Hence dim[ker M g ](ζ) dim ker g(ζ). Consequently dim Θ(ζ)E a = dim[ker M g ](ζ) dim ker g(ζ) = dim E c rank g(ζ), where the last equality follows from the Rank-Nullity Theorem. Since Θ is inner, we have that the boundary values of Θ satisfy Θ(ζ) Θ(ζ) = I Ec for almost all ζ T n. So there is an open set U D n such that for all ζ U dim E a = dim Θ(ζ)E a.

12 12 J. SARKAR, A. SASANE, AND B. D. WICK But from the definition of the rank of g, we know that there is a ζ D n such that we have k := rank g = rank g(ζ ). So there is a k k submatrix of g(ζ ) which is invertible. Now look at the determinant of this k k submatrix of g. This is a holomorphic function, and so it cannot be identically zero in the open set U. So there must exist a point ζ 1 U D n such that rank g = rank g(ζ 1 ) and dim E a = dim Θ(ζ 1 )E a. Hence dim E a dim E c rank g. For the proof of the opposite inequality, let us consider a principal minor g 1 (ζ 1 ) of the matrix of the operator g(ζ 1 ) (with respect to two arbitrary fixed bases in E c and E respectively). Then det g 1 H, det g 1 0. Let E c = E c,1 E c,2, E = E 1 E 2 (dim E c,1 = dim E 1 = rank g(ζ 1 )) be the decompositions of the spaces E c and E corresponding to this minor, and let [ ] g1 (ζ) g g(ζ) = 2 (ζ), ζ D γ 1 (ζ) γ 2 (ζ) n, be the matrix representation of g(ζ) with respect to this decomposition. Owing to our assumption on the rank, it follows that there is a matrix function ζ W (ζ) such that [ γ1 (ζ) γ 2 (ζ) ] = W (ζ) [ g 1 (ζ) g 2 (ζ) ]. So γ 2 (ζ) = W (ζ)g 2 (ζ) = (γ 1 (ζ)(g 1 (ζ)) 1 )g 2 (ζ). Thus with g co 1 := (det g 1 )g 1 1, we have γ 2 det g 1 = γ 1 g co 1 g 2, and using this we get the inclusion M Ω HE 2 c,2 (D n ) ker M g, where Ω HE c,2 E c (D n ) is given by [ ] g co Ω = 1 g 2. det g 1 We have rank Ω = dim E c,2 = dim E c rank g = dim ker(g(ζ 1 )). Consequently, we obtain dim[ker M g ](ζ 1 ) dim ker(g(ζ 1 )). We now turn to the extension of Tolokonnikov s Lemma to the polydisc. Proof of Theorem 1.4. (ii) (i): If g := P E F 1, then gf = I. It only remains to show that the operators M z1,..., M zn are doubly commuting on the space ker M g. Let Θ, Γ be such that: F = [ f Θ ] [ ] g and F 1 =. Γ Since F F 1 = I Ec, it follows that fg + ΘΓ = I Ec. Thus if h H 2 E c (D n ) is such that gh = 0, then Θ(Γh) = h, and so h ΘH 2 E c E) (Dn ). Hence ker M g ran M Θ. Also, since F 1 F = I, it follows that gθ = 0, and so ran M Θ ker M g. So ker M g = ran M Θ = ΘH 2 E c E (D2 ). By Theorem 1.3, the operators M z1,..., M zn must doubly commute on the subspace ker M g.

13 DOUBLY COMMUTING SUB-HARDY-MODULES 13 (i) (ii): Let S := {h H 2 E c (D n ) : g(z)h(z) 0} = ker g. S is a closed non-zero invariant subspace of HE 2 c (D n ). Also, by assumption, M z1,..., M zn are doubly commuting operators on S. Then by the above Theorem 1.3, there exists an auxiliary Hilbert space E a and an inner function Θ with values in L(E a, E c ) with dim E a dim E c such that S = ΘH 2 E a (D n ). By the Lemma on Local Rank, dim E a = dim E c rank g = dim E c dim E = dim(e c E). Let U be a (constant) unitary operator from E c E to E a and define Θ := ΘU. Then Θ is inner, and we have that ker g = ΘHE 2 c E (Dn ). To get F HE c E c (D n ) define the function F for z D n by { f(z)e if e E F (z)e := Θ(z)e if e E c E. We note that F H (D n ) and F E = f. We now show that F is invertible. With this in mind, we first observe that (I fg)h 2 E c (D n ) ΘH 2 E c E(D n ) = ker M g. This follows since g(i fg)h = gh gh = 0 for all h H 2 E c (D n ). Thus we have that Θ (I fg) H E c E c E (Dn ). Now, define Ω = g Θ (I fg). Clearly Ω H E c E c (D n ). Next, note that Similarly, F Ω = fg + ΘΘ (I fg) = I. ΩF = gfp E + Θ (I fg)(fp E + ΘP Ec E) = P E + Θ (fp E fgfp E + ΘP Ec E) = P E + Θ ΘP Ec E = I. Thus we have that F 1 H (D n ; E c E c ). Acknowledgements. The authors thank the anonymous referee for the careful review, for the help in improving the presentation of the paper, and also for suggesting Example 1.6. References [BLTT] J. Ball, W.S. Li, D. Timotin, T. Trent, A commutant lifting theorem on the polydisc, Indiana Univ. Math. J., 48 (1999), [B] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math., 81 (1948), 17.

14 14 J. SARKAR, A. SASANE, AND B. D. WICK [DMS] R. Douglas, G. Misra and J. Sarkar, Contractive Hilbert modules and their dilations, Israel J. Math., 187, (2012), [GS] D. Gaşpar and N. Suciu, Wold decompositions for commutative families of isometries, An. Univ. Timişoara Ser. Ştiinţ. Mat., 27 (1989), [L] P.D. Lax, Translation invariant spaces, Acta Math., 101 (1959), [M] V. Mandrekar, The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc., 103 (1988), [N] N.K. Nikolskiĭ, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften, 273, Springer-Verlag, Berlin, (1986). [R] W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, (1969). [Sa] J. Sarkar, Wold decomposition for doubly commuting isometries, preprint, arxiv: [S] M. S lociński, On the Wold-type decomposition of a pair of commuting isometries, Ann. Polon. Math., 37 (1980), [NF] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam (1970). [To] V. Tolokonnikov, Extension problem to an invertible matrix, Proc. Amer. Math. Soc., 117 (1993), [T] S. Treil, An operator Corona theorem, Indiana Univ. Math. J., 53 (2004), J. Sarkar, Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, , India address: jay@isibang.ac.in, jaydeb@gmail.com URL: A. Sasane, Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, U.K. address: sasane@lse.ac.uk Brett D. Wick, School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA USA , U.S.A. address: wick@math.gatech.edu URL: