= P HV n. T n 1 T m 2

Transcription

1 ON A GENERALIZATION OF ANDO S DILATION THEOREM NIRUPAMA MALLICK AND K SUMESH Abstract We introduce the notion of Q-commuting operators which is a generalization of commuting operators We prove a generalized version of commutant lifting theorem and Ando s dilation theorem in the context of Q-commuting operators arxiv: v1 [mathfa 10 Oct Introduction Through out this article H, K denote complex Hilbert spaces and B( ) denote the space of all bounded linear maps Let H i K i, i = 1, 2 be Hilbert spaces and T B(H 1, H 2 ) Then an operator S B(K 1, K 2 ) is said to be an (i) extension of T if S(H 1 ) H 2 and Sh = T h for all h H 1 (In such cases we write S H1 = T ) (ii) lifting of T if S(H 1 ) H 2 and T = P H 2 S H1 (equivalently S H2 = T ) (iii) dilation of T if T n = P H2 S n H1 for all n 0, where P H2 B(K 2 ) is the projection onto H 2 In each case, with respect to the decompositions K i = H i Hi, i = 1, 2, the operator S has the matrix form [ [ T T 0 S =, S =, S n = 0 [ T n n 0 respectively Note that S is a lifting of T if and only if S is an extension of T Clearly extension and lifting are dilations Existence of isometric lifting and unitary dilation of a contraction are well known (see [3, 7) Sz-Nagy proved that given any contraction T B(H) there exists a Hilbert space K H and an isometry V B(K) such that V is a dilation of T Such a pair (V, K) is called an isometric dilation of T, and is unique up to unitary equivalence if it is minimal, ie, K = span{v n (H) : n 0} (11) Using the Wold decomposition of the isometry V B(K) one can construct a Hilbert space K K and a unitary U B(K ) which is an extension and hence a dilation of V Thus every contraction T B(H) has an unitary dilation (U, K), which is unique up to unitary equivalence if it is minimal, ie, K = span{u n (H) : n Z} (12) In [9 Schaffer gave an elementary proof of the existence of minimal unitary dilation of a contraction Given a contraction T B(H) the lower right-hand corner of the matrix form of the Schaffer s unitary dilation of T gives a co-isometric extension (W, K) of T, which is said to be minimal if K = span{w n (H) : n 0} (13) Now, by taking adjoints, it follows that every contraction T B(H) has an isometric lifting (hence a dilation) which is minimal in the sense that (11) holds From the uniqueness property it follows that H is invariant for every minimal isometric dilation of T Thus (V, K) is a minimal isometric dilation of Date: October 12, Mathematics Subject Classification 46A20 Key words and phrases Isometric dilation, unitary dilation, co-isometric extension, commutant lifting, Ando s dilation theorem 1

2 2 NIRUPAMA MALLICK AND K SUMESH T if and only if (V, K) is a minimal isometric lifting of T if and only if (V, K) is a minimal co-isometric extension of T Ando ([1) proved that given any two commuting contractions T 1, T 2 B(H) there exist a Hilbert space K 0 H and commuting isometries V 1, V 2 B(K 0 ) such that T n 1 T m 2 = P HV n 1 V m 2 H for all n, m 0 In fact, V i can be chosen to be a lifting of T i, i = 1, 2 Further using Ito s theorem [2 he concluded that there exists a Hilbert space K H and commuting unitary operators U 1, U 2 B(K) such that T1 n T 2 m = P HU1 n U 2 m H for all n, m 0 Let T i B(H i ) be a contraction with isometric lifting (V i, K i ) (respectively co-isometric extension (W i, K i )) i = 1, 2 Suppose X B(H 1, H 2 ) intertwine T 1 and T 2, ie, XT 1 = T 2 X Then due to Sz-Nagy and Foias ([6,[7) there exists a norm-preserving lifting (respectively extension) Y of X which intertwine V 1 and V 2 (respectively W 1 and W 2 ) This is called intertwining lifting (respectively intertwining coextension) theorem The case when T 1 = T 2 and V 1 = V 2 (respectively W 1 = W 2 ) is known as commutant lifting (respectively commutant co-extension) theorem In [4 Sebestyen proved analogues of commutant lifting theorem and Ando s dilation theorem for anticommuting pair (ie, T 2 T 1 = T 1 T 2 ) of contractions Two operators T 1, T 2 B(H) are said to be q-commuting if T 2 T 1 = qt 1 T 2, where q C Recently Keshari and Mallick ([10) extended results of Ando, Sebestyen, Sz-Nagy and Foias into the context of q-commuting operators with q = 1 They proved a generalized version of commutant lifting theorem, intertwining lifting theorem and Ando s dilation theorem in the context of q-commuting operators, and called them as q-commutant lifting theorem, q-intertwining lifting theorem and q-commutant dilation theorem respectively In this article we introduce Q-commuting operators (see Definition 21) which generalizes the notion of q-commuting operators Our main aim is to prove an analogue of Ando-dilation theorem and commutant lifting theorem for this new class of operators Theorem 26 characterizes Q-commutant of a contraction in terms of the Q-commutant of its minimal isometric dilation This is a generalized version of (q- )commutant lifting theorem ([6, 10) The proof uses Schaffer construction Further using ideas from [8 we prove generalized versions (see Theorem 28, 214) of (q-)intertwining lifting theorem Theorem 213 characterizes Q-commutant of a contraction in terms of the Q-commutant of its minimal unitary dilation Finally we prove our main theorems that Q-commuting contractions can be dilated into Q- commuting isometries (Theorem 215) and further into Q-commuting unitaries (Theorem 219) These results generalize Ando s dilation theorem and q-commutant dilation theorem 2 Main results Suppose H, K are Hilbert spaces such that H K Given any Q B(H) we let Q K (or simply Q) denotes any operator on K such that H is a reducing subspace for Q and Q H = Q Note that (Q I H ) B(K) is an example for such an operator Q If Q is a contraction or (co-)isometry or unitary, then we require Q also to be a contraction or (co-)isometry or unitary respectively Definition 21 Given Q B(H) two operators T 1, T 2 B(H) are said to be Q-commuting if one of the following happens: T 2 T 1 = QT 1 T 2 or T 2 T 1 = T 1 QT 2 or T 2 T 1 = T 1 T 2 Q If Q = I H (respectively Q = I H ), then Q-commuting means commuting (respectively anti-commuting)

3 ON A GENERALIZATION OF ANDO S DILATION THEOREM 3 [ [ 2 1 Example 22 Let T 1 = and T 2 = in M 2 (C) Note that T 1, T 2 are not commuting In fact, [ [ there does not exists any q C such that T 2 T 1 = qt 1 T 2 But Q = and Q = in M 2 (C) are 1 0 such that T 2 T 1 = QT 1 T 2 = T 1 Q T 2 Note that there is no Q such that T 2 T 1 = T 1 T 2 Q Example 23 Suppose L, R, Q, Q B(l 2 ) are the linear operators given by L(x 1, x 2, x 3, ) := (x 2, x 3, x 4, ) R(x 1, x 2, x 3, ) := (0, x 1, x 2, x 3, ) Q(x 1, x 2, x 3, ) := (0, x 2, x 3, x 4, ) Q (x 1, x 2, x 3, ) := (0, 0, x 3, x 4, ) Clearly qlr RL for all q C Note that RL = QLR = LQ R = LRQ [ Example 24 Suppose T 1 = [ and T 2 = such that T 1 and T 2 are Q-commuting 2 0 in M 2 (C) Note that there does not exists any Q M 2 (C) The above example shows that given two operators S, T B(H), there may not exists always an operator Q B(H) such that S and T are Q-commuting However, the next Lemma says that given two operators T, Q B(H), under some suitable conditions, there always exists an operator S B(H) such that S and T are Q-commuting This is a generalization of [10, Lemma 35 Recall that T B(H) is called a pure co-isometry if T T = I and T n 0 in strong operator topology Lemma 25 Suppose T B(H) is a pure co-isometry and Q B(H) is an isometry If T (H) is invariant for Q, then there exists a co-isometry S B(H) such that ST = T SQ Proof Let W = (T H) Since T is an isometry W H is a wandering subspace for T, ie, T m (W) T n (W) for all m n N {0} Moreover, since T is pure co-isometry H = n=0 T n (W) Since T (H) W and Q (W) W, for m < n and w, w W we have (QT ) n w, (QT ) m w = (QT ) n m w, w = T (QT ) n m 1 w, Q w = 0 Thus (QT ) m (W) (QT ) n (W) for all m n N {0} Define S 0 : H H by S 0 ( T n w n ) = (QT ) n+1 w n n=0 for all w n W, n 0 Then, S0 ( T n w n ) 2 = (QT ) n+1 w n, (QT ) m+1 w m m 0 = (QT ) n+1 w n, (QT ) n+1 w n n=0 = wn, w n = T n w n, T n w n = T n w n, T m w m m 0

4 4 NIRUPAMA MALLICK AND K SUMESH = T n w n 2 Thus S 0 is a well defined isometry Moreover, for w n W, n 1 we have S 0 T ( T n w n ) = S 0 ( T (n+1) w n ) = (QT ) n+2 w n = QT (QT ) n+1 w n = QT S 0 ( T n w n ), hence S 0 T = QT S 0 Take adjoint on both sides to get ST = T SQ where S = S 0 21 Lifting theorems We recall some basic facts which we will be using frequently Suppose T i B(H i ) is a contraction with dilation (V i, K i ), i = 1, 2, and let Y B(K 1, K 2 )[ be an extension [ of the X B(H 1, H 2 ) Then wrt to the decomposition K i = H i Hi X we have Y = and V n T 0 i = n i [ for all n 0 and i = 1, 2 Note that V2 ny m =, hence T2 nxm = P H2 V2 ny m H1 for all n, m 0 T n 2 Xm Similarly if Y is any lifting of X, then we have X n T2 m = P H1 Y n V2 m H2 for all n, m 0 Now we prove an analogue of commutant lifting theorem for Q-commuting operators Theorem 26 (Q-commutant lifting) Suppose Q, T B(H) are contractions and (V, K) is any isometric lifting of T Let Q B(K) and X B(H) (i) If XT = QT X, then there exists a lifting Y B(K) of X such that Y V = QV Y (ii) If XT = T QX, then there exists a lifting Y B(K) of X such that Y V = V QY Further assume that Q is a unitary (iii) If XT = T XQ, then there exists a lifting Y B(K) of X such that Y V = V Y Q In all cases T n X m = P H V n Y m H and X n T m = P H Y n V m H for all n, m 0 Moreover, Y can be chosen such that Y = X [ Proof (i) Set T = QT 0 0 T [ and X = 0 X on H H Let D = (I Q Q) 1 2 B(K) and K 0 = ran(dv ) K Let K = K ( 1 K 0) We consider H K K through the canonical identification Now define Ṽ B( K) by QV DV Ṽ = 0 I K I K0 Note that Ṽ is an isometry Also since Q H = Q and V H = T we have Ṽ h = (QV ) h = V (Q h) = T Q h = (QT ) h for all h H, ie, Ṽ H = (QT ) Thus Ṽ is an isometric lifting of QT Set V = [Ṽ 0 B( K K) 0 V Clearly V is an isometric [ lifting of the contraction T Since T X = X T, by commutant lifting theorem, there exists Ŷ = B B( K K), where B B(K, K), such that V Ŷ = Ŷ V, Ŷ H H = X and Ŷ = X Let B = [ Y Y 1 Y 2 Y 3 tr with respect to the decomposition K = K ( 1 K 0 ) Then V Ŷ = Ŷ V = Ṽ B = BV = QV Y = Y V

5 ON A GENERALIZATION OF ANDO S DILATION THEOREM 5 where Y B(K) Also Ŷ H H = X = B H = X = Y H = X so that Y is [ a lifting of X Hence X Y B Ŷ = X = X, so that X = Y (ii) Set T = on H H Let X, D, K be as in case (i) with K 0 = ran(v D) K Define Ṽ B( K) T Q 0 0 T by V Q V D Ṽ = 0 I K I K0 [Ṽ Note that V = 0 B( K K) is an isometric lifting of the contraction T, and since T X = X T, by 0 V proceeding as in case (i), we get Y B(K) [ such that V QY [ = Y V, Y H = X and Y[ = X (iii) Suppose Q is a unitary Set T = and X = on H H, and V = on K K T Q 0 0 T X 0 V Q 0 0 V Note that ( V, K K) [ is an isometric lifting of T Since T X = X T, by commutant lifting theorem, there exists a lifting Ŷ = B(K K) of X such that Ŷ V = V Ŷ and Ŷ = X Observe that Y is the Y required lifting of X This completes the proof Remark 27 In Theorem 26(iii) suppose Q is only a co-isometry, so that T X = XT Q The above proof shows that, in such case also we can get a lifting Y of X satisfying all properties except the equality V Y Q = Y V, but we get V Y = Y V Q Theorem 28 (Q-intertwining lifting) Let T i B(H i ) be a contraction with isometric lifting (V i, K i ), i = 1, 2, and let X B(H 1, H 2 ) Suppose Q B(H 2 ) and Q B(K 2 ) are contractions (i) If XT 1 = QT 2 X, then there exists a lifting Y B(K 1, K 2 ) of X such that Y V 1 = QV 2 Y (ii) If XT 1 = T 2 QX, then there exists a lifting Y B(K 1, K 2 ) of X such that Y V 1 = V 2 QY Suppose Q B(H 1 ) and Q B(K 1 ) are unitary (iii) If XT 1 = T 2 XQ, then there exists a lifting Y B(K 1, K 2 ) of X such that Y V 1 = V 2 Y Q In all cases T n 2 X m = P H2 V n 2 Y m H1 and X n T m 1 = P H2 Y n V m 1 H1 for all n, m 0 Moreover, Y can be chosen such that Y = X Proof First assume that XT 1 = QT 2 X Set [ [ [ T T = 1 0, X =, Q = [ Q = 0 T 2 I K1 0 0 Q X 0, V = [ V V 2 I H1 0 0 Q B(K 1 K 2 ) B(H 1 H 2 ), and Note that H 1 H 2 is reducing for the contraction Q and Q H1 H 2 [ = Q Since X T = Q T X and V is an isometric lifting of T, by Theorem 26, there exists a lifting Ŷ = B(K 1 K 2 ) of X such that Ŷ V = Q V Ŷ and Ŷ = X As Ŷ V = Q V Ŷ we get Y V 1 = QV 2 Y Also since Ŷ H1 H 2 = X we have Y H2 = X, ie, Y is a lifting of X Hence X Y Ŷ = X = X The case when T 2 QX = XT 1 can be proved [ similarly since TQ X = X T For[ the case when XT 1 = T 2 XQ repeat the Q 0 Q 0 above process by taking Q = B(H 1 H 2 ) and Q = B(K 1 K 2 ) 0 I H2 0 I K2 Y

6 6 NIRUPAMA MALLICK AND K SUMESH Suppose T B(H) Recall that (V, K) is an isometric lifting of a T if and only if (V, K) is an co-isometric extension of T So we can restate the Theorems 26, 28 as follows We will be using this versions later Theorem 29 (Q-commutant extension) Suppose Q, T B(H) are contractions and (W, K) is any coisometric extension of T Let Q B(K) and X B(H) (i) If XT Q = T X, then there exists an extension Y B(K) of X such that Y WQ = WY (ii) If XQT = T X, then there exists an extension Y B(K) of X such that Y QW = WY Further assume that Q is a unitary (iii) If QXT = T X, then there exists an extension Y B(K) of X such that WY = QY W In all cases T n X m = P H W n Y m H and X n T m = P H Y n W m H for all n, m 0 Moreover, Y can be chosen such that Y = X Theorem 210 (Q-intertwining extension) Let T i B(H i ) be a contraction with co-isometric extension (W i, K i ), i = 1, 2 and let X B(H 1, H 2 ) Suppose Q B(H 1 ) and Q B(K 1 ) are contractions (i) If XT 1 Q = T 2 X, then there exists an extension Y B(K 1, K 2 ) of X such that Y W 1 Q = W 2 Y (ii) If XQT 1 = T 2 X, then there exists an extension Y B(K 1, K 2 ) of X such that Y QW 1 = W 2 Y Suppose Q B(H 2 ) and Q B(K 2 ) are unitary (iii) If QXT 1 = T 2 X, then there exists an extension Y B(K 1, K 2 ) of X such that QY W 1 = W 2 Y In all cases T2 n X m = P H2 W2 n Y m H1 and X n T1 m = P H2 Y n W1 m H1 for all n, m 0 Moreover, Y can be chosen such that Y = X Remark 211 Note that case (i) is a stronger version of [5, Theorem 3 In [5 Sebestyen considered Q B(K 1 ) with the additional assumption that span{w k 1 h : h H 1, 0 k n} reduces Q for every n 0 Recall that minimal isometric dilation is an isometric lifting Thus, Theorem 26 characterizes the operators X which are Q-commutant to T in terms of the operators Y which are Q-commutant to the minimal isometric dilation V of T Next theorem characterizes such operators X in terms of the operators Y which are Q-commutant to the minimal unitary dilation of T, provided Q is a unitary To prove our result we use the following lemma Lemma 212 ([8) Suppose T B(H) is a contraction with the unique minimal co-isometric extension (W, K 0 ) Let (U, K) be the unique minimal co-isometric extension of W Then U is a unitary, and (U, K) is the unique minimal unitary dilation of T Theorem 213 Let T B(H) be a contraction with the minimal unitary dilation (U, K) Suppose Q B(H) is an unitary and let X B(H) (i) If XT = QT X, then there exists a dilation Y B(K) of X such that Y U = (Q I H )UY (ii) If XT = T XQ, then there exists a dilation Y B(K) of X such that Y U = UY (Q I H ) In all cases X n T m = P H Y n U m H and T n X m = P H U n Y m H for all n, m 0 Moreover, Y can be chosen such that Y = X

7 ON A GENERALIZATION OF ANDO S DILATION THEOREM 7 Proof We prove only the case (i), and case (ii) can be proved similarly Suppose (W, K 0 ) the minimal co-isometric extension of T From Lemma 212 and the uniqueness of minimal unitary dilation we can assume that (U, K) is the minimal co-isometric extension of W Note that H K 0 K Let Q 0 = Q I K0 H B(K 0 ) Since Q XT = T X, from Theorem 29 (iii), there exists an extension Y 0 B(K 0 ) of X with Y 0 = X such that Q 0Y 0 W = WY 0, T n X m = P H W n Y0 m H and X n T m = P H Y0 n W m H for all n, m 0 Again since Y0 W Q 0 = W Y0, from Theorem 29 (i), there exists an extension Y B(K) of Y0 with Y = Y0 such that Y U (Q I K H ) = U Y, W n Y0 m = P K0 U n Y m K0 and Y0 nw m = P K0 Y n U m K0 for all n, m 0 Note that Y has the required properties Theorem 214 Let T i B(H i ) be a contraction with the minimal unitary dilation (U i, K i ), i = 1, 2, and let X B(H 1, H 2 ) (i) Suppose Q B(H 2 ) is an unitary and XT 1 = QT 2 X Then there exists a Y B(K 1, K 2 ) such that Y U 1 = (Q I H 2 )U 2 Y (ii) Suppose Q B(H 1 ) is an unitary and XT 1 = T 2 XQ Then there exists a Y B(K 1, K 2 ) such that Y U 1 = U 2 Y (Q I H ) In all cases XT1 n = P H 2 Y U1 n H 1 and T2 nx = P H 2 U2 ny H 1 for all n 0 Moreover, Y can be chosen such that Y = X Proof is similar to that of Theorem Dilation theorems In this section we prove an analogue of Ando s dilation theorem for Q- commuting contractions Theorem 215 (Q-commuting isometric dilation) Let T 1, T 2 B(H) be contractions and Q B(H) be an unitary such that T 2 T 1 = QT 1 T 2 Then there exists a Hilbert space K H and isometries V 1, V 2 B(K) such that (i) V 2 V 1 = (Q I H )V 1 V 2 (ii) V i is a lifting (and hence a dilation) of T i so that T1 nt 2 m for all n, m 0 = P HV n 1 V m 2 H and T n 2 T m 1 = P HV n 2 V m 1 H Proof Let ( V 1, K) be the minimal isometric dilation of T 1 Since T 2 T 1 = QT 1 T 2, by Theorem 26 (i), there exists V 2 B( K) such that V 2 V1 = (Q I K H ) V 1 V2, V 2 H = T 2, V 2 = T 2 1 Suppose (V 2, K) is the minimal isometric dilation of V 2 Note that H K K Since V 1 V2 = (Q I K H ) V 2 V1, by Theorem 26 (i), we get V 1 B(K) such that Let V 1 = V 1 V 2 = (Q I K H )V 2 V 1, V1 = K V 1, V 1 = V 1 1 [ V1 0 wrt the decomposition K = K K Since A B 0 V 1 V 1 + A A V 1 V 1 + A A I V 1 V 1 I I and V 1 is isometry we have A = 0, so that V 1 K = V 1 Since Q, V 2 are isometries and V 1 K = V 1 we have V 1 V n 2 k = (Q I H )V 1 V n 2 k = V2 V 1 V n 1 2 k = V1 V n 1 2 k = (Q IH )V 1 V n 1 2 k = = V2 V 1 V 2 k = V 1 V 2 k = (Q I H )V 1 V 2 k = V 2 V 1 k = V 1 k = V 1 k = k

8 8 NIRUPAMA MALLICK AND K SUMESH = V n 2 k for every k K and n 0 Consequently (I V1 V 1) 1 2 V n 2 k 2 = V2 n k, (I V1 V 1)V2 n k = V n 2 k, V n 2 k V n 2 k, V 1 V 1 V n 2 k = V n 2 k 2 V 1 V n 2 k 2 = 0 Since K = span{v2 n k : k K, n 0}, from above equation, we get (I V1 V 1 ) = 0, ie, V 1 is an isometry Moreover, since minimal isometric dilations are liftings we have Vi H = (Vi K ) H = V i H = Ti for i = 1, 2 Thus V 1, V 2 are isometric lifting of T 1, T 2 respectively, so that (ii) follows This completes the proof Corollary 216 (Q-commuting co-isometric dilation) Let T 1, T 2 B(H) be contractions and Q B(H) be a unitary such that T 2 T 1 = T 1 T 2 Q Then there exists a Hilbert space K H and co-isometries W 1, W 2 B(K) such that (i) W 2 W 1 = W 1 W 2 (Q I H ) (ii) W i is an extension (and hence a dilation) of T i so that T1 nt 2 m = P H W1 nw 2 m H and T2 nt 1 m = P H W2 n W1 m H for all n, m 0 Proof Since T1 T 2 = Q T2 T 1, from Theorem 215, there exists Hilbert space K H and isometric lifting Wi B(K) of Ti such that W1 W2 = (Q I H ) W2 W1 Note that W i B(K), i = 1, 2 are the required co-isometric extensions Theorem 217 Let V 1, V 2 B(H) be isometries and Q B(H) be an unitary such that V 2 V 1 = QV 1 V 2 Then there exists a Hilbert space K H and unitaries U 1, U 2 B(K) such that (i) U 2 U 1 = (Q I H )U 1 U 2 (ii) U i is an extension (and hence a dilation) of V i so that V n 1 V m 2 = P H U n 1 U m 2 H and V n 2 V m 1 = P H U n 2 U m 1 H for all n, m 0 Proof Suppose ( V 2, Ĥ) is the minimal unitary dilation of V 2 Then Ĥ = span{ V n 2 H : n Z} and V 2 is an extension of V 2 Define V 1 : Ĥ Ĥ by V 1 ( V n 2 h) = ( Q V2 ) nv1 h h H, n Z, where Q = (Q IĤ H ) B(Ĥ) Then for all h, h H and n m, V 1 ( V n 2 h), V 1 ( V m 2 h ) = ( Q V2 ) nv1 h, ( Q V2 ) mv1 h = ( Q V2 ) n mv1 h, V 1 h = ( Q V2 ) n m 1 Q V2 V 1 h, V 1 h = ( Q V2 ) n m 1Q V 2 V 1 h, V 1 h = ( Q V2 ) n m 1V1 V 2 h, V 1 h = V 1 V n m 2 h, V 1 h ( by repeating above process) = V n m 2 h, h ( V 1 is isometry) = V n m 2 h, h ( V 2 H = V 2 and n m 0)

9 ON A GENERALIZATION OF ANDO S DILATION THEOREM 9 = V n 2 h, V m 2 h Thus V 1 is a well defined isometry Clearly V 1 is an extension of V 1 Moreover, Q V1 V2 = V 2 V1 on Ĥ Suppose (U 1, K) is the minimal unitary dilation (and hence an extension) of V 1, so that K = span{u n 1 (Ĥ) : n Z} Define U 2 : K K by U 2 (U1 n ĥ) = ( QU 1 ) n V2 ĥ ĥ Ĥ, n Z, where Q = (Q I K H ) B(K) is unitary As in the case of V 1, it can be verified that U 2 is also a well defined isometric extension of V 2 Clearly U 2 U 1 = QU 1 U 2 Now we shall prove that U 2 is onto, hence an unitary For, if n > 0 let K n = span{u j 1 j ĥ : 0 j n, ĥ Ĥ} = span{u1 ĥ : 0 j n, ĥ Ĥ} We prove, by induction, that U 2 maps K n onto K n for every n > 0 Suppose n = 1 Then for all 0 j 1 j and ĥ Ĥ we have U V 1 2 Q jĥ K 1 and U 2 (U j 1 V 2 Q j ĥ) = ( QU 1 ) j V2 V 2 Qj ĥ = U j 1 Q j Qj ĥ = U j 1 ĥ Thus U 2 (K 1 ) = K 1 Assume that U 2 maps K n onto K n To prove that U 2 maps K n+1 onto K n+1 it is enough to prove that U (n+1) n 1 ĥ has a pre-image for every ĥ Ĥ Since U1 ĥ K n there exists x K n such that U 2 (x) = U1 nĥ Note that H Ĥ K n K, hence K n is reducing for Q Therefore there exists z K n such that U 2 (z) = QU 2 x Clearly U1 (z) K n+1, and U 2 (U1 z) = U 1 Q U 2 (z) = U1 U 2x = U1 n+1 ĥ Thus U 2 maps K n+1 onto K n+1 By induction we conclude that U1 nĥ has a pre-image under U 2 for all n > 0, ĥ Ĥ But, as U 2 ( V 2 V n 1 ĥ) = V 2 ( V V 2 1 n n ĥ) = V 1 ĥ = U 1 n ĥ we have U1 nĥ also has pre-image for every n 0, ĥ H Since K = span{u 1 n (Ĥ) : n Z} it implies that U 2 is onto Note that (U 1, U 2, K) is the required triple Corollary 218 Let W 1, W 2 B(H) be co-isometries and Q B(H) be an unitary such that W 2 W 1 = W 1 W 2 Q Then there exist a Hilbert space K H and unitaries U 1, U 2 B(K) such that (i) U 2 U 1 = U 1 U 2 (Q I H ) (ii) U i is a lifting (and hence a dilation) of W i so that W1 nw 2 m = P H U1 nu 2 m H and W2 nw 1 m = P H U2 n U1 m H for all n, m 0 Proof Since Wi s are isometries satisfying W 2 W 1 = QW 1 W 2, from above theorem there exists Hilbert space K H and unitaries U 1, U 2 B(K) such that Ui s are extensions of Wi s with U 2 U1 = (Q I H )U1 U 2 Combining Theorems 215, 217 and Corollaries 216, 218 we have the following analogue of Ando s theorem for Q-commuting contractions Theorem 219 (Q-commuting unitary dilation) Let T 1, T 2 B(H) be contractions and Q B(H) be an unitary such that T 2 T 1 = QT 1 T 2 (respectively T 2 T 1 = T 1 T 2 Q) Then there exist Hilbert space K H and unitaries U 1, U 2 B(K) such that (i) U 2 U 1 = (Q I H )U 1 U 2 (respectively U 2 U 1 = U 1 U 2 (Q I H )) (ii) T1 nt 2 m = P HU1 nu 2 m H and T2 nt 1 m = P HU2 nu 1 m H for all n, m 0

10 10 NIRUPAMA MALLICK AND K SUMESH Acknowledgment The first author thanks the Department of Atomic Energy (DAE), Government of India for financial support and IMSc Chennai for providing necessary facilities to carry out this work We thank B V R Bhat for helpful suggestions References [1 T Ando, On a pair of commutative contractions, Acta Sci Math (Szeged) 24 (1963), [2 T Ito, On the commutative family of subnormal operators, J Fac Sci Hokkaido Univ Ser I 14 (1958), 1 15 [3 B Sz-Nagy, Sur les contractions de l espace de Hilbert, Acta Sci Math Szeged 15 (1953), [4 Z Sebestyen, Anticommutant lifting and anticommuting dilation, Proc Amer Math Soc 121 (1994), no 1, [5 Z Sebestyen, Lifting intertwining operators, Period Math Hungar 28 (1994), no 3, [6 B Sz-Nagy and C Foias, Dilatation des commutants d opérateurs, C R Acad Sci Paris Sér A-B 266 (1968), A493 A495 MR [7 B Sz-Nagy and C Foias, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co, Amsterdam, 1970 [8 R G Douglas, P S Muhly and C Pearcy, Lifting commuting operators, Michigan Math J 15 (1968), [9 J J Schaffer, On unitary dilations of contractions, Proc Amer Math Soc 6 (1955), 322 [10 D K Keshari and N Mallick, q-commuting dilation, Proc Amer Math Soc, to appear, DOI: Nirupama Mallick: The Institute of Mathematical Sciences, IV Cross Road, CIT Campus, Tharamani, Chennai , India nirumallick@gmailcom, nirupamam@imscresacin K Sumesh: Indian Institute of Technology Madras, Sardar Patel Road, Opposite to C LRI, Adyar, Chennai, , India sumeshkpl@gmailcom, sumeshkpl@iitmacin