PAIRS OF COMMUTING ISOMETRIES - I

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1 PAIRS OF COMMUTING ISOMETRIES - I AMIT MAJI, JAYDEB SARKAR, AND SANKAR T. R. Abstract. We present an explicit version of Berger, Coburn and Lebow's classication result for pure pairs of commuting isometries in the sense of an explicit recipe for constructing pairs of commuting isometric multipliers with precise coecients. We describe a complete set of (joint) unitary invariants and compare the Berger, Coburn and Lebow's representations with other natural analytic representations of pure pairs of commuting isometries. Finally, we study the defect operators of pairs of commuting isometries. 1. Introduction A very general and fundamental problem in the theory of bounded linear operators on Hilbert spaces is to nd classications and representations of commuting families of isometries. In the case of single isometries this question has a complete and explicit answer: If V is an isometry on a Hilbert space H, then there exist a Hilbert [ space H] u and a unitary operator U on H u such that V on H S IW 0 and on (l 0 U 2 (Z + ) W) H u are unitarily equivalent, where W = ker V is the wandering subspace for V and S is the forward shift operator on l 2 (Z + ) [H]. This fundamental result is due to J. von Neumann [VN] and H. Wold [W] (see Theorem 2.1 for more details). The case of pairs (and n-tuples) of commuting isometries is more subtle, and is directly related to the commutant lifting theorem [FF] (in terms of an explicit, and then unique solution), invariant subspace problem [HH] and representations of contractions on Hilbert spaces in function Hilbert spaces [NF]. For instance: (a) Let S be a closed joint (M z1, M z2 )-invariant subspace of the Hardy space H 2 (D 2 ). Then (M z1 S, M z2 S ) on S is a pure (see Section 3) pair of commuting isometries. Classication of such pairs of isometries is largely unknown (see Rudin [R]) Mathematics Subject Classication. Primary 47A05, 47A13, 47A20, 47A45, 47A65; Secondary 46E22, 46E40. Key words and phrases. Isometries, commuting pairs, commutators, multipliers, Hardy space, defect operators. 1

2 2 A. MAJI, J. SARKAR, AND SANKAR T. R. (b) Let T be a contraction on a Hilbert space H. Then there exists a pair of commuting isometries (V 1, V 2 ) on a Hilbert space K such that T and P ker V 2 V 1 ker V 2 are unitarily equivalent (see Bercovici, Douglas and Foias [BDF]). (c) The celebrated Ando dilation theorem (see Ando [A]) states that a commuting pair of contractions dilates to a commuting pair of isometries. Again, the structure of Ando's pairs of commuting isometries is largely unknown. The main purpose of this paper is to explore and relate various natural representations of a large class of pairs of commuting isometries on Hilbert spaces. The geometry of Hilbert spaces, the classical Wold-von Neumann decomposition for isometries, the analytic structure of the commutator of the unilateral shift, and the Berger, Coburn and Lebow [BCL] representations of pure pairs of commuting isometries are the main guiding principles for our study. The Berger, Coburn and Lebow theorem states that: Let (V 1, V 2 ) be a pair of commuting isometries on a Hilbert space H, and let V = V 1 V 2 be a shift (or, a pure isometry - see Section 2). Then there exist a Hilbert space W, an orthogonal projection P and a unitary operator U on W such that Φ 1 (z) = U (P + zp ) and Φ 2 (z) = (P + zp )U (z D), are commuting isometric multipliers in HB(W) (D), and (V 1, V 2, V ) on H and (M Φ1, M Φ2, M z ) on HW 2 (D) are unitarily equivalent (see Bercovici, Douglas and Foias [BDF] for an elegant proof). Here and further on, given a Hilbert space H and a closed subspace S of H, P S denotes the orthogonal projection of H onto S. We also set P S = I H P S. In this paper we give a new and more concrete treatment, in the sense of explicit representations and analytic descriptions, to the structure of pure pairs of commuting isometries. More specically, we provide an explicit recipe for constructing the isometric multipliers (Φ 1 (z), Φ 2 (z)), and the operators U and P involved in the coecients of Φ 1 and Φ 2 (see Theorems 3.2 and 3.3). Then we compare the Berger, Coburn and Lebow representations with other possible analytic representations of pairs of commuting isometries. In Section 6, which is independent of the remaining part of the paper, we analyze defect operators for (not necessarily pure) pairs of commuting isometries. We provide a list of characterizations of pairs of commuting

3 PAIRS OF COMMUTING ISOMETRIES - I 3 isometries with positive defect operators (see Theorem 6.2). Our results hold in a more general setting with somewhat simpler proofs (see Theorem 6.5 for instance) than the one considered by He, Qin and Yang [HQY]. Moreover, we prove that for a large class of pure pairs of commuting isometries the defect operator is negative if and only if the defect operator is the zero operator. The paper is organized as follows. In Section 2 we review the classical Wold-von Neumann theorem for isometries and then prove a representation theorem for commutators of shifts. In Section 3 we discuss some basic relationships between wandering subspaces for commuting isometries, followed by a new and explicit proof of the Berger, Coburn and Lebow characterizations of pure pairs of commuting isometries. Section 4 is devoted to a short discussion about joint unitary invariants of pure pairs of commuting isometries. Section 5 ties together the explicit Berger, Coburn and Lebow representation and other possible analytic representations of a pair of commuting isometries. Then, in Section 6, we present a general theory for pairs of commuting isometries and analyze the defect operators. Concluding remarks, future directions and a close connection of our consideration with the Sz.-Nagy and Foias characteristic functions for contractions are discussed in Section Wold-von Neumann decomposition and commutators We begin this section by briey recalling the construction of the classical Wold-von Neumann decomposition of isometric operators on Hilbert spaces. Here our presentation is more algebraic and geared towards the main theme of the paper. First, recall that an isometry V on a Hilbert space H is said to be pure, or a shift, if it has no unitary direct summand, or equivalently, if lim m V m = 0 in the strong operator topology (see Halmos [H]). Let V be an isometry on a Hilbert space H, and let W(V ) be the wandering subspace [H] for V, that is, W(V ) = H V H. The classical Wold-von Neumann decomposition is as follows: Theorem 2.1. (Wold-von Neumann decomposition) Let V be an isometry on a Hilbert space H. Then H decomposes as a direct sum of V -reducing subspaces H s (V ) = V m W(V ) and H u (V ) = H H s (V ) and [ ] Vs 0 V = B(H 0 V s (V ) H u (V )), u

4 4 A. MAJI, J. SARKAR, AND SANKAR T. R. where V s = V Hs(V ) is a shift operator and V u = V Hu(V ) is a unitary operator. We will refer to this decomposition as the Wold-von Neumann orthogonal decomposition of V. Recall that the Hardy space H 2 (D) is the Hilbert space of all analytic functions on the unit disc D with square summable Taylor coecients (cf. [H], [RR]). The Hardy space is also a reproducing kernel Hilbert space corresponding to the Szegö kernel S(z, w) = (1 z w) 1 (z, w D). For any Hilbert space E, the E-valued Hardy space with reproducing kernel D D B(E), (z, w) S(z, w)i E, can canonically be identied with the tensor product Hilbert space H 2 (D) E. To simplify the notation, we often identify H 2 (D) E with the E-valued Hardy space HE 2 (D). The space of B(E)-valued bounded holomorphic functions on D will be denoted by HB(E) (D). Let Mz E denote the multiplication operator by the coordinate function z on HE 2(D), that is (Mz E f)(w) = wf(w) (f HE(D), 2 w D). Then M E z is a shift operator and W(M E z ) = E. To simplify the notation we often omit the superscript and denote M E z by M z, if E is clear from the context. We now proceed to give an analytic description of the Wold-von Neumann construction. Let V be an isometry on H, and let H = H s (V ) H u (V ) be the Wold-von Neumann orthogonal decomposition of V. Dene by Then Π V Π V : H s (V ) H u (V ) H 2 W(V )(D) H u (V ) Π V (V m η f) = z m η f (m 0, η W(V ), f H u (V )). is a unitary and [ ] Vs 0 Π V = 0 V u [ ] W(V ) M z 0 Π V. 0 V u In particular, if V is a shift, then H u (V ) = {0} and hence Π V V = M W(V ) z Π V.

5 PAIRS OF COMMUTING ISOMETRIES - I 5 Therefore, an isometry V on H is a shift operator if and only if V is unitarily equivalent to Mz E on HE 2 (D), where dim E = dim W(V ). In the sequel we denote by (Π V, Mz W(V ) ), or simply by (Π V, M z ), the Woldvon Neumann decomposition of the pure isometry V in the above sense. Let E be a Hilbert space, and let C be a bounded linear operator on H 2 E (D). Then C {M z}, that is, CM z = M z C, if and only if (cf. [NF]) C = M Θ for some Θ HB(E) (D) and (M Θf)(w) = Θ(w)f(w) for all f HE 2 (D) and w D. Now let V be a pure isometry, and let C {V }. Let (Π V, M z ) be the Wold-von Neumann decomposition of V, and let W = W(V ). Since Π V CΠ V on H2 W (D) is the representation of C on H and (Π V CΠ V )M z = M z (Π V CΠ V ), it follows that Π V CΠ V = M Θ, for some Θ HB(W) (D). The main result of this section is the following explicit representation of Θ. Theorem 2.2. Let V be a pure isometry on H, and let C be a bounded operator on H. Let (Π V, M z ) be the Wold-von Neumann decomposition of V. Set W = W(V ), M = Π V CΠ V and let Θ(w) = P W (I H wv ) 1 C W (w D). Then if and only if Θ HB(W) (D) and CV = V C, M = M Θ. Proof. Let h H. One can express h as h = V m η m, for some η m W, m 0 (as H = V m W). Applying P W V l to both sides and using the fact that W = W(V ) = ker V, we obtain η l = P W V l h for all l 0. This implies, for any h H, (2.1) h = V m P W V m h.

6 6 A. MAJI, J. SARKAR, AND SANKAR T. R. Now let CV = V C. Then there exists a bounded analytic function Θ H B(W) (D) such that Π V CΠ V = M Θ. For each w D and η W we have Θ(w)η = (M Θ η)(w) = (Π V CΠ V η)(w) = (Π V Cη)(w), as Π V η = η. Since in view of (2.1) Cη = V m P W V m Cη, it follows that Therefore Θ(w)η = (Π V ( = ( = V m P W V m Cη))(w) Mz m (P W V m Cη))(w) w m (P W V m Cη) = P W (I H wv ) 1 Cη. Θ(w) = P W (I H wv ) 1 C W (w D), as required. Finally, since the sucient part is trivial, the proof is complete. Note that in the above proof we have used the standard projection formula (see, for example, Rosenblum and Rovnyak [RR]) I H = SOT V m P W V m. It may also be observed that wv = w V < 1 for all w D, and so it follows that the function Θ dened in Theorem 2.2 is a B(W)-valued holomorphic function in the unit disc D. However, what is not guaranteed in general here is that the function Θ is in HB(W) (D). The above theorem says that this is so if and only if CV = V C. 3. Berger, Coburn and Lebow representations This section is devoted to a detailed study of Berger, Coburn and Lebow's representation of pure pairs of commuting isometries. Our approach is different and yields sharper results, along with new proofs, in terms of explicit coecients of the one variable polynomials associated with the class of pure pairs of commuting isometries. Before dealing more specically with pure

7 PAIRS OF COMMUTING ISOMETRIES - I 7 pairs of commuting isometries we begin with some general observations about pairs of commuting isometries. Let (V 1, V 2 ) be a pair of commuting isometries on a Hilbert space H. In the sequel, we will adopt the following notations: and V = V 1 V 2, W = W(V ) = W(V 1 V 2 ) = H V 1 V 2 H, W j = W(V j ) = H V j H (j = 1, 2). A pair of commuting isometries (V 1, V 2 ) on H is said to be pure if V is a pure isometry. The following useful lemma on wandering subspaces for commuting isometries is simple. Lemma 3.1. Let (V 1, V 2 ) be a pair of commuting isometries on a Hilbert space H. Then W = W 1 V 1 W 2 = V 2 W 1 W 2, and the operator U on W dened by U(η 1 V 1 η 2 ) = V 2 η 1 η 2, for η 1 W 1 and η 2 W 2, is a unitary operator. Moreover, P W V i = V i P Wj Proof. The rst equality follows from (i j). I V V = (I V 1 V 1 ) V 1 (I V 2 V 2 )V 1 = V 2 (I V 1 V 1 )V 2 (I V 2 V 2 ). The second part directly follows from the rst part, and the last claim follows from (I V V )V i = V i (I V j V j ) for all i j. This concludes the proof of the lemma. Let (V 1, V 2 ) be a pure pair of commuting isometries on a Hilbert space H, and let (Π V, M z ) be the Wold-von Neumann decomposition of V. Since V V i = V i V (i = 1, 2), there exist isometric multipliers (that is, inner functions [NF]) Φ 1 and Φ 2 in H B(W) (D) such that Π V V i = M Φi Π V (i = 1, 2). In other words, (M Φ1, M Φ2 ) on H 2 W (D) is the representation of (V 1, V 2 ) on H. Following Berger, Coburn and Lebow [BCL], we say that (M Φ1, M Φ2 ) is the BCL representation of (V 1, V 2 ), or simply the BCL pair corresponding to (V 1, V 2 ).

8 8 A. MAJI, J. SARKAR, AND SANKAR T. R. We now present an explicit description of the BCL pair (M Φ1, M Φ2 ). Theorem 3.2. Let (V 1, V 2 ) be a pure pair of commuting isometries on a Hilbert space H, and let (M Φ1, M Φ2 ) be the BCL representation of (V 1, V 2 ). Then for all z D. Φ 1 (z) = V 1 W2 V 2 V2 W 1 z, Φ 2 (z) = V 2 W1 V 1 V1 W 2 z, Proof. Let η in W = V 2 W 1 W 2, and let w D. Then there exist η 1 W 1 and η 2 W 2 such that η = V 2 η 1 η 2. Then V 1 η = V η 1 + V 1 η 2, and hence Φ 1 (w)η = (M Φ1 η)(w) = (Π V V 1 Π V η)(w) = (Π V V 1 η)(w) = (Π V V η 1 +Π V V 1 η 2 )(w). This along with the fact that V 1 η 2 W (see Lemma 3.1) gives for all w D. Therefore Φ 1 (w)η = (M z Π V η 1 + V 1 η 2 )(w) = (M z η 1 + V 1 η 2 )(w) = wη 1 + V 1 η 2 = wv 2 η + V 1 η 2, Φ 1 (z) = V 1 W2 V 2 V2 W 1 z, for all z D, as W 2 = Ker(V 2 ). The representation of Φ 2 follows similarly. In the following, we present Berger, Coburn and Lebow's version of representations of pure pairs of commuting isometries. This yields an explicit representations of the auxiliary operators U and P (see Section 1). The proof readily follows from Lemma 3.1 and Theorem 3.2. Theorem 3.3. Let (V 1, V 2 ) be a pure pair of commuting isometries on H. Then the BCL pair (M Φ1, M Φ2 ) corresponding to (V 1, V 2 ) is given by Φ 1 (z) = U (P W2 + zp W 2 ), and where U = is a unitary operator on W. Φ 2 (z) = (P W 2 + zp W2 )U, [ ] V2 W1 0 0 V1 : V1 W 2 W 1 V 1 W 2 V 2 W 1 W 2, Therefore, (V 1, V 2, V 1 V 2 ) on H and (M Φ1, M Φ2, Mz W ) on HW 2 (D) are unitarily equivalent, where W is the wandering subspace for V = V 1 V 2.

9 PAIRS OF COMMUTING ISOMETRIES - I 9 4. Unitary invariants In this short section we present a complete set of joint unitary invariants for pure pairs of commuting isometries. Recall that two commuting pairs (T 1, T 2 ) and ( T 1, T 2 ) on H and H, respectively, are said to be (jointly) unitarily equivalent if there exists a unitary operator U : H H such that UT j = T j U for all j = 1, 2. First we note that, by virtue of Theorem 2.9 of [BDF], the orthogonal projection P W2 and the unitary operator U on W, as in Theorem 3.3, form a complete set of (joint) unitary invariants of pure pairs of commuting isometries. More specically: Let (V 1, V 2 ) and (Ṽ1, Ṽ2) be two pure pairs of commuting isometries on H and H, respectively. Let W j be the wandering subspace for Ṽj, j = 1, 2. Then (V 1, V 2 ) and (Ṽ1, Ṽ2) are unitarily equivalent if and only if [ ] ] V2 ( W1 0 [Ṽ2 0 V1, P W2 ) and ( W1 0 V1 W 2 0 Ṽ1, P W2 ) Ṽ1 W2 are unitarily equivalent. In addition to the above, the following unitary invariants are also explicit. The proof is an easy consequence of Theorem 3.2. Here we will make use of the identications of A on H 2 W (D) and AM z on H 2 W (D) with I H 2 (D) A on H 2 (D) W and M z A on H 2 (D) W, respectively, where A B(W) (see Section 2). Theorem 4.1. Let (V 1, V 2 ) and (Ṽ1, Ṽ2) be two pure pairs of commuting isometries on H and H, respectively. Then (V 1, V 2 ) and (Ṽ1, Ṽ2) are unitarily equivalent if and only if (V 1 W2, V 2 V2 W 1 ) and (Ṽ1 W2, Ṽ 2 Ṽ2 W1 ) are unitarily equivalent. Proof. Let (M Φ1, M Φ2 ) and, ) be the BCL pairs corresponding to (M Φ1 M Φ2 (V 1, V 2 ) and (Ṽ1, Ṽ2), respectively, as in Theorem 3.2. Let C 1 = V 1 W2 and C 2 = V2 V2 W 1 be the coecients of Φ 1. Similarly, let C 1 and C 2 be the coecients of Φ 1. Now let Z : W W be a unitary such that ZC j = C j Z, j = 1, 2. Then M Φ1 = I H 2 (D) C 1 + M z C 2 = I H 2 (D) Z C1 Z + M z Z C2 Z = (I H 2 (D) Z )(I H 2 (D) C 1 + M z C 2 )(I H 2 (D) Z) = (I H 2 (D) Z (I )M Φ1 H 2 (D) Z).

10 10 A. MAJI, J. SARKAR, AND SANKAR T. R. Because M Φ2 = M z M Φ 1 and M Φ2 = M z M Φ1, it follows that (M Φ1, M Φ2 ) and (M Φ1, M Φ2 ) are unitarily equivalent, that is, (V 1, V 2 ) and (Ṽ1, Ṽ2) are unitarily equivalent. To prove the necessary part, let (M Φ1, M Φ2 ) and (M Φ1, M Φ2 ) are unitarily equivalent. Then there exists a unitary operator X : H 2 W (D) H2 W(D) [RR] such that Since XM Φj = M Φj X (j = 1, 2). XM W z = XM Φ1 M Φ2 = M Φ1 XX M Φ2 X = M Φ1 M Φ2 X = M W z X, there exists a unitary operator Z : W W such that X = I H 2 (D) Z. This and XM Φ1 = M Φ1 X implies that (I H 2 (D) Z)(I H 2 (D) C 1 + M z C 2 ) = (I H 2 (D) C 1 + M z C 2 )(I H 2 (D) Z). Hence (C 1, C 2 ) and ( C 1, C 2 ) are unitarily equivalent. This completes the proof of the theorem. Observe that the set of joint unitary invariants {V 1 W2, V 2 V2 W 1 }, as above, is associated with the coecients of Φ 1 of the BCL pair (M Φ1, M Φ2 ) corresponding to (V 1, V 2 ). Clearly, by duality, a similar statement holds for the coecients of Φ 2 as well: {V 2 W1, V 1 V1 W 2 } is a complete set of joint unitary invariants for pure pairs of commuting isometries. 5. Pure isometries In this section we will analyze pairs of commuting isometries (V 1, V 2 ) such that either V 1 or V 2 is a pure isometry, or both V 1 and V 2 are pure isometries. We begin with a concrete example which illustrates this particular class and also exhibits its complex structure. Recall that the Hardy space H 2 (D 2 ) over the bidisc D 2 is the Hilbert space of all analytic functions on the bidisc D 2 with square summable Taylor coecients (see Rudin [R]). Let M zj on H 2 (D 2 ) be the multiplication operator by the coordinate function z j, j = 1, 2. Note that (M z1, M z2 ) on H 2 (D 2 ) can be identied with (M z I H 2 (D), I H 2 (D) M z ) on H 2 (D) H 2 (D), and consequently, (M z1, M z2 ) on H 2 (D 2 ) is a pair of doubly commuting (that is, Mz 1 M z2 = M z2 Mz 1 ) pure isometries. Now let S be a joint (M z1, M z2 )-invariant closed subspace of H 2 (D 2 ), that is, M zj S S. Set V j = M zj S (j = 1, 2).

11 PAIRS OF COMMUTING ISOMETRIES - I 11 It follows immediately that V j is a pure isometry and V 1 V 2 = V 2 V 1, and hence (V 1, V 2 ) is a pair of commuting pure isometries on S. If we assume, in addition, that (V 1, V 2 ) is doubly commuting (that is, V1 V 2 = V 2 V1 ), then it follows that (V 1, V 2 ) on S and (M z1, M z2 ) on H 2 (D 2 ) are unitarily equivalent. See Slocinski [S] for more details. In general, however, the classication of pairs of commuting isometries, up to unitary equivalence, is complicated and very little seems to be known. For instance, see Rudin [R] for a list of pathological examples (also see Qin and Yang [QY]). We now turn our attention to the general problem. Let (V 1, V 2 ) be a pair of commuting isometries on H, and let V 1 be a pure isometry. Then, in particular, V = V 1 V 2 is a pure isometry, and hence (V 1, V 2 ) is a pure pair of commuting isometries. Since V 1 V 2 = V 2 V 1, by Theorem 2.2, it follows that (5.1) Π V1 V 2 = M ΘV2 Π V1, where Θ V2 HB(W 1 )(D) is an inner multiplier and (5.2) Θ V2 (z) = P W1 (I H zv 1 ) 1 V 2 W1 (z D). Let (M Φ1, M Φ2 ) be the BCL pair (see Theorem 3.3) corresponding to (V 1, V 2 ), that is, Π V V i = M Φi Π V for all i = 1, 2. Set Π 1 = Π V1 Π V. Then Π 1 : HW 2 (D) H2 W 1 (D) is a unitary operator such that Π 1 M Φ1 = M W 1 z Π 1 and Π 1 M Φ2 = M ΘV2 Π1. Therefore, we have the following commutative diagram: H Π V H 2 W (D) Π V1 Π 1 H 2 W 1 (D) where (M Φ1, M Φ2 ) on HW 2 (D) and (M W 1 z, M ΘV2 ) on HW 2 1 (D) are the representations of (V 1, V 2 ) on H. We now proceed to settle the non-trivial part of this consideration: An analytic description of the unitary map Π 1. To this end, observe rst that since Π V1 V 1 = M W 1 z Π V1, (5.1) gives Then that is, Π V1 V = M W 1 z M ΘV2 Π V1. Π 1 M W z = Π V1 V Π V = M W 1 z M ΘV2 Π V1 Π V, (5.3) Π1 M W z = (M W 1 z M ΘV2 ) Π 1.

12 12 A. MAJI, J. SARKAR, AND SANKAR T. R. Let η W. By Equation (2.1) we can write η = which yields that is (Π V1 η)(w) = ( = ( = Π 1 η = Π V1 Π V η = Π V1 η = V m Π V1 V m 1 P W1 V m 1 η)(w) M m z P W1 V m 1 η)(w) w m (P W1 V m 1 η), 1 P W1 V1 m η. Therefore z m (P W1 V m 1 η), Π 1 η = P W1 [I H + z(i H zv 1 ) 1 V 1 ]η, for all η W. It now follows from (5.3) that Π 1 (z m η) = (zθ V2 (z)) m P W1 [I H + z(i H zv1 ) 1 V1 ]η, for all m 0, and so, by S(, w)η = z m w m η, it follows that Π 1 (S(, w)η) = Π 1 ( z m w m η) = (I W1 wzθ V2 (z)) 1 P W1 [I H + z(i H zv 1 ) 1 V 1 ]η, for all w D and η W. Finally, from Π 1M W 1 z = M Φ1 Π 1 and Π 1η 1 = η 1 for all η 1 W 1, it follows that Π 1(z m η 1 ) = M m Φ 1 η 1 for all m 0, and hence Π 1(S(, w)η 1 ) = (I W Φ 1 (z) w) 1 η 1, for all w D and η 1 W 1. We summarize the above observations in the following theorem. Theorem 5.1. Let (V 1, V 2 ) be a pair of commuting isometries on H. Let i, j {1, 2} and i j. If V i is a pure isometry, then is a unitary operator, Π i = Π Vi Π V B(H 2 W(D), H 2 W i (D)), Π i M W z = M zθvj Πi, Π i M W i z = M Φi Π i, and Π i (S(, w)η) = (I Wi wzθ Vj (z)) 1 P Wi [I H + z(i zv i ) 1 V i ]η,

13 for all w D and η W, where for all z D. Moreover for all w D and η i W i. PAIRS OF COMMUTING ISOMETRIES - I 13 Θ Vj (z) = P Wi (I H zv i ) 1 V j Wi Π i (S(, w)η i ) = (I W Φ i (z) w) 1 η i, Note that the inner multipliers Θ Vi HB(W j )(D) above satisfy the following equalities: Π Vj V i = M ΘVi Π Vj. Now let (V 1, V 2 ) be a pair of commuting isometries such that both V 1 and V 2 are pure isometries. The above result leads to an analytic representation of such pairs. Corollary 5.2. Let (V 1, V 2 ) be a pair of commuting pure isometries on a Hilbert space H. If (M Φ1, M Φ2 ) is the BCL representation corresponding to (V 1, V 2 ), then M Φ1 and M Φ2 are pure isometries, Π 1 M Φ2 = M ΘV2 Π1, Π2 M Φ1 = M ΘV1 Π2, Π = Π 2 Π 1 : H 2 W 1 (D) H 2 W 2 (D) is a unitary operator, and ΠM W 1 z = M ΘV1 Π and ΠMΘV2 = M W 2 Π. Moreover, for each w D and η j W j, j = 1, 2, and Π(S(, w)η 1 ) = (I W2 wθ V1 (z)) 1 P W2 (I H zv 2 ) 1 η 1, Π (S(, w)η 2 ) = (I W1 wθ V2 (z)) 1 P W1 (I H zv 1 ) 1 η 2. Proof. A repeated application of Theorem 5.1 yields Π 1 M Φ2 = Π 1 M Φ 1 (M Φ1 M Φ2 ) = Π 1 M Φ 1 M W z = (M W 1 z ) Π1 Mz W = (M W 1 z ) M zθv2 Π1, that is, Π 1 M Φ2 = M ΘV2 Π1 and similarly Π 2 M Φ1 = M ΘV1 Π2. For η 1 W 1, we have Π V2 η 1 = P W2 (I H zv 2 ) 1 η 1. Since Π 1η 1 = η 1 and Π V η 1 = η 1, it follows that Πη 1 = Π 2 η 1 = Π V2 Π V η 1 = Π V2 η 1, that is Πη 1 = P W2 (I H zv 2 ) 1 η 1. Now using the identity Π(zη 1 ) = M ΘV1 Πη1, we have Π(z m η 1 ) = Θ V1 (z) m P W2 (I H zv 2 ) 1 η 1, z

14 14 A. MAJI, J. SARKAR, AND SANKAR T. R. for all m 0 and η 1 W 1. Finally S(, w)η 1 = w m z m η 1 gives Π(S(, w)η 1 ) = (I W2 wθ V1 (z)) 1 P W2 (I H zv 2 ) 1 η 1. The nal equality of the corollary follows from the equality Π (z m η 2 ) = Θ V2 (z) m ( Π η 2 ) = Θ V2 (z) m P W1 (I H zv 1 ) 1 η 2, for all m 0 and η 2 W 2. This concludes the proof. In the nal section, we will connect the analytic descriptions of Π 1 and Π 2 as in Theorem 5.1 with the classical notion of the Sz.-Nagy and Foias characteristic functions of contractions on Hilbert spaces [NF]. 6. Defect Operators Throughout this section, we will mostly work on general (not necessarily pure) pairs of commuting isometries. Let (V 1, V 2 ) be a pair of commuting isometries on a Hilbert space H. The defect operator C(V 1, V 2 ) of (V 1, V 2 ) (cf. [HQY]) is dened as the self-adjoint operator C(V 1, V 2 ) = I V 1 V 1 V 2 V 2 + V 1 V 2 V 1 V 2. Recall from Section 3 that given a pair of commuting isometries (V 1, V 2 ), we write V = V 1 V 2, and denote by W j = W(V j ) = ker V j = H V j H, the wandering subspace for V j, j = 1, 2. The wandering subspace for V is denoted by W. Finally, we recall that (see Lemma 3.1) W = W 1 V 1 W 2 = V 2 W 1 W 2. This readily implies (6.1) P W = P W1 P V1 W 2 = P V2 W 1 P W2. The following lemma is well known to the experts, but for the sake of completeness we provide a proof of the statement. Lemma 6.1. Let (V 1, V 2 ) be a commuting pair of isometries on H. Then H s (V ) and H u (V ) are V j -reducing subspaces, for all j = 1, 2. H s (V j ) H s (V ), and H u (V j ) H u (V ), Proof. For the rst part we only need to prove that H s (V ) is a V 1 -reducing subspace. Note that since (see Lemma 3.1) V 1 W W V W, it follows that V 1 V m W V m (W V W) H s (V ),

15 PAIRS OF COMMUTING ISOMETRIES - I 15 for all m 0. This clearly implies that V 1 H s (V ) H s (V ). On the other hand, since V 1 W = W 2 W and V 1 V m W = V m 1 (V 2 W) V m 1 (W V W), it follows that V 1 H s (V ) H s (V ). To prove the second part of the statement, it is enough to observe that V m H = V m 1 (V m 2 H) = V m 2 (V m 1 H) V m 1 H, V m 2 H, for all m 0, and as n V n 1 h 0, or V n 2 h 0 V n h 0, for any h H. This concludes the proof of the lemma. The following characterizations of doubly commuting isometries will prove important in the sequel. Lemma 6.2. Let (V 1, V 2 ) be a pair of commuting isometries on a Hilbert space H. Then the following are equivalent: (i) (V 1, V 2 ) is doubly commuting. (ii) V 2 W 1 W 1. (iii) V 1 W 2 W 2. Proof. Since (i) implies (ii) and (iii), by symmetry we only need to show that (ii) implies (i). Let V 2 W 1 W 1. Let H = H s (V ) H u (V ) be the Wold-von Neumann orthogonal decomposition of V (see Theorem 2.1). Then H s (V ) and H u (V ) are joint (V 1, V 2 )-reducing subspaces, and the pair (V 1 Hu(V ), V 2 Hu(V )) on H u is doubly commuting, because V j Hu(V ), j = 1, 2, are unitary operators, by Lemma 6.1. Now it only remains to prove that V1 V 2 = V 2 V1 on H s (V ). Since (V 1 V 2 V 2 V 1 )V m = V 1 V m V 2 V 2 V 1 V m = V m 1 V 2 2 V 2 2 V m 1 = 0, it follows that V1 V 2 V 2 V1 = 0 on V m W for all m 1. In order to complete the proof we must show that V1 V 2 = V 2 V1 on W. To this end, let η = η 1 V 1 η 2 W for some η 1 W 1 and η 2 W 2. Then V 1 V 2 (η 1 V 1 η 2 ) = V 1 V 2 η 1 + V 1 V 2 V 1 η 2 = V 2 η 2, as V 2 W 1 W 1, and on the other hand This completes the proof. V 2 V 1 (η 1 V 1 η 2 ) = V 2 V 1 η 1 + V 2 V 1 V 1 η 2 = V 2 η 2. The key of our geometric approach is the following simple representation of defect operators.

16 16 A. MAJI, J. SARKAR, AND SANKAR T. R. Lemma 6.3. C(V 1, V 2 ) = P W1 P V2 W 1 = P W2 P V1 W 2. Proof. The result readily follows from (6.1) and C(V 1, V 2 ) = (I V 1 V 1 ) + (I V 2 V 2 ) (I V V ) = P W1 + P W2 P W. The nal ingredient to our analysis is the fringe operator F 2. The notion of fringe operators plays a signicant role in the study of joint shift-invariant closed subspaces of the Hardy space over D 2 (see the discussion at the beginning of Section 5). Given a pair of commuting isometries (V 1, V 2 ) on H, the fringe operators F 1 B(W 2 ) and F 2 B(W 1 ) are dened by F j = P Wi V j Wi (i j). Of particular interest to us are the isometric fringe operators. Note that F 2 F 2 = P W1 V 2 P W1 V 2 W1. Lemma 6.4. The fringe operator F 2 on W 1 is an isometry if and only if V 2 W 1 W 1. Proof. As I W1 F 2 F 2 = I W1 P W1 V 2 P W1 V 2 W1, (6.1) implies that I W1 F 2 F 2 = P W1 V 2 P V1 W 2 V 2 W1. Then F 2 F 2 = I W1 if and only if P V1 W 2 V 2 W1 = 0, or, equivalently, if and only if V 2 W 1 V 1 W 2 = W 1, by Lemma 3.1. This completes the proof. Therefore, the fringe operator F 2 is an isometry if and only if the pair (V 1, V 2 ) is doubly commuting. We are now ready to formulate a generalization of Theorem 3.4 in [HQY] by He, Qin and Yang. Here we do not assume that (V 1, V 2 ) is pure. Theorem 6.5. Let (V 1, V 2 ) be a pair of commuting isometries on H. Then the following are equivalent: (a) C(V 1, V 2 ) 0. (b) V 2 W 1 W 1. (c) (V 1, V 2 ) is doubly commuting. (d) C(V 1, V 2 ) is a projection. (e) The fringe operator F 2 is an isometry.

17 PAIRS OF COMMUTING ISOMETRIES - I 17 Proof. The equivalences of (a) and (b), (b) and (c), and (b) and (e) are given in Lemma 6.3, Lemma 6.2 and Lemma 6.4, respectively. The implication (c) implies (d) follows from C(V 1, V 2 ) = P W1 P W2 = P W2 P W1. Clearly (d) implies (a). This completes the proof. We now prove that for a large class of pairs of commuting isometries negative defect operator always implies the zero defect operator. Theorem 6.6. Let (V 1, V 2 ) be a pair of commuting isometries on H. Suppose that V 1 or V 2 is pure. Then C(V 1, V 2 ) 0 if and only if C(V 1, V 2 ) = 0. Proof. With out loss of generality assume that V 2 is pure. If C(V 1, V 2 ) 0, then by Lemma 6.3, we have P W1 P V2 W 1, or, equivalently and hence for all m 0. Therefore W 1 V 2 W 1, W 1 V 2 m W 1 V 2 m H, W 1 = V 2 m W 1 V 2 m H = {0}, as V 2 is pure. Hence W 1 = {0} and V 2 W 1 = {0}. This gives C(V 1, V 2 ) = P W1 P V2 W 1 = 0. The same conclusion holds if we allow dim W j < for some j {1, 2}. Theorem 6.7. Let (V 1, V 2 ) be a pair of commuting isometries on H. Suppose that dim W j < for some j {1, 2}. Then C(V 1, V 2 ) 0 if and only if C(V 1, V 2 ) = 0. Proof. We may suppose that dim W 1 <. Let C(V 1, V 2 ) 0. Since W 1 V 2 W 1 and V 2 is an isometry, it follows that W 1 = V 2 W 1. Hence C(V 1, V 2 ) = P W1 P V2 W 1 = 0. This completes the prove. The same conclusion also holds for positive defect operators.

18 18 A. MAJI, J. SARKAR, AND SANKAR T. R. 7. Concluding Remarks As pointed out in the introduction, a general theory for pairs of commuting isometries is mostly unknown and unexplored (however, see Popovici [P]). In comparison, we would like to add that a great deal is known about the structure of pairs (and even of n-tuples) of commuting isometries with nite rank defect operators (see [BKS], [BKPS1], [BKPS2]). A complete classication result is also known for n-tuples of doubly commuting isometries (cf. [GS], [S], [JS]). It is now natural to ask whether the present results for pure pairs of commuting isometries can be extended to arbitrary pairs of commuting isometries (see [D], [GG] and [GS] for closely related results). Another relevant question is to analyze the joint shift invariant subspaces of the Hardy space over the unit bidisc [ACD] from our analytic and geometric point of views. More detailed discussion on these issues will be given in forthcoming papers. Also we point out that some of the results of this paper can be extended to n-tuples of commuting isometries and will be discussed in a future paper. We conclude this paper by inspecting a connection between the Sz.-Nagy and Foias characteristic functions of contractions on Hilbert spaces [NF] and the analytic representations of Π 1 and Π 2 as described in Theorem 5.1. Let T be a contraction on a Hilbert space H. The defect operators of T, denoted by D T and D T, are dened by D T = (I T T ) 1/2, D T = (I T T ) 1/2. The defect spaces, denoted by D T and D T, are the closure of the ranges of D T and D T, respectively. The characteristic function [NF] of the contraction T is dened by θ T (z) = [ T + zd T (I zt ) 1 D T ] DT (z D). It follows that θ T HB(D T,D T ) (D) [NF]. The characteristic function is a complete unitary invariant for the class of completely non-unitary contractions. This function is also closely related to the Beurling-Lax-Halmos inner functions for shift invariant subspaces of vector-valued Hardy spaces. For a more detailed discussion of the theory and applications of characteristic functions we refer to the monograph by Sz.-Nagy and Foias [NF].

19 PAIRS OF COMMUTING ISOMETRIES - I 19 Now let us return to the study of pairs of commuting isometries. Let (V 1, V 2 ) be a pair of commuting isometries on H. We compute P W1 [I H + z(i H zv 1 ) 1 V 1 ] W = [P W1 + zp W1 (I H zv 1 ) 1 V 1 ] W Since V 1 W = W 2, it follows that = [I H V 1 V 1 + zp W1 (I H zv 1 ) 1 V 1 ] W = I W + [ V 1 + zp W1 (I H zv 1 ) 1 ]V 1 W. [ V 1 +zp W1 (I H zv 1 ) 1 ]V 1 W = [ V 1 +zd V 1 (I H zv 1 ) 1 D V 2 ] DV 2 (V 1 W ). Therefore, setting (7.1) θ V1,V 2 (z) = [ V 1 + zd V 1 (I H zv 1 ) 1 D V 2 ] DV 2, for z D, we have P W1 [I H + z(i H zv 1 ) 1 V 1 ] W = I W + θ V1,V 2 (z)v 1 W, for all z D. Therefore, if V 1 is a pure isometry, then the formula for Π 1 in Theorem 5.1(i) can be expressed as Π 1 (S(, w)η) = (I W1 wθ V2 (z)) 1 P W1 [I W + θ V1,V 2 (z)v 1 W ]η. for all w D and η W. Similarly, if V 2 is a pure isometry, then the formula for Π 2 in Theorem 5.1 (ii) can be expressed as Π 2 (S(, w)η) = (I W2 wθ V1 (z)) 1 P W2 [I W + θ V2,V 1 (z)v 2 W ]η, for all w D and η W, where (7.2) θ V2,V 1 (z) = [ V 2 + zd V 2 (I H zv 2 ) 1 D V 1 ] DV 1, for all z D. It is easy to see that θ Vi,V j (z) B(W j, W) for all z D and i j. Note that since the defect operator D Vj = 0, the characteristic function θ Vj of V j, j = 1, 2, is the zero function. From this point of view, it is expected that the pair of analytic invariants {θ Vi,V j : i j} will provide more information about pairs of commuting isometries. Subsequent theory for pairs of commuting contractions and a more detailed connection between pairs of commuting pure isometries (V 1, V 2 ) and the analytic invariants {θ Vi,V j : i j} as dened in (7.1) and (7.2) will be exhibited in more details in future occasion.

20 20 A. MAJI, J. SARKAR, AND SANKAR T. R. Acknowledgements. The authors are grateful to the anonymous reviewers for their critical and constructive reviews and suggestions that have substantially improved the manuscript. The rst author's research work is supported by NBHM Post Doctoral Fellowship No. 2/40(50)/2015/ R & D - II/ The research of the second author was supported in part by (1) National Board of Higher Mathematics (NBHM), India, grant NBHM/R.P.64/2014, and (2) Mathematical Research Impact Centric Support (MATRICS) grant, File No : MTR/2017/000522, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. [ACD] References O. P. Agrawal, D. N. Clark and R. G. Douglas, Invariant subspaces in the polydisk, Pacic J. Math., 121 (1986), 111. [A] T. Ando, On a pair of commutative contractions, Acta Sci. Math. (Szeged), 24 [BDF] [BCL] [BKS] (1963), H. Bercovici, R.G. Douglas and C. Foias, On the classication of multiisometries, 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), Z. Burdak, M. Kosiek and M. Slocinski, The canonical Wold decomposition of commuting isometries with nite dimensional wandering spaces, Bull. Sci. Math. 137 (2013), [BKPS1] Z. Burdak, M. Kosiek, O. Pagacz and M. Slocinski, On the commuting isometries, Linear Algebra Appl. 516 (2017), [D] [FF] [GG] [GS] [BKPS2] Z. Burdak, M. Kosiek, O. Pagacz and M. Slocinski, Shift-type properties of commuting, completely non doubly commuting pairs of isometries, Integral Equations Operator Theory 79 (2014), R. G. Douglas, On the C -algebra of a one-parameter semigroup of isometries, Acta Math., 128 (1972), C. Foias and A. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, 44. Birkhäuser Verlag, Basel, D. Gaspar and P. Gaspar, Wold decompositions and the unitary model for bi isometries, Integral Equations Operator Theory 49 (2004), D. Gaspar and N. Suciu, Wold decompositions for commutative families of isometries, An. Univ. Timisoara Ser. Stint. Mat., 27 (1989), [H] P. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961)

21 PAIRS OF COMMUTING ISOMETRIES - I 21 [HQY] W. He, Y. Qin, and R. Yang, Numerical invariants for commuting isometric [HH] [NF] [P] [QY] [RR] pairs, Indiana Univ. Math. J. 64 (2015), 119. H. Helson, Lectures on invariant subspaces, Academic Press, New York-London B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space. North-Holland, Amsterdam-London, 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), M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Corrected reprint of the 1985 original. Dover Publications, Inc., Mineola, NY, [R] W. Rudin, Function Theory in Polydiscs, Benjamin, New York, [S] [JS] [VN] M. Slocinski, On the Wold-type decomposition of a pair of commuting isometries, Ann. Polon. Math. 37 (1980), J. Sarkar, Wold decomposition for doubly commuting isometries. Linear Algebra Appl. 445 (2014), J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann., 102 (1929), [W] H. Wold, A study in the analysis of stationary time series, Stockholm, Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, , India address: amit.iitm07@gmail.com Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, , India address: jay@isibang.ac.in, jaydeb@gmail.com Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, , India address: sankartr90@gmail.com