On C 0 multi-contractions having a regular dilation

Transcription

1 SUDIA MAHEMAICA 170 (3) (2005) On C 0 mult-contractons havng a regular dlaton by Dan Popovc (mşoara) Abstract. Commutng mult-contractons of class C 0 and havng a regular sometrc dlaton are studed. We prove that a polydsc contracton of class C 0 s the restrcton of a backwards mult-shft to an nvarant subspace, extendng a partcular case of a result by R. E. Curto and F.-H. Vaslescu. A new condton on a commutng mult-operator, whch s equvalent to the exstence of a regular sometrc dlaton and mproves a recent result of A. Olofsson, s obtaned as a consequence. 1. Introducton. he problem of fndng an sometrc (or a untary) dlaton of a commutng system of contractons was proposed by Sz.-Nagy [8] n the early fftes: for a gven commutng system = ( 1,..., n ) of bounded operators (called commutng mult-operator; brefly c.m.) on a Hlbert space H fnd condtons for the exstence of a c.m. V = (V 1,...,V n ) consstng of sometrc operators (or, equvalently [5], untary operators) on a Hlbert space K contanng H (as a closed subspace) such that m h = P H V m h, m Z n +, h H (f = ( 1,..., n ) and m = (m 1,...,m n ) Z n + we use the notaton m = n =1 m ; P H s just the orthogonal projecton of K onto H ). hen V s sad to be an sometrc (respectvely a untary) dlaton of. Ando [1] proved that arbtrary pars of commutng contractons have sometrc dlatons. Unfortunately, accordng to a counter-example gven by Parrott [7], the exstence s not always ensured for systems of at least three commutng contractons. By contrast to the sngle operator case, for n 2, the mnmalty condton (1) K = V m H m Z n + does not ensure the unqueness, up to untary equvalence, of the sometrc dlaton V (f such a dlaton does exst). However, f V s a mnmal sometrc 2000 Mathematcs Subject Classfcaton: 47A20, 47A13, 47A45. Key words and phrases: regular sometrc dlaton, commutng mult-contracton, polydsc contracton, class C 0. [297]

2 298 D. Popovc dlaton of then H s nvarant under V := (V1,...,V n ) and V H =, = 1,...,n: V s sad to be a mnmal co-sometrc extenson of. In addton, f U s the mnmal untary extenson of V then the restrcton of U to m Z n U m H s a mnmal sometrc dlaton of. It wll be referred + to as the mnmal sometrc dlaton of correspondng to V. It was the dea of Brehmer [2] to ntroduce a specal class of sometrc or untary dlatons: an sometrc or untary dlaton V (on K ) of (on H ) s called regular f t satsfes ( m ) m+ h = P H (V m ) V m+ h, m Z n, h H. Brehmer provded necessary and suffcent condtons on ensurng the exstence of a regular sometrc (hence also untary) dlaton, namely, (2) β := ( 1) α α α 0 0 α β for every β Z n + wth β e := (1,...,1). If t exsts, a mnmal regular sometrc dlaton s unque up to untary equvalence. A c.m. s sad to be a polydsc contracton f e 0. If s a commutng mult-contracton wth e = 0 then s a polydsc sometry ([3]). For a vector r = (r 1,...,r n ) R n and a c.m. = ( 1,..., n ) let r := (r 1 1,...,r n n ). It s not hard to observe that, for any mult-ndex β = (β 1,..., β n ) wth 0 β e, n n ( k ) β r = β + β p (1 r 2 p ) p β k p=1 β p e ( k p p ). k=1 1,..., k =1 p=1 If β s fxed and α 0 for every α wth 0 α β, we deduce that α r α, 0 α β, e r e. he followng result, whch can be seen as a partcular case of [3, Lemma 3.6] (t was recently redscovered n [6, Proposton 2.1]), s then obtaned: Proposton 1.1. If has a regular sometrc dlaton then so does r for every vector r wth e r e. It s our am n the followng to study c.m. of class C 0, namely the class of c.m. = ( 1,..., n ) whch satsfy kh k 0 (h H, = 1,...,n). s of class C 0 f belongs to the class C 0. For a c.m. of class C 0 only one of the Brehmer postvty condtons (2) must a pror be satsfed n order to obtan a regular sometrc dlaton. More precsely, extendng a partcular case of a result by R. E. Curto and F.-H. Vaslescu [3, heorem 3.16], we prove that a polydsc contracton of class C 0 s the restrcton of a backwards multshft to an nvarant subspace (heorem 2.1). Improvng a recent result by A. Olofsson [6] (Corollary 2.5), p=1

3 Mult-contractons havng a regular dlaton 299 we deduce a new necessary and suffcent condton on a c.m. n order to ensure the exstence of a regular sometrc dlaton (Corollary 2.4). By contrast to the methods n [6] (where powerful tools of measure theory are used), our proofs are drastcally smplfed by applyng (only) standard operator theory. 2. C 0 mult-contractons and dlatons. Recall from [3] that a multshft s just a doubly commutng tuple of shfts; a backwards mult-shft s the adjont of a mult-shft. Polydsc contractons of class C 0 are restrctons of backwards multshfts. More precsely, extendng a partcular case of a result by R. E. Curto and F.-H. Vaslescu [3, heorem 3.16], we have: heorem 2.1. Let = ( 1,..., n ) be a c.m. on H. he followng condtons are equvalent: () s a polydsc contracton of class C 0 ; () s the restrcton of a backwards mult-shft to an nvarant subspace; () has a (mnmal) regular sometrc dlaton and the correspondng mnmal sometrc dlaton of s a mult-shft. Proof. Let m (1 m n) be a fxed postve nteger and suppose that β 0 for every β = (β 1,...,β n ) Z n + wth β e and β := β β n = m. Easy computatons show that α α = α+e for any α Z n + wth α e and α = m 1 and any {1,...,n} \ suppα. hen α α, snce 0 α + e e and α + e = m. By teraton we deduce that (3) p α p α for every p Z +. If () holds true then p α p h, h α p h 2 p 0 for every h H. Hence α 0 by (3). We proceed nductvely to prove that a polydsc contracton of class C 0 has a regular sometrc dlaton. he correspondng mnmal sometrc dlaton V = (V 1,..., V n ) of s then doubly commutng ([4], [9]). Moreover, V p V m h = V m V p h = V p h = p h p 0 for every {1,...,n}, m Z n + and h H. he mnmalty condton (1)

4 300 D. Popovc forces V C 0 for any. herefore V s a mult-shft, and ths completes the proof of () (). If V s the mnmal sometrc dlaton of gven by () (correspondng to the mnmal regular sometrc dlaton of ) then V (whch s a backwards mult-shft) s a mnmal co-sometrc extenson of ( ) =. Condton () s obtaned. Fnally, observe that a backwards mult-shft s of class C 0 and has a regular dlaton (beng doubly commutng). he same s then true for ts restrcton to an nvarant subspace. he mplcaton () () s proved. he restrcton of a c.m. consstng of backwards shfts to an nvarant subspace obvously belongs to the class C 0, but t s not necessarly a polydsc contracton: Example 2.2. Let S be the standard (unlateral) shft on l 2 Z +, that s, S(c 0, c 1,...) = (0, c 0, c 1,...), (c p ) p 0 l 2 Z +. hen = (S,..., S ) s a c.m. on l 2 Z + consstng of backwards shfts. In addton, f B s the standard blateral shft on l 2 Z, namely B(c p ) p Z = (c p 1 ) p Z, (c p ) p Z l 2 Z, then, under the obvous dentfcaton l 2 Z + l 2 Z, U = (B,...,B ) s a untary dlaton of. However, does not have a regular sometrc dlaton (equvalently, by heorem 2.1, s not a polydsc contracton). We only have to observe that e 1+e 2 = I 2SS + S 2 S 2 fals to be a postve operator snce, for example, e 1+e 2 (0, 1, 0,...), (0, 1, 0,...) = 1. Corollary 2.3. A polydsc contracton consstng of strct contractons has a regular sometrc dlaton. A new condton on a c.m., equvalent to the exstence of a regular sometrc dlaton, s also obtaned: Corollary 2.4. has a regular sometrc dlaton f and only f there exsts a sequence r k R n +, r k e, such that r k e as k and r k s a polydsc contracton of class C 0 for all k. Proof. If has a regular sometrc dlaton then so does r for every r R n + wth r e (by Proposton 1.1). Conversely, f r k s a polydsc contracton of class C 0 then, by heorem 2.1, r k has a regular sometrc dlaton. Snce r k e as k we conclude that has a regular sometrc dlaton.

5 Mult-contractons havng a regular dlaton 301 In partcular, our methods provde a result of Olofsson [6, heorem 2.1]: Corollary 2.5. Let be a c.m. consstng of contractons. hen has a regular sometrc dlaton f and only f there exsts a sequence r k = (r k 1,...,rk n) R n +, r k < 1, such that r k e as k and r k s a polydsc contracton for all k. A smlar argument can be appled for polydsc sometres: Proposton 2.6. A c.m. of class C 0 on a non-null Hlbert space cannot be a polydsc sometry. Proof. Let = ( 1,..., n ) be a c.m. of class C 0 on a Hlbert space H. Suppose that s a polydsc sometry. Proceed as n the proof of heorem 2.1() (). Suppose that β = 0, for every β Z n + wth β e and β = m (m s fxed; 1 m n). hen, for any α Z n + wth α e and α = m 1, and for all {1,...,n} \ suppα and p Z +, (4) p α p = α. We let p n (4) to deduce that α = 0. We repeat ths step nductvely to obtan e = I = 0, {1,...,n}. Hence each s sometrc, a contradcton. Corollary 2.7. here are no polydsc sometres consstng of strct contractons. References [1]. Ando, On a par of commutatve contractons, Acta Sc. Math. (Szeged) 24 (1963), [2] S. Brehmer, Über vertauschbare Kontraktonen des Hlbertschen Raumes, bd. 22 (1961), [3] R. E. Curto and F.-H. Vaslescu, Standard operator models n the polydsc, Indana Unv. Math. J. 42 (1993), [4] D. Gaşpar and N. Sucu, On the ntertwnngs of regular dlatons, Ann. Polon. Math. 66 (1997), [5]. Itô, On the commutatve famly of subnormal operators, J. Fac. Sc. Hokkado 58 (1958), 1 5. [6] A. Olofsson, Operator valued n-harmonc measure n the polydsc, Studa Math. 163 (2004), [7] S. Parrott, Untary dlatons for commutng contractons, Pacfc J. Math. 34 (1970), [8] B. Sz.-Nagy, ransformatons de l espace de Hlbert, fonctons de type postf sur un groupe, Acta Sc. Math. (Szeged) 15 (1953),

6 302 D. Popovc [9] D. motn, Regular dlatons and models for mult-contractons, Indana Unv. Math. J. 47 (1998), Department of Mathematcs and Computer Scence Unversty of the West mşoara Bd. Vasle Pârvan nr. 4 RO mşoara, Romana E-mal: popovc@math.uvt.ro Receved January 17, 2005 Revsed verson May 10, 2005 (5567)