On the intertwinings of regular dilations

Transcription

1 ANNALES POLONICI MATHEMATICI LXVI (1997) O the itertwiigs of regular dilatios by Dumitru Gaşpar ad Nicolae Suciu (Timişoara) W lodzimierz Mlak i memoriam Abstract. The aim of this paper is to fid coditios that assure the existece of the commutat liftig theorem for commutig pairs of cotractios (briefly, bicotractios) havig ( -)regular dilatios. It is kow that i such geerality, a commutat liftig theorem fails to be true. A positive aswer is give for cotractive itertwiigs which doubly itertwie oe of the compoets. We also show that it is possible to drop the doubly itertwiig property for oe of the compoets i some special cases, for istace for semi-subormal bicotractios. As a applicatio, a result regardig the existece of a uitary (isometric) dilatio for three commutig cotractios is obtaied. 0. Itroductio. It is well kow that the theorem of B. Sz.-Nagy ad C. Foiaş regardig the liftig of the commutat of a pair of cotractios plays a importat role i the applicatios of dilatio theory i operator iterpolatio problems, optimizatio ad cotrol, i geology ad geophysics. This is excelletly illustrated i the book [5] of C. Foiaş ad A. E. Frazho. Lately, the dilatio theory method was exteded to the study of commutig multioperators by may authors (W. Mlak, M. S lociński, M. Kosiek, M. Ptak, E. Albrecht, V. Müller, R. E. Curto, F. H. Vasilescu, A. Octavio, B. Chevreau ad others). I 1993, at the B. Sz.-Nagy Aiversary Iteratioal Coferece i Szeged, C. Foiaş raised the problem of obtaiig a commutat liftig theorem for a pair of bicotractios havig regular uitary dilatios. I 1994, at the XV-th Iteratioal Coferece o Operator Theory i Timişoara, V. Müller proved that i such a geerality, the commutat liftig theorem fails. I the preset work it is our aim to fid coditios that assure the existece of such a liftig. I this frame a structure for regular (or -regular) dilatios is eeded Mathematics Subject Classificatio: Primary 47A20; Secodary 47A13. Key words ad phrases: commutig multioperator, -regular dilatio, cotractive itertwiig, (semi-)subormal pair. [105]

2 106 D. Gaşpar ad N. Suciu 1. Prelimiaries. For a complex separable Hilbert space H, B(H) meas the C -algebra of all bouded liear operators o H (with Hilbert adjoit as ivolutio). The elemets of the (closed) uit ball i B(H) are called cotractios o H. A -tuple of operators will be called a multioperator. If the members of the -tuple commute, the we have a commutig multioperator. A commutig multioperator cosistig of cotractios will be called a multicotractio (bicotractio if = 2) o H. For a multicotractio T := (T 1,..., T ) we defie T := (T 1,..., T ). We shall also use the multiidex otatio T m := T m T m, m = (m 1,..., m ) Z +, where Z (resp. Z + ) is the set of all (resp. positive) itegers. A multicotractio T o H will be briefly deoted by [H, T ]. A isometric (resp. uitary) dilatio of a multicotractio [H, T ] is a multicotractio [K, U] cosistig of isometric (resp. uitary) operators, such that K cotais H as a closed subspace ad (1) T m = P H U m H (m Z +), where P H = P K,H is the orthogoal projectio of K o H. It is kow (see [1]) that each bicotractio has a isometric (ad uitary) dilatio, ad geerally speakig, a -tuple cosistig of more tha three commutig cotractios has o isometric dilatio (see [15]). A isometric, respectively uitary, dilatio [K, U] of [H, T ] is called miimal if (2) K = U m H, or respectively, m Z + (2 ) K = m Z U m H. Let us first ote that if [K, V ] is a isometric miimal dilatio of the multicotractio [H, T ], the by (2), H is ivariat with respect to V ad Vi H = T i (i = 1,..., ). Let us also metio that if [K, V ] is a isometric miimal dilatio of [H, T ], ad [ K, U] is the miimal uitary extesio (see [18]) of [K, V ], the it is the uitary miimal dilatio of [H, T ]. O the other had, it is kow that i case of a sigle cotractio, the miimality coditio (2) or (2 ) implies that the isometric (resp. uitary) dilatio is uiquely determied up to a uitary equivalece which fixes H. But for > 1 this is ot true ([1], [18]). A isometric (resp. uitary) miimal dilatio [K, U] of the multicotractio [H, T ] is called regular (respectively -regular) if it satisfies (3) T m T m + = P H U m U m + H (m Z ),

3 or respectively, O the itertwiigs of regular dilatios 107 (3 ) T m + T m = P H U m U m + H (m Z ), where m + := (m + 1,..., m+ ), m = (m 1,..., m ) ad m + i := max{m i, 0}, m i := max{ m i, 0}. Regular dilatios were studied i [3], [10], [18] ad recetly i [4] ad [8]. Their existece is ot assured for ay multicotractio, ot eve for = 2. However, if such a dilatio exists, the by the miimality coditio (2) or (2 ) it is uiquely determied up to uitary equivalece (see [18]). It is easy to see from (3) ad (3*) that [K, U] is a regular (resp. -regular) uitary dilatio of [H, T ] iff [K, U ] is a -regular (resp. regular) uitary dilatio of [H, T ]. We also ote that if [K, V ] is a regular (resp. -regular) isometric dilatio of [H, T ] the the miimal uitary extesio [ K, U] of [K, V ] is a regular (resp. -regular) uitary dilatio. O the other had, if [K, U] is a regular uitary dilatio, by puttig K + := m Z + U m H, K + := m Z + U m H, V i := U i K+ ad V i := U i K + (i = 1,..., ), the [K +, V ] is a regular isometric dilatio of [H, T ], whereas [K +, V ] is a -regular isometric dilatio of [H, T ]. Let us also recall that a multicotractio [H, T ] has a regular isometric dilatio iff T := ( 1) m T m T m + 0, m where m := m m (see [18], [4]). A multicotractio is called a polydisc isometry ([4]) whe T = 0. It is easily see that if T i (i = 1,..., ) are isometries (i.e. T is a -toral isometry [2]), the T is a polydisc isometry. Now if I i=1 T i T i 0, the T 0. If i=1 T i T i = I, the [H, T ] is called a spherical isometry ([2]). Whe [H, T ] is a polydisc or a spherical isometry, we say that [H, T ] is a polydisc or a spherical coisometry, respectively. If the multicotractio is doubly commutig, the obviously T = (I T1 T 1 )... (I TT ) 0. Furthermore, it is easy to verify Propositio 0. For a multicotractio [H, T ] the followig statemets are equivalet: (i) [H, T ] is doubly commutig; (ii) [H, T ] has a regular isometric dilatio which is doubly commutig; (iii) [H, T ] has a regular uitary dilatio [K, U] such that [K, U ] is a regular uitary dilatio for [H, T ]. I particular, a multicotractio cosistig of coisometries has a regular isometric (or uitary) dilatio iff it is doubly commutig.

4 108 D. Gaşpar ad N. Suciu Let us observe that Propositio 0(iii) meas more tha that T has a regular ad a -regular dilatio. For example, for a bicotractio T = (T 0, T 1 ) with T T 1 2 1, we have T 0 ad T 0 but it is possible that T 0 T 1 T 1 T 0. Fially, also recall that a isometric pair [H, V ] is called a shift -tuple (see [7], [8]) or a multishift (see [4]) if there exists a waderig (closed) subspace E i H (i.e. V m E V p E, m p, m, p Z +) such that H = m Z + V m E. For the sake of simplicity we shall work i the case = Regular isometric dilatios. The isometric dilatios cosistig of doubly commutig isometries are i some sese coected with regular dilatios. Precisely this is give i Theorem 1. For a bicotractio [H, T ] with T = (T 0, T 1 ) the followig assertios are equivalet: (i) T has a doubly commutig miimal isometric dilatio; (ii) T has a miimal isometric dilatio of the form [M G, W V ], where W is a bishift o M ad V is a bidisc coisometry o G; (iii) If [K 0, S 0 ] is the miimal isometric dilatio of T 0, the there exists a cotractio S 1 o K 0 which doubly commutes with S 0, such that P H S 1 = T 1 P H ; (iv) T has a -regular isometric (uitary) dilatio. P r o o f. (i) (ii). Let [K, U] be a miimal isometric dilatio of T with U 0, U 1 doubly commutig isometries o K. By the Wold decompositio ([17], [7]) we have K = K u K s K s0 K s1, so that U 0 ad U 1 reduce each subspace ad U 0, U 1 are uitary o K u, U is a shift pair o K s ad U i is uitary (resp. a shift) o K s1 i (resp. K si ), i = 0, 1. Put G = K u K s0 K s1, V 1 i i with Vi i = U i K u K s0, V 1 i i = U i K s1 (i = 0, 1) ad V = (V 0, V 1 ), W = (V0 0, V1 1 ), W = (V0 1, V1 0 ). Because V1 1 is uitary, W is a bidisc isometry o K u K s0, ad sice V0 1 is uitary, W is a bidisc V i = V i i isometry o K s1. The V = W + W = 0, so V is a bidisc isometry o G. Therefore sice W := (U 0 K s, U 1 K s ) is a bishift o M = K s ad K = M G, W V = U, we see that the dilatio [K, U] of T has the form described i (ii). (ii) (iii). Let [M G, W V ] be as i (ii). Sice W is a bishift o M, the isometries W 0 ad W 1 doubly commute o M ([16]). Also the isometries V 0 ad V 1 doubly commute o G, because V has a -regular dilatio. Therefore the isometries U 0 = W 0 V 0 ad U 1 = W 1 V 1 doubly commute o M G.

5 Put K 0 = O the itertwiigs of regular dilatios 109 m Z + W m 0 H, S 0 = U 0 K 0, S 1 = P K0 U 1 K 0. The [K 0, S 0 ] is the miimal isometric dilatio of T 0, S 0 S 1 = S 1 S 0 ad P H S 1 = T 1 P H. Sice U 0 H = T 0 = S 0 H, we also have U 0 K 0 = S 0. Furthermore, for k = p Z + S p 0 h p K with the sequece {h p } H with fiite support, we obtai S 1 S 0k = S 1 S 0h 0 + p 1 = P K0 U 1 U 0 h 0 + p 1 S 1 S p 1 0 h p S p 1 0 S 1 h p = P K0 U 0 U 1 h 0 + p 1 S 0S p 0 S 1h p = S0S 1 h 0 + S0 S 1 S p 0 h p = S0S 1 k, where we have used the fact that P K0 U0 U 1 K 0 = S0S 1. Cosequetly, S 0 ad S 1 doubly commute o K 0. (iii) (iv). If K 0, S 0 ad S 1 are as i (iii), the Si H = T i (i = 0, 1) ad sice S 0 ad S 1 doubly commute, we have p 1 T = I T 0 T 0 T 1 T 1 + T 0 T 1 T 0 T 1 = P H (I S 0 S 0 S 1 S 1 + S 0 S 1 S 0S 1) H = P H (I S 0 S 0)(I S 1 S 1) H 0. Cosequetly, T has a -regular isometric (or uitary) dilatio. (iv) (i). Suppose T 0. Deote by M = H 2 (T 2, H) ad Z = (Z 0, Z 1 ) the shift pair (that is, Z 0 ad Z 1 are the operators of multiplicatio with the coordiate fuctios) o M. Usig Theorem 3.15 of [4], there are a Hilbert space H 1, a bicotractio N = (N 0, N 1 ) o H 1 with N 0 ad N 1 ormal operators ad with N a bidisc isometry, ad a isometry A of H i M H 1 such that AH is ivariat for (Z i N i ) ad (Z i N i ) A = AT i, i = 0, 1. The it results that T p 0 T q 1 = A (Z 0 N 0 ) p (Z 1 N 1 ) q A (p, q Z + ). Let ow [K 1, (M 0, M 1 )] be a miimal isometric dilatio of N with M 0 ad M 1 doubly commutig isometries o K 1. Put K = M K 1, U i = Z i M i (i = 0, 1). The U 0 ad U 1 are doubly commutig isometries o K. Deotig by J the embeddig of M H 1 i K, we fid that JA is a isometry of H ito K.

6 110 D. Gaşpar ad N. Suciu For m = (p, q) Z 2 + ad h H we obtai (JA) U m JAh = (JA) U p 0 U q 1 (P H 0 Ah P H1 Ah) = (JA) (Z m P H0 Ah M m P H1 Ah) = A (P H0 Z m P H0 Ah P H1 M m P H1 Ah) = A (Z m P H0 Ah N m P H1 Ah) = A (Z 0 N 0 ) p (Z 1 N 1 ) q Ah = T m h. Let us observe that the subspace AH is ivariat for Ui for h H we have Now defie U i Ah = U i (P H0 Ah P H1 Ah) = Z i P H0 Ah M i P H1 Ah (i = 0, 1), because = Z i P H0 Ah N i P H1 Ah = (Z i N i ) Ah = AT i h. K + = m Z 2 + U m Ah, V i = U i K + (i = 0, 1) ad B = J + A, where J + is the embeddig of M H 1 i K +. The B is a isometry of H ito K + ad we have T m = B V m B (m Z 2 +). Idetifyig H with BH i K +, we deduce that [K +, V ] is a miimal isometric dilatio of T. It remais to prove that V 0 V1 = V1 V 0. First, sice U 0 ad U 1 doubly commute o K, it results that K + is ivariat for Ui, i = 0, 1. Ideed, for k = m Z U m Ah 2 m with the sequece {h m } H with fiite + support, we obtai U 0 k = q 0 = q 0 = q 0 U 0 U q 1 Ah 0q + p 1 q 0 U q 1 U 0 Ah 0q + p 1 q 0 U q 1 AT 0 h 0q + p 1 q 0 U 0 U p 0 U q 1 Ah pq U p 1 0 U q 1 Ah pq U p 1 0 U q 1 Ah pq. Therefore U0 K + K + ad aalogously U1 K + K +. The Vi = Ui K + (i = 0, 1), ad cosequetly, V 0 V1 = V1 V 0. Hece [K +, (V 0, V 1 )] is a doubly commutig miimal isometric dilatio of T. Corollary 2. A bicotractio T o H has a regular isometric (uitary) dilatio if ad oly if T has a doubly commutig miimal coisometric extesio.

7 O the itertwiigs of regular dilatios 111 P r o o f. If T 0, the T has a doubly commutig miimal isometric dilatio [K, (W 0, W 1 )]. Hece (W0, W1 ) is a miimal coisometric extesio of T ad W0, W1 doubly commute o K. The coverse is obvious. R e m a r k. If K = K i u K i s is the Wold decompositio of K relative to W i (i the previous proof), the W 1 i reduces K i u ad K i s, i = 0, 1. Thus, the matrix of W 1 i relative to the decompositio K = Ki u K i s has diagoal form for i = 0, 1, that is, T is diagoally extedable (see [11]). Now we ca give the followig characterizatio of the double commutativity of a isometric dilatio of [H, T ]. Propositio 3. Suppose T = (T 0, T 1 ) is a bicotractio o H ad [K, (U 0, U 1 )] a miimal isometric dilatio of T. The the isometries U 0 ad U 1 doubly commute o K iff [K, U] is a -regular isometric dilatio of [H, T ]. I particular, the doubly commutig miimal isometric dilatio of T (if it exists) is uique up to uitary equivalece. P r o o f. It is ot difficult to see that the coditio (3 ) is equivalet to (4) T p i T q 1 i = P HU q 1 i U p i H (p, q Z +; i = 0, 1). Suppose that the dilatio U = (U 0, U 1 ) satisfies (4) ad let [ K, (Ũ0, Ũ1)] be the miimal uitary extesio of U. The for p, q Z + ad i = 0, 1 we have T p i T q 1 i = P HŨ q 1 ĩ U p i H = P HŨ p i Ũ q 1 i H ad we deduce that [ K, (Ũ 0, Ũ 1 )] is a regular miimal uitary dilatio for T. By Theorem 1, T has a doubly commutig miimal isometric dilatio [M, (V 0, V 1 )]. Obviously, V i satisfies (4) (i place of U i ) ad cosequetly ( ) P H U q i U p 1 i H = P HV q i V p 1 i H (p, q Z +; i = 0, 1). Let us prove that the dilatios U = (U 0, U 1 ) ad V = (V 0, V 1 ) are uitarily equivalet. Let {h } Z 2 + H be a sequece with fiite support. Sice U ad V are dilatios of T ad satisfy ( ), by defiig m := (i, j) Z 2 + ad := (p, q) Z 2 + we obtai U 2 h = (U h, U m h m ) Z 2 + m, Z 2 + = j<q (U0 i U p 0 U q j 1 h, h m ) + (U0 i U (j q) 1 U p 0 h, h m ) j q = (U p i 0 U q j i<p j<q 1 h, h m ) + i p j<q (U (i p) 0 U q j 1 h, h m )

8 112 D. Gaşpar ad N. Suciu + (U (j q) 1 U p i i<p j q = (V p i 0 V q j i<p j<q 0 h, h m ) + i p j q 1 h, h m ) + i p j<q + (V (j q) 1 V p i i<p j q = m, Z h, h m ) + i p (V m V h, h m ) = (U (i p) 0 U (j q) 1 h, h m ) (V (i p) 0 V q j 1 h, h m ) j q Z 2 + (V (i p) 0 V (j q) 1 h, h m ) V h 2. Usig the miimality coditios of the spaces K ad M ad the orm equalities above, we deduce that there exists a uitary operator W from K to M satisfyig W U h = V h Z 2 + Z 2 + for {h } H with fiite support. Cosequetly, W H = I ad W U i = V i W, i = 0, 1, ad i particular, it results that U 0 ad U 1 doubly commute o K. Sice the other assertios were also implicitly proved, the proof is fiished. Now, havig i mid the coditio (iii) of Theorem 1, we obtai Corollary 4. Let T = (T 0, T 1 ) be a bicotractio o H ad [K 0, S 0 ] (respectively [K 0, S 0 ]) the miimal isometric dilatio of T 0 (resp. T0 ). The T has a -regular (resp. regular) isometric dilatio if ad oly if T1 (resp. T 1 ) has a cotractive extesio o K 0 (resp. K 0 ) which doubly commutes with S 0 (resp. S 0 ). It is obvious (by the proof of Propositio 3) that if [K, V ] is a -regular (resp. regular) isometric dilatio of [H, T ] ad if [ K, U] is the miimal uitary extesio of V the the regular (resp. -regular) isometric dilatio of T is [K, (V 0, V 1 )], where (5) K = 0 U1 H, V i = Ui K (i = 0, 1). m, Z + U m Furthermore, with the otatios of Theorem 1(iii), the -regular isometric dilatio [K, V ] of T is the regular ad -regular isometric dilatio of the doubly commutig bicotractio S = (S 0, S 1 ) (see Propositio 0), ad i fact, [K, V 1 ] is the miimal isometric dilatio of S Itertwiigs of regular dilatios. Let H ad H be two Hilbert spaces ad T = (T 0, T 1 ) ad T = (T 0, T 1) two bicotractios o H ad H respectively. A bouded liear operator A : H H itertwies T ad

9 O the itertwiigs of regular dilatios 113 T if AT i = T i A, i = 0, 1. The operator A doubly itertwies T ad T if AT i = T i A ad AT i = T i A, i = 0, 1. V. Müller has show i [14] that if A itertwies two bicotractios which have regular dilatios, the i geeral, A caot be lifted i the sese of [5], [18] to a operator which itertwies these dilatios. I order to give coditios uder which this is possible, we will first prove Theorem 5. Let [H, T ] ad [H, T ] be two bicotractios havig - regular isometric dilatios [K, U] ad [K, U ] respectively. Let A be a cotractio from H i H such that AT i = T i A (i = 0, 1) ad AT 0 = T 0 A. The there is a cotractio B from K i K with BU i = U i B (i = 0, 1), BU0 = U 0 ad P H B = AP H. P r o o f. Let A : H H be a cotractio which satisfies AT i = T i A (i = 0, 1) ad AT0 = T 0 A. Let [K 0, S 0 ] ad [K 0, S 0] be the miimal isometric dilatios of T 0 ad T 0 respectively. By Theorem 1(iii) there are cotractios S 1 o K 0 ad S 1 o K 0 such that S 1 doubly commutes with S 0 ad P H S 1 = T 1 P H, while S 1 doubly commutes with S 0 ad P H S 1 = T 1P H. Sice K 0 = H S p 0 (S 0 T 0 )H, K 0 = H S p 0 (S 0 T 0 )H p Z + p Z + (see [5], [18]), ad A doubly itertwies T 0 ad T 0, we ca defie a cotractio A 0 : K 0 K 0 by settig A 0 k 0 := Ah + p 0 S p 0 (S 0 T 0)Ah p for k 0 = h + p Z + S p 0 (S 0 T 0 )h p, where h, h p H. We have A 0 H = A, ad for k 0 K 0 as above, [ A 0 S 0 k 0 = A 0 T 0 h + (S 0 T 0 )h + ] S p+1 0 (S 0 T 0 )h p p 0 = AT 0 h + (S 0 T 0)Ah + S p+1 0 (S 0 T 0)Ah p p 0 [ = S 0 Ah + ] S p 0 (S 0 T 0)Ah p = S 0A 0 k 0. p 0 Therefore A 0 S 0 = S 0A 0. Also, A 0 S 0 = S 0 A 0 ad A 0 H = A, because for k = p 0 Sp 0 h p (fiite sum) with h p H, we have A 0 S0k = AT0 h 0 + p 1 = S 0 Ah 0 + S 0 A 0 S p 1 0 h p = T 0 Ah 0 + S p 1 0 Ah p p 1 p 1 S p 0 Ah p = S 0 p 0 S p 0 Ah p = S 0 A 0 k,

10 114 D. Gaşpar ad N. Suciu ad for h H, (A 0h, k) = p = p (h, S p 0 Ah p) = p (T p 0 A h, h p ) = p (T p 0 h, Ah p ) = p (A h, S p 0 h p) = (A h, k). (A T p 0 h, h p ) Next, we also have A 0 S 1 = S 1A 0, because for {h p} p 0 H with fiite support, A 0S 1 S p 0 h p = A 0S p 0 S 1 h p = S p 0 A 0T 1 h p = S p 0 A T 1 h p p p p p = p S p 0 T 1 A h p = p S p 0 S 1A 0h p = S 1A 0 p S p 0 h p, ad cosequetly, A 0S 1 = S 1A 0, whece A 0 S 1 = S 1A 0. We coclude that A itertwies S 1 ad S 1 ad doubly itertwies S 0 ad S 0, ad A 0 is a extesio for A, while A 0 is a extesio for A. Hece P H A 0 = AP H. Now let [K, U 1 ], [K, U 1] be the miimal isometric dilatios of S 1, S 1 respectively, ad let U 0, U 0 be the -extesios of S 0 (o K) ad of S 0 (o K ), respectively, such that U 0 doubly commutes with U 1 ad U 0 doubly commutes with U 1. Usig the sequeces of -step dilatios for S 1 ad S 1 ad the correspodig -step itertwiig liftigs of A, we ca defie a cotractio B : K K by Bk = lim A P K k (k K), where {K } ad {A } are iductively defied with K 1 = K 0 D S1 ad A 1 : K 0 D S1 K 0 D S 1 of the form ( ) A0 0 A 1 =, X 1 D A0 Y 1 D C beig the defect space of the operator C. Here the operator (X 1, Y 1 ) : D A0 D S1 D S 1 is (X 1, Y 1 ) = Γ 0 P 0, where P 0 is the orthogoal projectio of D A0 D S1 o the subspace {D A0 S 1 k D S1 k : k K 0 } ad Γ 0 (D A0 S 1 k D S1 k) = D S 1 A 0 k, k K 0. The B satisfies BU 1 = U 1B, BU 0 = U 0B, BU0 = U 0 B ad P K 0 B = A 0 P K0 (see [9] for details). Hece P H B = BP H ad sice [K, (U 0, U 1 )] ad [K, (U 0, U 1)] are the -regular isometric dilatios for T ad T respectively, B is the desired operator. The proof is fiished. Corollary 6. Let [H, T ] ad [H, T ] be two bicotractios which have -regular (or regular) isometric dilatios with the miimal uitary extesios [ K, Ũ] ad [ K, Ũ ] respectively. If A is a cotractio from H to H which satisfies AT i = T i A (i = 0, 1) ad AT 0 = T 0 A, the there exists a cotractio à from K to K such that ÃŨi = Ũ iã (i = 0, 1) ad P H à H = A.

11 O the itertwiigs of regular dilatios 115 P r o o f. Suppose that T ad T have the -regular isometric dilatios [K, U] ad [K, U ] ad let [ K, Ũ] ad [ K, Ũ ] be the miimal uitary extesios of U ad U respectively. If A is a itertwiig cotractio of T ad T ad B is a itertwiig cotractio of U ad U with P H B = AP H give by Theorem 5, the there exists (see [12]) a cotractio B from K ito K which itertwies Ũ ad Ũ, such that B K = B. It results that P H B H = A, whece PH B H = A ad B itertwies Ũ ad Ũ. Obviously, [ K, Ũ ] ad [ K, Ũ ] are the regular uitary dilatios of T ad T respectively. Theorem 7. Let [H, T ] ad [H, T ] be two bicotractios havig regular isometric dilatios [K, U] ad [K, U ] respectively, such that T1 or T 1 is a isometry. If A is a cotractio of H ito H such that AT i = T i A (i = 0, 1) ad AT0 = T 0 A, the there exists a cotractio B from K to K with BU i = U i B (i = 0, 1), ad P H B = AP H. P r o o f. Suppose that the bicotractios T = (T 0, T 1 ) ad T = (T 0, T 1) have regular isometric dilatios. The T = (T0, T1 ) has a -regular isometric dilatio ad therefore if [K 0, S 0 ] is the miimal isometric dilatio of T0, the there is a cotractio S 1 o K 0 which doubly commutes with S 0, such that P H S 1 = T1 P H. Let [ K 0, S 0 ] be the miimal isometric dilatio of the coisometry S0 ad let S 1 be the -extesio of S1 to K 0 which doubly commutes with S 0. But S 0 is a uitary operator o K 0 ad [K 0, S 0 ] give by K 0 = S 0 H, S 0 = S 0 K 0, Z + is the miimal isometric dilatio of T 0. We have S1 H = T 1 ad therefore S 1 H = T 1. Hece K 0 is a ivariat subspace for S 1 ad S 1 = S 1 K 0 is a cotractio o K 0 which satisfies S 0 S 1 = S 1 S 0 ad S 1 H = T 1. Aalogously, if [K 0, S 0] is the miimal isometric dilatio of T 0, the there is a cotractio S 1 o K 0 which satisfies S 0S 1 = S 1S 0 ad S 1 H = T 1. Now let A : H H be a cotractio which itertwies T 1 ad T 1 ad doubly itertwies T 0 ad T 0. As i the proof of Theorem 5 there is a cotractio A 0 : K 0 K 0 which doubly itertwies S 0 ad S 0, such that A 0 H = A. The for ay sequece {h } H with fiite support we have A 0 S 1 S0 h = S 0 A 0 S 1 h = S 0 AT 1 h = S 0 T 1Ah = S 0 S 1A 0 h = S 1A 0 S0 h, therefore A 0 S 1 = S 1A 0. Let us remark that if T1 or T 1 is a isometry, the so is S1 (respectively S 1). I this case it is kow (see [5]) that A 0 has a uique cotractive itertwiig liftig of the miimal isometric dilatios

12 116 D. Gaşpar ad N. Suciu of S 1 ad S 1. Now as i the proof of Theorem 5 (see [9]) we ca obtai a cotractio B : K K, where [K, U 1 ] ad [K, U 1] are the miimal isometric dilatios of S 1 ad S 1 respectively, such that P K 0 B = A 0 P K0 ad BU 1 = U 1B, BU 0 = U 0B, U 0 ad U 0 beig the isometric extesios of S 0 ad S 0 to K ad K which commute with U 1 ad U 1 respectively. Fially, it is easy to see that [K, (U 0, U 1 )] ad [K, (U 0, U 1)] are the regular isometric dilatios of T ad T respectively. The proof is fiished. Now we ca obtai the versios of Theorems 5 ad 7 for double itertwiigs which complete those obtaied i [14]. Propositio 8. Let [H, T ] ad [H, T ] be two bicotractios havig regular (or -regular) isometric dilatios [K, V ] ad [K, V ] respectively. If A is a cotractio from H ito H which doubly itertwies T ad T, the there exists a (uique) -extesio of A from K to K which preserves the orm of A ad doubly itertwies V ad V. P r o o f. Suppose first that T ad T have -regular isometric dilatios. Let [K 0, (S 0, S 1 )] ad [K 0, (S 0, S 1)] be as i the proof the Theorem 5. Cosider A : H H a cotractive double itertwiig of T ad T ad A 0 : K 0 K 0 with A 0 S i = S i A 0 (i = 0, 1), A 0 S0 = S 0 A 0, A 0 H = A, A 0 H = A ad A 0 = A. The for {h p } p 0 H with fiite support, we have A 0 S1 S p 0 h p = p p = p S p 0 A 0S 1h p = p S p 0 S 1 A 0 h p = S S p 0 AT 1 h p = p 1 A 0 S p 0 h p, p S p 0 T 1 Ah p ad so A 0 S1 = S 1 A 0. Now let [K, V 1 ] ad [K, V 1] be the miimal isometric dilatios of S 1 ad S 1 ad let V 0, V 0 be the extesios of S 0, S 0 to K, K which doubly commute with V 1, V 1 respectively. As above, there exists a cotractio B : K K with BV i = V i B, BV i = V i B, (i = 0, 1), B K 0 = A 0, B K 0 = A 0, whece B H = A, B H = A ad B = A 0 = A. So the coclusio holds for the -regular isometric dilatios [K, V ] ad [K, V ] of T ad T. Next let [ K, Ũ], [ K, Ũ ] be the miimal uitary extesios of V, V ad [K, V ], [K, V ] be the regular isometric dilatios of T, T respectively (as i (5)). The there exists ([12], [7]) a cotractio à : K K such that ÃŨi = Ũ iã (i = 0, 1), à K = B, à K = B ad à = B. Because à H = A ad ÃŨ i = Ũ i à (i = 0, 1), we have ÃK K. But à H = A ad à Ũ i = Ũ i A (i = 0, 1) imply à K K. So we ca defie the operator C : K K by C = à K. The C = à K ad C H = A, C H = A ad sice V i = Ũ i K, V i = Ũ i K (i = 0, 1), it results that

13 O the itertwiigs of regular dilatios 117 CV i = V i C ad CV i = V i C (i = 0, 1). Fially, A C Ã = B = A ad so C = A. Thus the coclusio holds for the regular isometric dilatios of T ad T, ad cosequetly, i the case whe T ad T have regular isometric dilatios. Uder certai coditios we ca drop the doubly itertwiig property o a compoet. The first fact i this cotext is cotaied i Propositio 9. Let [H, T ] be a bicotractio with the first compoet T 0 a coisometry, ad [H, T ] be aother bicotractio which has a -regular isometric dilatio. Let [K, U] ad [K, U ] be the -regular isometric dilatios of T ad T respectively. If A is a cotractio from H to H with AT i =T i A (i = 0, 1), the there exists a cotractio B from K to K such that BU i = U i B (i = 0, 1), ad P H B = AP H. P r o o f. Let A : H H be a cotractive itertwiig of T ad T. Preservig the otatios of the proof of Theorem 5, we fid (by the liftig theorem) that there exists a cotractio A 0 from K 0 to K 0 which satisfies A 0 S 0 = S 0A 0 ad P H A 0 = AP H. Because T 0 is a coisometry, its miimal isometric dilatio S 0 is a uitary operator o K 0. The from Theorem B of [6] (which ca be exteded to operators actig o differet spaces) it results that A 0 S0 = S 0 A 0. So A 0 doubly itertwies S 0 ad S 0. Furthermore, A 0 itertwies S 1 ad S 1, the doubly commutig commutats of S 0 ad S 0 which lift T 1 ad T 1 respectively, give by Theorem 1(iii). By Theorem 5, A 0 has a cotractive lift, which itertwies the -regular isometric dilatios of S = (S 0, S 1 ) (of T ) ad S = (S 0, S 1) (of T ), whece the coclusio follows. The dual versio of Propositio 9 is i fact a extesio of Propositio 5.2 from [12] (for bicotractios) ad of Propositio 10 from [2]. Corollary 10. Let [H, T ] be a bicotractio which has a regular isometric dilatio, ad [H, T ] be a bicotractio with T 0 a isometry. We also suppose that T 1 or T 1 is a isometry. If A is a cotractio of H ito H which itertwies T ad T, the there exists a cotractio B which itertwies the regular isometric dilatios of T ad T ad satisfies P H B = AP H. P r o o f. If [K 0, S 0 ] is the miimal isometric dilatio of T 0 ad S 1 is a cotractio o K 0 with S 0 S 1 = S 1 S 0 ad S 1 H = T 1, the A 0 = AP H is a cotractio from K 0 ito H ad satisfies A 0 S 0 = T 0A 0, A 0 S 1 = T 1A 0, ad A 0 is a liftig for A. Sice S 0 ad T 0 ad respectively S1 or T 1 are isometries, there is a liftig for A 0 which itertwies the regular isometric dilatios for (S 0, S 1 ) ad (T 0, T 1). Recall ([13]) that a bouded liear operator S o H is subormal if there exists a ormal operator N o a Hilbert space K H such that H is

14 118 D. Gaşpar ad N. Suciu ivariat for N ad S = N H. If, furthermore, K = p 0 N p H, the N is said to be the miimal ormal extesio of S. I this case, N is uique (up to uitary equivalece) ad N = S. A bicotractio T = (T 0, T 1 ) will be called semi-subormal if oe of the cotractios is subormal ad the other oe has a extesio which commutes (therefore doubly commutes) with the miimal ormal extesio of the subormal oe. Such a bicotractio T has a regular isometric dilatio because, if T 0 is subormal ad N 1 is a extesio of T 1 commutig with the miimal ormal extesio N 0 of T 0, the we have T = P H (N0,N 1 ) H 0. It is easy to see that every subormal bicotractio is semi-subormal. Now we have the followig completio of Corollary 10. Propositio 11. Let T = (T 0, T 1 ) ad T = (T 0, T 1) be two semisubormal bicotractios o H ad H respectively, such that T 0 ad T 0 are subormal ad T 1 is a isometry. If A is a cotractio from H to H which itertwies T ad T ad A has a extesio which itertwies the miimal ormal extesios of T 0 ad T 0, the A has a extesio which itertwies the regular isometric dilatios of T ad T. P r o o f. Let T, T ad A be as i the hypothesis ad let [ H, N] ad [ H, N ], where N = (N 0, N 1 ) ad N = (N 0, N 1), be such that N 0 (resp. N 0) is the miimal ormal extesio o H (resp. H ) of T 0 (resp. T 0) ad N 1 (resp. N 1) is a cotractio o H (resp. H ) which exteds T 1 (resp. T 1) ad doubly commutes with N 0 (resp. N 0). From the hypothesis ad the Fuglede Putam Theorem, there is a operator à : H H which doubly itertwies N 0 ad N 0, ad à H = A. The for q 0 ad h H we have ÃN 1 N q q 0 h = ÃN = N q 0 N 1h = N q 0 0 T 1Ah = N q 0 ÃT 1 h = N q 0 AT 1h N 1Ãh = N 1ÃN q 0 h. Usig the structure of the space H, it results that ÃN 1 = N 1Ã. Let us remark that because T 1 is a isometry o H, N 1 is also a isometry o H, hece the miimal uitary extesio [K, V 1] of N 1 is just the miimal coisometry extesio of N 1. Let [K, V 1 ] be the miimal coisometry extesio of N 1 ad let V 0, V 0 be the -extesios of N 0, N 0 which doubly commute with V 1, V 1 respectively. Sice à itertwies N 1 ad N 1, there exists a cotractio Ã1 from K ito K which satisfies Ã1V 1 = V 1Ã1 ad Ã1 H = Ã. I fact, à 1 doubly itertwies V 1 ad V 1 (see [6]). Moreover, à 1 doubly itertwies the ormal operators V 0 ad V 0, because for {h } 0 H with fiite support we have

15 Ã 1 V 0 V 1 h = O the itertwiigs of regular dilatios 119 = V 1 Ã 1 V 0 h = V 1 N 0Ãh = V 1 ÃN 0 h V 1 V 0Ã1h = V 0Ã1 V 1 h. Hece Ã1 doubly itertwies the bicotractios V = (V 0, V 1 ) ad V = (V 0, V 1). The by Propositio 8, Ã 1 has a -extesio B which itertwies the regular isometric dilatios M = (M 0, M 1 ) o M of V ad M = (M 0, M 1) o M of V respectively. Settig K = M H, K = M H Z 2 + Z 2 + ad U i = M i K, U i = M i K (i = 0, 1), we deduce that [K, (U 0, U 1 )] ad [K, (U 0, U 1)] are the regular isometric dilatios of T ad T respectively. Sice BH = ÃH H ad B itertwies M i ad M i (i = 0, 1), it results that BK K. Cosequetly, B = B K is a cotractio of K i K with BU i = U i B (i = 0, 1) ad B H = A. Propositio 11 ca be applied, i particular, to the case whe T is a subormal bicotractio, which icludes the spherical isometries ([2]) ad the bidisc isometries ([4]). Now we ca give sufficiet coditios for three commutig cotractios i order to have uitary dilatios. Theorem 12. Let T 0, T 1, T 2 be three pairwise commutig cotractios o H. If the bicotractios (T 0, T 1 ) ad (T 0, T 2 ) have regular (or -regular) isometric dilatios, the (T 0, T 1, T 2 ) has a isometric (uitary) dilatio. P r o o f. Suppose first that (T 0, T 1 ) ad (T 0, T 2 ) have -regular isometric dilatios. Let [K 0, S 0 ] be the miimal isometric dilatio of T 0 ad S 1, S 2 be cotractios o K 0 which doubly commute with S 0, such that P H S i = T i P H, i = 1, 2. The it results that S1S 2 = S2S 1, ad cosequetly, S 1 S 2 = S 2 S 1 o K 0. Let [K 1, (V 0, V 1 )] be the -regular isometric dilatio of (S 0, S 1 ), therefore with V 0 ad V 1 doubly commutig isometries o K 1. By Theorem 5 there exists a cotractio V 2 : K 1 K 1 with P K0 V 2 = S 2 P K0 ad V 2 V i = V i V 2 (i = 0, 1), V 2 V 0 = V 0 V 2. Therefore V 0 doubly commutes with (V 1, V 2 ) ad by Propositios 8 or 11, if [K, (U 1, U 2 )] is the regular isometric dilatio of (V 1, V 2 ), the there is a isometry U 0 o K such that U 0 K 1 = V 0 ad U 0 commutes with U 1 ad respectively with U 2. So for m,, j Z + ad h H we have P H U m 0 U 1 U j 2 h = P K 1,HP K1 U 1 U j 2 V m 0 h = P K1,HV 1 V j 2 V m 0 h = P K0,HP K0 V j 2 V m 0 V 1 h

16 120 D. Gaşpar ad N. Suciu = P K0,HS j 2 P K 0 V m 0 V 1 h = P H S m 0 S 1 S j 2 h = T m 0 P H S 1 S j 2 h = T m 0 T 1 P H S j 2 h = T m 0 T 1 T j 2 h. Therefore [K, (U 0, U 1, U 2 )] is a isometric dilatio of (T 0, T 1, T 2 ). Now if (T 0, T 1 ) ad (T 0, T 2 ) have regular isometric dilatios, the (T 0, T 1 ) ad (T 0, T 2 ) have -regular isometric dilatios, so (T 0, T 1, T 2 ) ad cosequetly (T 0, T 1, T 2 ) have isometric dilatios. Corollary 13. Let T 0, T 1 ad T 2 be three pairwise commutig cotractios o H, such that T 0 is subormal ad (T 0, T 1 ) ad (T 0, T 2 ) are semi-subormal bicotractios. The (T 0, T 1, T 2 ) has a uitary dilatio. P r o o f. The bicotractios (T 0, T 1 ) ad (T 0, T 2 ) have regular isometric dilatios ad we apply Theorem 12. Ackowledgemets. The work of the first amed author at this research is partially supported by the Alexader vo Humboldt-Stiftug. The fial form of the article was preseted durig his visit to the Uiversität des Saarlades i Saarbrücke. The work of the secod amed author was partially completed durig his visit to the Uiversity La Sapieza i Rome. We would like to thak Professors Erst Albrecht ad Stefao Marchiafava for their hospitality. We are also very grateful to the reviewer for poitig out some misprits ad laguage errors i the first variat of the paper. Refereces [1] T. Ado, O a pair of commutative cotractios, Acta Sci. Math. 24 (1963), [2] A. Athavale, O the itertwiig of joit isometries, J. Operator Theory 23 (2) (1990), [3] S. Brehmer, Über vertauschbare Kotraktioe des Hilbertsche Raumes, Acta Sci. Math. (Szeged) 22 (1961), [4] R. E. Curto ad F. H. Vasilescu, Stadard operator models i the polydisc, Idiaa Uiv. Math. J. 42 (1993), [5] C. Foiaş ad A. E. Frazho, The Commutat Liftig Approach to Iterpolatio Problems, Birkhäuser, Basel, [6] T. Furuta, A extesio of the Fuglede Putam theorem to subormal operators usig a Hilbert Schmidt orm iequality, Proc. Amer. Math. Soc. 81 (2) (1981), [7] D. Gaşpar ad N. Suciu, Itertwiig properties of isometric semigroups ad Wold-type decompositios, i: Oper. Theory Adv. Appl. 24, Birkhäuser, Basel, 1987, [8],, O the Geometric Structure of Regular Dilatios, Oper. Theory Adv. Appl., Birkhäuser, [9],, O itertwiig liftigs of the distiguished dilatios of bicotractios, to appear.

17 O the itertwiigs of regular dilatios 121 [10] I. H a l p e r i, Sz.-Nagy Brehmer dilatios, Acta Sci. Math. (Szeged) 23 (1962), [11] M. Kosiek, A. Octavio ad M. Ptak, O the reflexivity of pairs of cotractios, Proc. Amer. Math. Soc. 123 (1995), [12] W. M l a k, Itertwiig operators, Studia Math. 43 (1972), [13], Commutats of subormal operators, Bull. Acad. Polo. Sci. Sér. Sci. Math. Astr. Phys. 19 (9) (1971), [14] V. M ü l l e r, Commutat liftig theorem for -tuples of cotractios, Acta Sci. Math. (Szeged) 59 (1994), [15] S. P a r r o t t, Uitary dilatios for commutig cotractios, Pacific J. Math. 34 (1970), [16] M. S lociński, Isometric dilatios of doubly commutig cotractios ad related models, Bull. Acad. Polo. Sci. Sér. Sci. Math. Astroom. Phys. 25 (12) (1977), [17], O the Wold-type decompositio of a pair of commutig isometries, A. Polo. Math. 37 (1980), [18] B. Sz.-Nagy ad C. Foiaş, Harmoic Aalysis of Operators o Hilbert Space, North-Hollad, Amsterdam Budapest, Departmet of Mathematics Uiversity of Timişoara Bv. V. Pârva Timişoara, Romaia gaspar@tim1.uvt.ro Reçu par la Rédactio le Révisé le