Wu, Pei Yuan

English

Wu Pei Yuan

University of Szeged

Acta Sci. Math., 42 (1980), 331—304 On a conjecture of Sz.-Nagy and Foia? PEI YUAN WU SZ.-NAGY and FOIA§ [ 1 0 ] conjectured that if T is a C 0 contraction with finite multiplicity and X£{T}' then T\ker X and (7,*|kerZ*)* are quasi-similar. Recently, BERCOVICI [ 1 ] and UCHIYAMA [ 1 1 ] have independently given counter-examples to this conjecture. However, in the present paper we want to establish a weaker form of this conjecture. More specifically, we will show that if T is a C0(N) contraction and X£ {T}' then there exists a 7€ {T}' such that T |kerZ and (r*|ker Y*)* are quasi-similar. Indeed, this follows from the follow-ing two main results of Section 1: If T is a C0(N) contraction, then (1) every invariant subspace for T is of the form ker X for some X£{T}' (Theorem 1.2) and (2) for every invariant subspace K for T there exists an invariant subspace L for T* such that T\K is quasi-similar to (T*\L)* (Theorem'1.3). In Section 2, we consider the corresponding question for C U contractions. It will be shown that the Sz.-Nagy and Foia§ conjecture holds for completely non-unitary (c.n.u.) CU contractions with finite defect indices. Moreover, result (1) also holds for such operators with bi-invariant subspaces replacing invariant subspaces. ((2) with the same modifications follows from the analogue of the Sz.-Nagy and Foia§ conjecture trivially.) The corresponding results for weak contractions will be considered in Section 3. In this paper only bounded linear operators on complex, separable Hilbert spaces will be considered. For an operator T, let {T}' and {T}" denote the commutant and double commutant of T, and let Lat T, Lat" T and Hyperlat T denote the lattices of invariant subspaces, bi-invariant subspaces and hyperinvariant subspaces of T, respectively. If Tx and T2 are operators on H1 and H2, respectively, (resp. T^T^ denotes that there exists an injection X: H^H2 (resp. an injection X: H^H^ with dense range, called quasi-affinity) which intertwines Tx and T2, i. e. T1X=XT2. 7\ is quasi-similar to T2 ( 7 \ ~ if TX<T2 Received May 10, 1979, and in revised form October 15, 1979. This research was partially supported by National Science Council of Taiwan. 332 Pei Yuan Wu and R 2 - < 7 \ . Readers are referred to S Z . - N A G Y and FOIA§ [6] for basic definitions and properties of contractions of various classes. 1. C0(N) contractions. If T is a C„(N) contraction, then T is quasi-similar to a uniquely determined Jordan operator of the form S(cp^)®...®S((p„), called the Jordan model of T, where <plt ..., <p„ are inner functions satisfying (Pj\q>j-x and S(<Pj) denotes the compression of the shift on H2Q<pjH2 for j— 1, 2, ..., n (cf. [7]). We start with the following lemma. L e m m a 1.1. Let T be a C0(N) contraction and let K£hatT. Assume that J=S(cp1)®...®S((pn) and 1/1=S(t/>1)ffi...©S'0/0 are the Jordan models of T and T\K, respectively. Then mS.n and ^jWj for j=\,2, ...,m. Proo f . Since T\K~<T~J, the conclusion follows from [9], Theorem 4. T h e o r e m 1.2. Let T be a C0(N) contraction on H and let K be a subspace of H. Then the following statements are equivalent: (1) tfiELat T\ (2) K= (Range X)~ for some X£{T}'; (3) tf=ker Y for some Y£{T}'. Proo f . It suffices to show (l)=>-(2). Let J=S(<p1)®...®S((pn) and Jr = = S(4/1)®...®S(tp^) be the Jordan models of T and T\K, respectively. Let V: = (H2Qcp1H2)®...®(H2®(pnH2) and TV: = (H2®^2)®... ... ©(H2®ipmH2)-^K be quasi-affinities intertwining T, J and Jx, T\K, respectively. Lemma 1.1 implies that m^n and if/j\q>j for j=\,2, ...,m\ say, (pj=\pjt]j for each j. Note that Sfoj)|(Raiige rij(S(fPj)))- a S(ij/j) (cf. [9], pp. 315—316). For each j, let Zj be the operator which implements this unitary equivalence and let Z.H^Kx be the operator Z1ri1(S(<p]))®...®Zmrim(S(<pm))®0®...®0. Then n — m Z intertwines J, Jt and has dense range in Kx. Finally, let X= WZV. It is obvious that X£ {T}' and (Range X)~. This completes the proof. It is interesting to contrast the preceding theorem with the main result in [14]. T h e o r e m 1.3. If T is a C0(N) contraction on H and K£Lat T, then there exists an Lat T* such that T\K is quasi-similar to (T*\L)*. P r o o f . Let J=S((p1)®...®S(<p„) and J1 = S(\J/1)®...®S(ipm) be the Jordan models of T and T\K, respectively. Then j=S(<p1)®...®S(cpn) is the Jordan model of T*, where <pj(z)=q>j(z) for each j. Let V: H^iH2®^^2)®... ...®(H2Q(pnH2)—H be the quasi-affinity intertwining J and T*. From Lemma 1.1 we have m ^ n and ij/j\q>j for j=l,2, ...,m; say, <pj=ij/jt]j for each j. Then (f>j—ij/jfjj, which implies that S(<p;)|(Range fjj(S(q>j)))~ for j = l , ...,m. On a conjecture of Sz.-Nagy and Foia? 333 For each j, let Zj be the operator which implements this unitary equivalence and let Z=Z 1 f f i . . .©Zm . Let J1=S(ij)1)®...®S({j/m) on K^iH2®^^2)®... ...®(H2QiJ/mH2) and let L=(VZKl)~. Then Lat T* and J^T*\L. It follows that J^T*\L and hence (T*\L)*~Jf^S(il/1)®...®S(\l/m)~T\K, complet-ing the proof. Coro l l a ry 1.4. If T is a C0(N) contraction and {T}', then there exists another 7<E {T}' such that T|kerZ is quasi-similar to (T*|ker Y*)*. Note that Theorems 1.2 and 1.3 generalize the corresponding results for opera-tors on finite-dimensional spaces proved by HALMOS (cf. [ 4 ] , Theorems 2 and 3).*) 2. C u contractions. Let T be a c.n.u. C u contraction with finite defect indices defined on H=(H2®AL2)e{0Tw®Aw:w£H2} by T(f®g)=P(e"f®e"g) for f®g£H, where A = (1 — 0£0T)1 / 2 and P denotes the (orthogonal) projection onto H. Let U be the operator of multiplication by e" on (zl^L2)-, where A + = (1 - 0 r0*) 1 / 2 , and let X: H^(AtL2n)- be the quasi-affinity X(f®g) =-A J+QTg which intertwines T and U (cf. [18], Lemma 3.4). Let 8 be an outer scaler multiple of 0T and let Q be a contractive analytic function such that Q0T= = 0TQ=S1. Note that we have QA^=A£2. Define the operator Y: by Yu=P(0®Qu) for ue(A^L2)-. Note that Qu is in (AL2n)~ for any uf(Aj¥L2n)^. Indeed, if u£(AtL2n)~, there exists a sequence of vectors {um} in L2 such that Ajum-*u in norm as m — <=°. Since QA^um=AQum£AL2 for all m and QA^um^-Qu, we conclude that Qu£(AL2)~ as asserted. Lemma 2.1. Let T, U, X and Y be as above. Then (1) YX=5(T) and XY=5(U); (2) Y is a quasi-affinity intertwining U and T. Proof . (1) For any P(0®g)£H, we have rX(i>(0©g)) = yx((O©£)-(0Tw©2lw)) = Y(-A^-0Tw) + 0T(g-Aw)) = v = Y(0Tg) = P(O®Q0Tg) = P(0®dg) = S(T)P(0®g), where wZH% and in the third equality we used the relation Ajf0T=0TA. Since {P(0®g): g^OdL2)-} is dense in H (cf. [19], proof of Lemma 2), we conclude that Editor's Note: The results in this section are actually true for arbitrary C0 contractions on (not necessarily separable) Hilbert spacer. We have only to refer to the existence of the Jordan model and the validity of Lemma 1.1 in the general case. These have been proved by H. BERCOVICI (On the Jordan model of C„ operators. II, Acta Sci. Math., 42 (1980), 43—56). He also proved Theorem 1.2 for arbitrary C0 contractions. 334 Pei Yuan Wu YX=8(T). In a similar fashion, for any u^iA^L^)', XYu = XP(0®Qu) = 6TQu = Su = 8(U)u. Hence XY=S(U). (2) Forany YUu = Y(e"u) = P(Q®Qe"u) = TP(0®Qu) = TYu. This shows that Y intertwines U and T. That Y is a quasi-affinity follows from (1) and the fact that both 5(T) and 8(U) are quasi-affinities (since <5^0 is an outer function; cf. [6], pp. 118 and 121, resp.). To prove the analogue of the Sz.-Nagy and Foia§ conjecture for C u contractions, we need the next theorem. It implies that for c.n.u. C u contractions with finite defect indices, the lattices of invariant subspaces, bi-invariant subspaces and hyper-invariant subspaces are all preserved under quasi-similarities. Note that the assertions concerning bi-invariant and hyperinvariant subspaces have been obtained before by more elaborate methods (cf. [18] and [15]). By introducing the operator Y we are able to prove all three assertions in one stroke. Theorem 2.2. Let T, U, X and Y be as above. Then Lat r = L a t U, L a t " T s ^ Lat" U and Hyperlat T= Hyperlat U. Moreover, in each case the (lattice) isomorphisms are implemented by the mappings K-<-XK and L-+YL where K is in Lat T, Lat" T or Hyperlat T and L in Lat U, Lat" U or Hyperlat U, and T\K is quasi-similar to U\XK. We first prove the following lemma. Lemma 2.3. Let U be an absolutely continuous unitary operator and let Lat U. If b is an outer function, then 5(U\L) is a quasi-affinity on L. Proof . Since U'=U\L is a contraction, we may consider the canonical decomposition U'=Ux®U2 of U', where Ux and U2 are the unitary and c.n.u. parts of U', respectively. Then 8(U')=8(U1)®8(U2). That <5^0 is an outer function implies that both ¿(£/x) and S(U2) are quasi-affinities (cf. [6], pp. 121 and 118, resp.). It follows that 8(U') is a quasi-affinity. P roof of Theorem 2.2. Since X and Y intertwine T and U, it is easily seen that XK£ Lat U and Lat T for any ^ L a t T and L6Lat U. Moreover, {YXK)~ = (YXK)~ = (8(T)K)~ = (8{T\K)K)~ = K and (XYL) - = (XYL)~ = =(8(U)L)-=(8(U\L)L)~=L by Lemmas 2.1 and 2.3. We infer that the mappings K-~XK and L-+YL implement the (lattice) isomorphisms between Lat T and Lat U and T\K is quasi-similar to U\XK. On a conjecture of Sz . -Nagy and Foia? 335 To complete the proof it suffices to show that for any K£ Lat" T or Hyperlat T, XK£ Lat" U or Hyperlat U and for any ¿6 Lat" U or Hyperlat U, FLgLat" T or Hyperlat T, respectively. If Lat" T, then a{T\K)Qo(T) (cf. [20], Theorem 3), and so T\K£C1V Therefore T\K is quasi-similar to a unitary operator. Since T\K<XJ\XK, we infer by Lemma 4.1 of [3] that XK is a reducing subspace of U, and so ~XK£Lat"U. On the other hand, if L£Lat" U, then YL={P(0® ®Qu):u£L}~. An operator S in {T}" must be of the form P ^ j , where A is a bounded analytic function, B and C are bounded measurable functions and C is scalar-valued satisfying A 0T= 0TAO and B 0T+ C A = A A „ for some bounded analytic function A0(cf. [19], Lemma 2). Hence S(P(0® Qu))=P = r o r r O 1 — 1 J L J = p \cQu\ =P QCi\ YL f o r a n y UeL' s i n c e L € L a t " U a n d C € { ^ r - 11 follows that SYLQYL, whence YL£Lat" T. If Hyperlat T, then XK£ Hyperlat U by [15], Corollary 1. Now let Hyperlat U and let S=P be an operator in {T}\ where A, B and C are as above except that C may not be scalar-valued (cf. [8]). As before, YL={P(0®Qu):u£L}~ and S P ( 0 ® f o r any u£L. Note that L, being hyperinvariant for U, is of the form xt-(A ̂ L^" for some Borel subset F of the unit circle. Assume that U=XFA*V for some v£L2n. Then CQu=/FCQA^v. Note that A0T=0TAo and B0T+CA=AAo imply that QA—AaQ and B3 + +CAQ=AA0Q. Thus CQA* = CAQ = A A0Q - B 5 = A Q A — Q 0TB = Q ( A * A - 0TB ) and we have C Q U = X F & ( A * A — 0TB ) V . Note that 0TBv£(A^L2N)~ and hence CQu = Qw, where w=xF(AJFA-0TB)v£L. This shows that SP (0®Qu)€ YL for any u^XpA^L. Since {xFA^v: vdL^} is dense in L, we conclude that SP (0®Qu)£YL for all u£L. Hence SYLQYL and YL£Hyperlat T, completing the proof. The next theorem is the analogue of Theorem 1.2 for C u contractions. It should be contrasted with [16], Theorem 3.6. Theorem 2.4. Let T be a c.n.u. C n contraction on H with finite defect indices and let KQH be a subspace. Then the following statements are equivalent: (1) Lat" T; (2) K= (Range S)~ for some S£ {T}'\ (3) tf=ker V for some V£ {T}'. 336 Pei Yuan Wu Proo f . It suffices to show (1)=»(2). Let U, X and Y be defined as before and L=XK. Then L, being in Lat" U, reduces U. Therefore L^(W(AJJfy')~ for some {U}'. (W may be taken to be the (orthogonal) projection from onto L) Let S= YWX. Then Si {T}' and (Range S)~=(YWXH)~ = ~(YW(/ iJ$- ) -=z{YL)-=OrXK)-**K by Theorem 2.2. In the remainder of this section we will show that the Sz.-Nagy and Foia§ conjecture holds for C u contractions. L e m m a 2.5. If T is a normal operator on II with finite multiplicity ancl X€{T}', then T |kerX is unitarily equivalent to (T*|ker J*)*. Recall that the multiplicity of an operator T on II is the least cardinal number of a set of vectors in H which, together with their transforms by T,T2,..., span II. Proo f . Note that (Range X)~ and (Range X*)~ reduce T and T|(Range X)~ is unitarily equivalent to T | (Range X*)~ (cf. [3], Lemma 4.1). Let si be the von Neumann algebra generated by T and I. Since T is normal, si is abelian, hence finite. On the other hand, T has finite multiplicity implies that si' is also finite, By [5], Theorem 3, we conclude that T|ker X is unitarily equivalent to T|ker X*=(T*|kcr X"f . T h e o r e m 2,6. If T is a c.n.u. Cn contraction on II with finite defect indices and X£ {71}', then T |kerX is quasi-similar to (T*|ker X*)*. P r o o f . Let U be the operator of multiplication by elt on (A^Ll)~ and let Z : and Y: ->-H be the quasi-affinities defined in the beginning of Section 2 such that YZ—5(T) and ZY=d(U), where <5 is some outer function. It is easily seen that ZXY£ {U}', (F(ker Z Z T ) ) - g k e r Z and (Z(kerX))~Q Qk&vZXY. From Theorem 2.2 we infer that ker ZXY^(ZY(ker ZXY))~Q ^(Z(kerX))~QkerZXY. Hence (Z(ker X))~ = ker ZXY and T\k&x X is quasi-similar to i/|ker ZXY. Note that Z* and Y* are also quasi-affinities satisfying Z*Y*=8(T*) and Y*Z*=S(U*) where S(z)=J^j is outer. A similar argument as above shows that T*|ker X* is quasi-similar to £/*|kcr (ZXY)*. Lemma 2.5 says that E/|ker ZXY is unitarily equivalent to C/|ker (ZXY)*. We conclude that T|ker X is quasi-similar to (T*|ker X*)* as asserted. 3. Weak contractions. In this section we generalize some results in Sections 1 and 2 to weak contractions. The next theorem is the generalization of Theorems 1.2 and 2.4. It should be contrasted with [17], Theorem 3.8 and Corollary 3.9. T h e o r e m 3.1. Let T be a c.n.u. weak contraction on H with finite defect indices and let KQH be a subspace. Then the following statements are equivalent: (1) Lat" T; On a conjecture of Sz.-Nagy and Foia? 337 (2) K= (Ranged)- for some {T}'; (3) K=ktt V for some V£ {T}'. Proof . We have only to prove (1)=»(2). Let H0, HXQH be the invariant subspaces of T on which the C0 and Cn parts of T act, respectively. Since K£Lat" T, T\K is also a weak contraction (cf. [1,8], Theorem 4.1). Hence we may also consider the subspaces K0, KtQK on which the C0 and C n parts of T\K act. Since i ^ L a t " T0 and Lat" Tx (cf. [18], Theorem 4.1), K0=S^H0 and K1=SJI1 for some S ^ W and {Ttf by Theorems 1.2 and 2.4. We also have Ih^VoH and H^VJI for some V0, V1£{T}" (cf. [17], Theorem 3.1 and [6], p. 334, resp.). Let S^SoPo+SiP i - It is easily seen that S£ {71}' and (Range S)- = (S0V0H+ S1V1 H)~ = ( S 0 H 0 + S ± H x ) ~ = K 0 \ / K , = K . This completes the proof. The next theorem (partially) generalizes Theorems 1.3 and 2.6. T h e o r e m 3.2. If T is a c.n.u. weak contraction on H with finite defect indices and Kd Lat" T, then there exists an LgLat" T* such that T\K is quasi-similar to (T*\L)*. Proof . As in the proof of the preceding theorem, let II0, H \ = H and K0, KXQK be the invariant subspaces for T, T\K such that T0=T\H0 and 711h3 7,|7/J are the C0 and Cn parts of T and T\K0 and T\KX are the C0 and Cn parts of T\K, respectively. Since Lat" T0 and K^L&t" T1, there exist L0€Lat" T* and Z^gLat" T* such that T(I\K0^(T*\L0)* and T^-iT^LJ* by Theorems 1.3 and 2.6. Let 1 /=L 0 © Lat" T*©Lat" T* = Lat" (T*© T*) (cf. [2], Prop. 1.3 and Lemma 4.4). By [12], Theorem 1, T | J 5 T — l - ^ o ) ® 1 ^ ) ~ ~(T0*|L0)*©(Ti)|Li)* = ((T*®T*)\L')*. Similarly, T~T0®TU whence T*~ ~7,0*©T1*. Note that quasi-similar c.n.u. weak contractions with finite defect indices have isomorphic bi-invariant subspacc lattices and the restrictions of the weak contractions to the corresponding bi-invariant subspaces are quasi-similar to each other (cf. [18], Added in proof). We infer that there is an L£Lat" T* such that T*\L~(Tf®T*)\L'. Hence T\I(~(T*\L)* as asserted. We conclude this paper by a simple observation that if T is a weak contraction and Xe{T}' then 7"|(Range X)~ is quasi-similar to (T*|(Range X*)")*. Indeed, this follows from the following Lemma 3.3. If T is an operator on H and {T}', then (T*|(RangeJSr*)-)*-< -<r|(Range X)~. Proof . Let Xx be the operator X|(RangeX*)- from (RangeX*)~ to (Range X)~. It is routine lo check that Xx is a quasi-affinity intertwining (r*|(Range X*)~)* and T| (Range X)~. We leave the details to the readers. 338 Pei Yuan Wu : On a conjecture of Sz.-Nagy and Foiaç Coro l l a ry 3.4. If T is a C„ contraction, a Cu contraction or a weak contrac-tion and X£{T}', then T|(Range X)~ is quasi-similar to (r*|(Range Z*)-)*. Proof . This follows from Lemma 3.3 and [13], Lemma 3. [1] H. BERCOVICI, Co-Fredholm operators. I, Acta Sci. Math., 41 (1979), 15—27. [2] J. B. CONWAY—P. Y. Wu, The splitting of sá(Tx®Ti) and related questions, Indiana Univ. Math. J., 26 (1977), 41—56. [3] R. G. DOUGLAS, On the operator equation S*XT=X and related topics, Acta Sci. Math., 30 (1969), 19—32. [4] P. R. HALMOS, Eigenvectors and adjoints, Linear Algebra and Appl., 4 (1971), 11—15. [5] R. V. KADISON—I. M. SINGER, Three test problems in operator theory, Pacific J. Math., 7 (1957), 1101—1106. [6] B. SZ.-NAGY—C. FOIAÇ, Harmonic analysis of operators on Hilbert space, North Holland— Akadémiai Kiadó (Amsterdam—Budapest, 1970). [7] B. SZ.-NAGY—C. FOIAÇ, Modèle de Jordan pour une classe d'opérateurs de l'espace de Hil-bert, Acta Sci. Math., 31 (1970), 91—115. [8] B. SZ.-NAGY—C. FOIAÇ, On the structure of intertwining operators, Acta Sci. Math., 35 (1973), 225—254. [9] B. SZ.-NAGY—C. FOIAÇ, Jordan model for contractions of class C.0) Acta Sci. Math., 36 (1974), 305—322. [10] B. SZ.-NAGY—C. FOIAÇ, On injections, intertwining operators of class Co, Acta Sci. Math., 40 (1978), 163—167. [11] M. UCHIYAMA, Quasi-similarity of restricted C0 contractions, Acta Sci. Math., 41 (1979), 429-433. [12] P. Y. Wu, Quasi-similarity of weak contractions, Proc. Amer. Math. Soc., 69 (1978), 277—282. [13] P. Y. Wu, Jot-dan model for weak contractions, Acta Sci. Math., 40 (1978), 189—196. [14] P. Y. Wu, The hyperinvariant subspace lattice of a contraction of class C.0 , Proc. Amer. Math. Soc., 72 (1978), 527—530. [15] P. Y. Wu, Hyperinvariant subspaces of C u contractions, Proc. Amer. Math. Soc., 75 (1979), 53—58. [16] P. Y. Wu, Hyperinvariant subspaces of C u contractions, II, Indiana Unio. Math. J., 27 (1978), 805—812. [17] P. Y. Wu, Hyperinvariant subspaces of weak contractions, Acta Sci. Math., 41 (1979), 259— 266. [18] P. Y. Wu, Bi-invariant subspaces of weak contractions, J. Operator Theory, 1 (1979), 261—272. [19] P. Y. Wu, Clt contractions are reflexive, Proc. Amer. Math. Soc., 77 (1979), 68—72 [20] D. A. HERRERO, On analytically invariant subspaces and spectra, Trans. Amer. Math. Soc., 233 (1977), 37—44. References DEPARTMENT OF APPLIED MATHEMATICS NATIONAL CHIAO TUNG UNIVERSITY HSINCHU TAIWAN, CHINA Current address: DEPARTMENT OF MATHEMATICS INDIANA UNIVERSITY BLOOMINGTON, INDIANA 47401 U.S.A. 

On a conjecture of Sz.-Nagy and Foiaş

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On a conjecture of Sz.-Nagy and Foiaş

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