Dilations and Commutant Lifting for Jointly Isometric OperatorsA Geometric Approach

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1 jounal of functonal analyss 140, (1996) atcle no Dlatons and Commutant Lftng fo Jontly Isometc OpeatosA Geometc Appoach K. R. M. Attele and A. R. Lubn Depatment of Mathematcs, Illnos Insttute of Technology, Chcago, Illnos Receved Febuay 2, 1993; evsed Octobe 18, 1994 A tuple of commutng contactons T=(T 1, T 2,..., T n ) s called a jont-somety f T* j T j =I. We gve a geometc poof that jont sometes have a egula untay dlaton and that ts commutant lfts. We also show that T s subnomal and that ts mnmal nomal extenson s also jontly sometc Academc Pess, Inc. 1. INTRODUCTION Let H be a Hlbet space and T=[T j ] be a system of commutng contactons on H. It sad to be a jont-somety f &T (x)& 2 =&x& 2 fo evey x # H, o, equvalently, f T *T =I n the WOT. Athavale [2], usng hs pevous wok on opeato-valued kenels, poved that fnte jontly sometc systems ae subnomal, and that the elements n the commutant lft to the commutatnt of the commutng nomal extenson. Poposton 3, whch may be vewed as a commutant lftng theoem [13, p. 96], follows easly fom Athavale's esults n the case of fnte systems, howeve, ou appoach s puely geometc, and conceptually moe elementay than Athavale's wok; moeove, t holds fo nfnte systems. Ou technques also gve a dect poof of the subnomalty of jontly-sometc systems. 2. DILATIONS OF JOINT-ISOMETRIES A commutatve system, fnte o nfnte, S=[S j ] j # J of bounded lnea opeatos on a Hlbet space K#H s sad to be a dlaton of a commutatve system T=[T j ] j # J of bounded lnea opeatos on a Hlbet space H f T n 1 j 1 }}}T n j =P(S n 1 j 1 }}}S n j ) fo all choces of nteges n 0, =1,..., and fo evey fnte set of subscpts j # J, whee P denotes the othogonal pojecton of K onto H. The Copyght 1996 by Academc Pess, Inc. All ghts of epoducton n any fom eseved.

2 JOINT-ISOMETRIC DILATIONS 301 ndex set J may be, a usually unstated, [1, 2,..., m] fo some postve ntege m o the ente set of postve nteges. The above dlaton S s called untay f each S j # S s untay, and t s well-known that a tuple of two contactons has a untay dlaton, but thee exst tuples of thee contactons fo whch no untay dlaton exsts. Commutant lftng theoems concen opeatos on H commutng wth T that have dlatons on K commutng wth S. Sz-Nagy and Foas's book, [13], povdes an excellent geneal efeence fo these topcs. Fo the est of ths secton gven mult-ndex I=(n 1,..., n ), I wll denote the sum n 1 + }}} + n, I!=n 1!}}}n!, I + =(n +, 1...,n+), I& = (n &,..., 1 n& ), whee n+ =sup(n, 0) and n & =sup(&n,0); = k denotes the mult-ndex whose kth enty s 1 and all othe entes ae 0. Gven fxed, we denote I=[I=(n,..., n )n s 0 o 1]. Fo a fnte set u of postve nteges, I<u wll mean I # I wth =max u and n =0 fo all u. Gven a commutng system [T j ] opeatos, denote T I =T n 1 1 }}}T n. Defnton. A untay dlaton S of a commutatve tuple of contactons T s called egula f fo all mult-ndces I of nteges (T I& )* (T I+ )=PS I. Ou poofs heavly ely upon the followng theoem. Theoem 1 [13, p. 37]. only f [T j ] j # J has a egula untay dlaton f and S(u)= I<u (&1) I (T I )* T I 0 fo evey fnte subset u of J. We may eque that the egula dlaton be mnmal,.e., that the subspaces (S I H), I # Z J, span K; n ths case, the egula dlaton s unque up to an somophsm. A coollay of ths esult states [13, p. 39] that T has a egula untay dlaton f &T & 2 1. We begn wth a staghtfowad genealzaton of ths esult. Poposton 2. Let T=[T j ] j # J be a commutatve system of jontcontactons,.e., &T j (x)& 2 &x& 2 fo evey x # H. Then T has a egula untay dlaton. Poof. Let u be a fnte subset of J, and wthout loss of genealty, let u=[1, 2,..., ]. Fo I # I, h # H, and p=1, 2,...,, let

3 302 ATTELE AND LUBIN a p (h)= I =p &T I h& 2 j=1 I =p&1 I =p&1\ j=1 = I =p&1 =a p&1 (h), &T I h& 2 &T j T I h& 2 &T j T I h& 2+ whee the second nequalty follows because T s an jont contacton. Hence, (S(u)h, h)= \ I # I(&1) I (T I )* T I h, h + = p=0 (&1) p a p (h) =[a 0 (h)&a 1 (h)]+[a 2 (h)&a 3 (h)]+ }} } a 0 (h)&a 1 (h) =&h& 2 & 0. j=1 &T j h& 2 Thus, S(u)0 whch mples T has a egula untay dlaton. K Snce von Neumann's nequalty, &p(t 1,..., T )&sup[ p(z) z 1] fo all -vaable polynomals p, follows fom the exstence of a untay dlaton fo T, we have as an mmedate coollay to Poposton 1 the esult poven n [7] that jont contactons satsfy von Neumann's nequalty. (Compae ths wth [12].) Ou next poposton eques a combnatoc lemma essentally concened wth pattonng the set of combnatons of objects p+1 at a tme nto classes each element of whch contans a common combnaton of the objects taken p at a tme. Ths s pobably well-known but ou lemma wll establsh the notaton used n the poof. Note that we use mult-ndex notaton n leu of combnatons.

4 JOINT-ISOMETRIC DILATIONS 303 Lemma 3. Let I p =[I # I I =p] fo p<. Then thee ae C p = [L 1,..., L np ]/I p and D L1,..., D Lnp /[1, 2,..., ] such that n p I p+1 =. B L =1 whee B L =[I # I p+1 I=L += k, k # D L ] and [B L ] ae pawse dsjont. Poof. Let L 1 # I p be abtay and let D L1 =[k L 1 has 0 n the kth coodnate]. Note that ths detemnes B L1. Suppose L 1,..., L l, D L1,..., D Ll have been chosen such that [B L =1,..., l] ae pawse dsjont and each &1 D L = {k L has 0 n the kth coodnate and (L += k ). j=1 B Lj=. If l B =1 L =I p+1, let n p =l and the constucton s complete. Othewse, choose I # I p+1, I l B =1 L and let L (l+1) =I&= k0 fo any = k0 such that I has 1 n that coodnate. D L(l+1) defned as above contans at least the one element k 0, and hence ou pocess must temnate afte fntely many steps. K Note that the constucton s not unque even up to pemutatons and the cadnaltes of the D L n geneal dffe. The followng theoem, poved by dffeent means, appeas n [3, Pop. 8] fo fnte tuples. Poposton 4. Let T=[T j ] j # J be a jont-somety on H. Let A be a contacton on H such that AT j =T j A fo all j. Then T 0 =(T_[A]) has a egula untay dlaton. Poof. Denote A=T 0 and J 0 =[0] _ J and let u be a fnte subset of J 0.If0u, then, usng the pevous notaton, S(u)0 by Poposton 8. Hence wthout loss of genealty, let u=[0, 1,..., ] and h # H. Then (S(u) h, h)=&h& 2 & \ + \ j=1 1j 1 <j 2 &T j h& 2 +&Ah& 2+ &T j1 T j2 h& (&1) +1 &T 1 }}}T Ah& 2 j=1 &T j Ah& +}}} 2+

5 304 ATTELE AND LUBIN =&h& 2 + p=1 (&1) p_ I # I p &T I h& 2 + I # I p&1 &T I Ah& 2& +(&1) +1 &T 1 }}}T Ah& 2 =(&h& 2 &&Ah& 2 )+ = p=0 s = p=0 s p=0 (&1) p Q p p=1 (Q 2p &Q 2p+1 )+Q 2s+2 (Q 2p &Q 2p+1 ), whee s s the geatest ntege stctly less than 2 and (&1) p\ I # I p &T I h& 2 & I # I p &T I Ah& 2+ Q p = I # I p (&T I h& 2 &&T I Ah& 2 ), p=0, 1,..., and Q 2s+2 s ethe Q o 0 dependng upon s even o odd. We note that Q p 0 fo all p snce &T I Ah& 2 =&AT I h& 2 &A& 2 &T I h& 2 and &A&1. We wll conclude the poof by showng that Q 2p &Q 2p+1 0 fo p=0, 1,..., s. We use the notaton fom the pevous lemma and let C$ p be the elatve complement C$ p =(I p "C p ), p=0,..., 2s; also let D$ L =J"D L (note that D L /u=[1,..., ]/J). Note that fo any x # H and any p, I # I 2p+1 &T I x& 2 = I # C 2p K # B I &T K x& 2 = I# C 2p l # D I &T l T I x& 2 by the pattonng of I 2p+1 n the lemma and that &x& 2 = j # J &T j x& 2 = j # D L &T j x& 2 + j # D$ L &T j x& 2.

6 JOINT-ISOMETRIC DILATIONS 305 Now, Q 2p &Q (2p+1) = \ I # I 2p &T I h& 2 & I # I (2p+1) &T I h& 2+ & \ 2+ &T I Ah& 2 & &T I Ah& I # I 2p I # I 2p = \ I # C 2p &T I h& 2 + I # C$ 2p &T I h& 2 & I # C 2p l # D I &T l T I h& 2+ & \ 2+ &T I Ah& 2 + &T I Ah& 2 & &T l T I Ah& I # C 2p I # C$ 2p I # C 2p l # D I = _ I # C 2p\ &T I h& 2 & &T l T I h& 2 l # D I + 2& + &T I h& I # C$ 2 p + 2& + &T I Ah& I # C$ 2p 2++ &T l T I Ah& l 2+ # D$ I &T I Ah& I # C$ 2p & _ I # C 2p\ &T I Ah& 2 & l # D I &T l T I Ah& 2 = \ I # C 2p\ &T l T I h& 2 l # D$ I + & I # C 2p\ + \ I # C$ 2p &T I h& 2 & = I # C 2p l # D$ I (&T l T I h& 2 &&AT l T I h& 2 ) + I # C$ 2p (&T I h& 2 &&AT I h& 2 ) (1&&A& 2 ) _ I # C 2p l # D$ I &T l T I h& 2 + I # C$ 2p &T I h& 2& 0, and the theoem s establshed. K 3. SUBNORMALITY A commutatve system T of opeatos on a Hlbet space s sad to be subnomal f thee exsts a system of commutng nomal opeatos N on a Hlbet space K$H such that N=T on H. A sngle somety s, clealy,

7 306 ATTELE AND LUBIN subnomal. We show that jont-sometes ae also subnomal; see Athavale papes [2, 3] fo a poof of ths and a dscusson of jont-sometes albet fom a dffeent vew pont fom ous. Now a wod about notaton Let n and m be postve nteges, m s fxed fo Poposton 1, and J=(j 1, j 2,..., j n ) be a mult-ndex whee 1j m. The collecton of all such mult-ndces J wll be denoted by F(n) (we have suppessed wtng m). Then T J stands fo the poduct of opeatos T j1 T j2 }}}T jn. To pove the subnomalty of jont-sometes, we wll use Agle's [1] cteon A contacton S s subnomal f and only f n (I&S*S) [n] = k=0 (&1) kn C k S* k S k 0, (n1). Poposton 5. Let (T 1, T 2,..., T m ) be a jont-somety. Then, (I&T *T ) [n] = T* J T J, J # F$(n) whee the pme ove F(n) ndcates that J fo J # F(n). Poof. The poof s by nducton on n, but fst notce that n+1 C k = n C k + n C k&1 nk1. Assume the clam of the poposton holds fo some n1, then (1&T* T ) [n+1] =(1&T* T ) [n] &T* (1&T* T ) [n] T = T* J T J &T* \ T* J T J+ T J # F$(n) J # F$(n) = T* J (1&T* T ) T J J # F$(n) = j{ J # F$(n) T* J T* j T j T J = J # F$(n+1) establshng the asseton fo n+1. T* J T J, We may note that fo the valdty of the poof, the commutatvty of the tuple T s not necessay; the poof stll holds f K T* J T J = T * J T J, whee J s any mult-ndex n F(n) and J s any pemutaton of J.

8 JOINT-ISOMETRIC DILATIONS 307 If (T 1, T 2,..., T m ) s a jont-somety, then T ae sub- Coollay 6. nomal. We may note that Poposton 4 holds n a C* algeba, and snce Agle's cteon s vald n the C* algeba settng, Coollay 2 holds n the C* algeba settng as well. The commutant of a subnomal opeato cannot be, n geneal, lfted to the commutant of ts mnmal nomal extenson, but n the case of jontsometes t can be done Poposton 7. Let (T 1, T 2,..., T m ) be a jont-somety. Then thee exsts nomal extensons M of T such that (M 1, M 2,..., M m ) s a jontsomety. Poof. Suppose (T 1, T 2,..., T m, S) s a jont-somety and let N K K be the mnmal nomal extenson of S. Consde the standad extenson of T,1m,toK N \ k N* k f k+ = \ k N* k T f k+ f 1,f 2,..., f n # H. We wll vefy the cteon [5], o [6, 10.4 Theoem, Chap. 2, p. 80], fo N to be well-defned and that n _c>0 % j, k=0 n (S j T f k, S k T f j )c j, k=0 (S j f k, S k f j ), (1) m N* N +N*N=I. (2) Snce S and the T 1 1m commute, the left-hand-sde of Eq. (1) s m j, k=0 (T* T S j f k, S k f j ) m = j, k=0\ 1& T* q T q &S*S +S j f k, S k f q{ j n = j, k=0 (S j f k, S k f j ) n & j, k=0 S j T q f k, S k T q f q{ j &(Sj+1 f k, S k+1 f j ) (3)

9 308 ATTELE AND LUBIN By Halmos's cteon fo subnomalty [8], o [6, Theoem 1.9, Chap. II, p. 30], the last two expessons of (3) ae non-negatve; thus nequalty (1) holds wth c=1. We wll now vefy Eq. (2) " 2 N N* k f k k" + "N k = " k 2 N* k T f k" + " k 2 N* k f k" 2 N* k Nf k" = (N* k T f k, N* l T f l )+(N* k Nf k,n* l Nf l ). Snce N s nomal, ths s = = = = (N l T f k, N k T f l )+(N l Nf k,n k Nf l ) (S l T f k, S k T f l )+(S l Sf k,s k Sf l ) (T S l f k, T S k f l )+(SS l f k, SS k f l ) (T* T S l f k, S k f l )+(S*SS l f k, S k f l ) = (S l f k, S k f l )= (N l f k, N k f l ). k, l Snce N s nomal the last expesson s &N k f k & 2, whch vefes Eq. (2). Moeove, Bam's theoem [5, Theoem 8] o [6, Theoem 10.5, Chap. II, p. 81], mples that the lftng of a nomal opeato (to the commutant of some mnmal nomal extenson) emans nomal. It s a tvalty to vefy that (N 1, N 2,..., N m, N ) s a commutng tuple, so t s, n fact, a jont-somety. Thus gven a jont-somety (T 1, T 2,..., T m ) we can now ntate an nductve pocess whee at each stage the opeatos get lfted to a jont somety; at the m&th stage the m&, m&+1,..., m opeatos ae nomal. Ths poves the poposton. K The M ae the mnmal jontly-sometc nomal extensons of (T 1, T 2,..., T m ).

10 JOINT-ISOMETRIC DILATIONS 309 Coollay 8. If (T 1, T 2,..., T m ) s an jont-somety, then T T* I. Poof. Let (M 1, M 2,..., M m T m ). Let P be the pojecton fom K to H. Then use T* =PM*, and M M* =I. K 4. DILATIONS OF JOINT-ISOMETRIES Recall that a commutatve system T=[T j ], fnte o nfnte, s called a jont-contacton f 7T* T I. The eade may want to see the example n the followng secton, n the context of the followng poposton. Poposton 9. Suppose (T 1, T 2,..., T m ) s a jont-contacton on H. Then the commutng tuple (T 1, T 2,..., T m ) s the dlaton of a jont-somety f and only f the unt ball n C m s a complete spectal set fo (T 1, T 2,..., T m ). Poof. Suppose T=(T 1, T 2,..., T n ) has a jontly-sometc dlaton N=(N 1, N 2,..., N m ) on a space K, then, by Poposton 4, we may assume that the dlaton s also nomal. The Waelboeck spectum of T, _ R (T), s contaned n the polynomally convex hull of _(N), the spectum of N n the C* algeba geneated by Nhee s the standad agument Wte P fo the pojecton of K to H, and let (* 1, * 2,..., * m )=* # _ R (T), and let p be a m vaate polynomal. Then, by the spectal mappng theoem [4], Hence p(*)#_(p(t)). p(*) &p(t)&=&p(p(n)) H & &p(n)&=&p&, whee the last nom s the sup nom ove _(N), the algebac spectum of N. Thus, n patcula, the _ R (T) s contaned n the unt ball of C m, and by Aveson's dlaton theoem [4] the unt ball s a complete spectal set. Convesely, f the unt ball n C m s a complete spectal set fo T, then, thee exsts a commutng tuple of nomal dlaton N of T such that _ R (N)B, whee B s the unt ball [4]. Thus, the C* algeba spectum of N s also contaned n the unt sphee, so N s a jont-somety. K

11 310 ATTELE AND LUBIN 5. EXAMPLES If (T 1,..., T m ) s any tuple of commutng bounded lnea opeatos on H, then (ct 1,..., ct m ) s a jont-contacton fo c suffcently small. (In fact, f [T j ] j # J s a commutve system, then [c j T j ] wll be an jont contacton f [c j ] s appopately chosen e.g., f c j <(2 &j &T j & &1 ). Thus, jont contactons ae exceedngly geneal. We gve below some examples of jontsometes and dscuss a case n whch a jont-contacton s nduced by a jont-somety. Let S be a jont-somety on H and let M/H be a jonly nvaant subspace fo S j. Let T j =PS j M = be the compesson of S j to M =, whee P s the othogonal pojecton of H onto M =. Then =PS M = s a jont contacton havng S as a jontly sometc dlaton. The followng s an example of a jont-contacton whch s not compesson of a jont-somety. Example. Let Z n =[I=( + 1,..., n )] be the set of mult-ndces of nonnegatve nteges and let B=[e I I # Z n +] be an othonomal bass fo the (abstact) Hlbet space H. Let [w I,k I # Z n +, k=1,..., n] be a bounded net of complex numbes such that w I, k w I+=k, l=w I, l w I+=l, k fo all I # Z n +, k, l=1,..., n. Defne T k on H by T k e I =w I, k e I+=k, k=1,..., n. Then T=(T 1,..., T n ) s a system of n-vaable commutng weghted shfts as dscussed n [11]. Note that T can also be consdeed as shfts on a weghted sequence space. Clealy, T s a jont-somety f n w k=1 I, k 2 =1 fo all I and many systems [w I, k ] satsfyng ths condton can be easly be constucted. Consde weghts such that T* k e I =( k I ) 12 e I&=k o 0 dependng upon f k >0 o k =0. Note that [T* k ] s the system of n-vaable weghted shfts that appeaed n [1011]. If I{0, n &T* j=1 j e I & 2 = n j=1 j I =1=&e I & 2 and T* j e 0 =0, j=1,..., n. Hence, [T* j ] s jontly-contactve. Suppose [S j ] s a jontly-sometc dlaton of T* on K#H. Snce PS j =T j, &S j x&&t j x& fo all x # H and hence f I{0, &e I & 2 = n &S j=1 je I & 2 n &T* j=1 j e I & 2 =&e I & 2. Thus, fo j=1,..., n, &S j e I &= &T* j e I & and consequently S j e I =T* J e I fo I{0, j=1,..., n. Howeve, fo 1kn, k{j, S k e 0 =S k T* j e =j =S k S j e =j =S j S k e =j =S j 0=0.

12 JOINT-ISOMETRIC DILATIONS 311 Hence, [S j ] cannot be jontly sometc. Note that ths example s vald fo n2. REFERENCES 1. J. Agle, Hypecontactons and subnomalty, J. Ope. Theoy 13 (1985), A. Athavale and S. Pedesen, Moment poblems and subnomalty, J. Math. Anal. Appl. 146 (1990), A. Athavale, On the ntetwnng of jont sometes, J. Ope. Theoy 23 (1990), W. B. Aveson, Subalgebas of C*-algebas, Acta Math. 123 (1972), J. Bam, Subnomal opeatos, Duke Math. J. 22 (1955), J. B. Conway, ``The Theoty of Subnomal Opeatos,'' Mathematcal Suveys and Monogaphs, Vol. 36, Ame. Math. Soc., Povdence, RI, S. W. Duy, A genealzaton of von Neumann's nequalty to the complex ball, Poc. Ame. Math. Soc. 68 (1978), P. R. Halmos, Nomal dlatons and extensons of opeatos, Summa Bas. Math. 2 (1950), N. P. Jewell and A. R. Lubn, Commutng weghted shfts and analytc functon theoy n seveal vaables, J. Ope. Theoy 1 (1979), A. R. Lubn, Models fo commutng contactons, Mchgan Math. J. 23 (1976), A. R. Lubn, Weghted shfts and poducts of subnomal opeatos, Indana Unv. Math. J. (1977) A. R. Lubn, On von Neumann's nequalty, Intenat. J. Math. Sc. 1 (1978), B. Sz-Nagy and C. Foas, ``Hamonc Analyss of Opeatos on Hlbet Space,'' Noth- Holland, Amstedam, 1970.