Foiaş, Ciprian

Szőkefalvi-Nagy, Béla

English

Szőkefalvi-Nagy Béla

Foiaş Ciprian

University of Szeged

Acta Sei. Math., 38 (1976), 311—341' Commutants and bicommutants of operators of class C0 BÉLA SZ.-NAGY and CIPRIAN FOIAÇ Dedicated to P. R. Halmos on his 60th birthday Introduction By operator we mean a linear and bounded one. For any operator T on a Hilbert space § we consider the following weakly (or equivalently, strongly) closed sub-algebras of $?(§): s4r\ the subalgebra generated by / and T; {T\: the commutant of T\ {T}": the bicommutant of T; i f r : the subalgebra consisting of those X£{T}' for which LatXz>Lat T (i.e. X leaves invariant every subspace of § invariant for T). If T is a completely non-unitary contraction on § we also define: JfT\ the set of operators on § which admit a representation X=v(T)~1u(T) with functions u, v£H°° such that v(T) is a quasi-affinity (i.e. an operator with zero kernel and dense range). From this definition it readily follows: We shall consider operators T of class C0, i.e. completely non-unitary contrac-tions such that w ( J ) = 0 for some inner function w; among these functions w there is a minimal one, denoted by mT. For TdC0 and the operator v(T) is a quasi-affinity if and only if vAmr = 1 (i.e. if v and mT have no non-constant inner divisor); cf. [H], Proposition III. 4.7. For T€ C0 we have equality in (0), i.e. (0) JÇ c {T}", cf. [H], Chapter IV. (1) = {T}" for o-Received October 30, 1975. 312 B. Sz.-Nagy and C. Foia? This was proved in [2] if the underlying space § is separable, by using the "Jordan model" of operators of class C„. A subsequent extension of the Jordan model to the non-separable case, given in [3], yields, by the same proof, the validity of (1) for non-separable § also. In Sections 1 and 2 of the present paper we shall prove the inclusions (2) JTT<ZJ*T for T£C0 , (3) serc.jfT for r e c 0 . As a consequence of (1), (2), (3), and of the trivial inclusion stT<zS£T we deduce {T}" = ^ c ^ c ^ c ^ = {T}" for r€C 0 . So we establish the following: Theorem. For any operator T of class C„ we have = {T}" = .AfT. For operators T of class C0 with finite defect indices (classes C0(N); N= 1, 2, . . .) these results were proved in the recent paper [4] by Wu (Theorems 3.2 and 3.3). It was this paper that suggested the present investigation. The proofs we are going to give for the general case employ quite different arguments as those in [4]. I'. Proof of JfT c siT Let T£C0 on Suppose there is an X£_JfT which is not contained in s/T . This means that there exist h l t ..., Ar6§ and e^-0 such that (1.1) ¿ WXhj-plDhjW* ^ £2 for all polynomials p. j=i •' Setting H = 0 & T = ©T, X = © X , b = @hj, 1 1 1 1 (1.1) can also be expressed as (1.2) ||Xh-p(T)h|| s e for all polynomials p. As t h e r e e x i s t such that i)Am r=l, v(T)X=u(T), and hence, (1.3) v(T)X = u(T). Denote by Hh the cyclic subspace for T generated by h and define (1.4) K = {k€H:i;(T)k<EHh}. Commutants and bicommutants of operators of class C, 313 Clearly, K is invariant for T and T0=T|K is of class C0. Its minimal function is a divisor of m? (—mT) so we also have t)A/nTs=l. Thus, u(Tt) is a quasi-affinity on K and so it has dense range in K. As by definition (1.4) o(T0)K = i)(T)KcHh we infer that <1.5) K c H h . Now, by (1.3) we have r(T)Xh=«(T)h€Hh, and therefore Xh6K; thus, by (1.5), Xh€H„. This implies that there is a polynomial p such that ||Xh—p(T)h|| «= ê  This contradicts (1.2), and hence achieves the proof. 2. Proof of se T ajr T Let T(i C0 on 5. By [2], Proposition 2, we have T>- S(m)®G, where m=mT and G is the restriction of T to some invariant subspace ©, i.e. there exists a quasi-affinity such that (2.1) TA = A(S(m)QG). Here, as usual, S(m) denotes the compression of the canonical shift on IP to the subspace 9>(m)=H2QmH2. Consider the cyclic vector e for S(m), given by e= I —m(Qi)m, and an arbitrarily chosen vector and set (2.2) ht = A((l-t)e®tg), t being a numerical parameter to be fixed later. Further, set Tt = T\f>„ and mt = mTt. From (2.1) and (2.2) we deduce w(T)h0 = A(w(S(m))e®0) for all H°°\ hence T0 has the same minimal function as S(m), i.e. m0=m—mT. 314 B. Sz.-Nagy and C. Foiaç While it may happen that m1 is a proper inner divisor of mT, it follows from a lemma due to M . SHERMAN that the values t for which m, is a proper divisor of mT are exceptional, that is, countable many; cf. [1]. Let T be a non-exceptional value of t, different from 0 and 1 ; thus mz=mT, (Mt?* 1. Let Then X9)t<z$>t and X\§te {T,}', for all t. Since T, is a C0 class oper-ator with cyclic vector h„ every operator in its commutant is a function of T, of the "Nevanlinna class" JfTt (cf . [H], Chapter IV, and [1], Théorème 2). Thus there exist functions u,, v,eH°° such that (2.3) vt/\mt = 1 and vt(T)Xht = ut(T)h,; in particular, (2.4) t>0A mT = 1, vtAmT = 1. Set (2.5) X' = v0(T)X-u0(T); X' also belongs to i f T and by (2.3) we have (2.6) X'h0 = 0 and v,(T)X'h, = u',(T)h, for u't = v0u,-u0vt. Hence, X'ht=X'((l—t)h0+th1)=tX'h1 and i v,(T) • tu{(T)hx = tiv.uiUT)^ <U M )* f , |Vi(T)Vt{T)x>hi = Vl(T)u',(T)ht = (tw'XDft 1 -O^o + ^i) so wè have By (2.1) and (2.2), and since A is injective, this implies (1 - 0 - ( i v O ( S ( m ) ) e © 0 = 0®t-(vtu'1-v1u',)(G)g-, so we have for any 0, 1, and in particular for t—x: (2.7) (Vlu'r)(S(m))e = 0, (vtui-Vlu't)(G)g = 0. The first equation (2.7) implies vxuxÇ.mHSince mG\m we infer (v1u^)(G)=0. Comparing this with the second equation (2.7) we deduce vt(G)ui(G)g = M(G)g = 0. On account of (2.4), vt(T) is a quasi-affinity so its restriction vT(G) is injective; thus u1(G)g=0. Hence, uiCT)*! = uî(T)A(0@g) = ^Mi(5(w)©G)(0©g) = ¿(Offii^G)*) = 0, and therefore, by (2.6), vx(T)X'hx=0. Now, the subspace being invariant for T is also invariant for X' ; thus X'h^v But ^ A m ^ l by (2.3), and thus u1(7\)=y1(7n)|§1 is a quasi-affinity on and in particular injective, so we conclude X'h1=0. Commutants and bicommutants of operators of class C0 315"» Combining this result with the equation X'h0=0, see (2.6), and recalling that by its definition (2.5) the operator X' is independent of the choice of g in © we readily conclude that X'A=0, X'=0, and therefore v0(T)X-u0(T) = 0 (»„AmT = 1), that is, X£JÇ. This concludes the proof. References IH] B. SZ.-NAGY—C. FOIAÇ, Harmonic analysis of operators on Hilbert space, North-Holland/' Akadémiai Kiadó (Amsterdam/Budapest, 1970). [1] B. SZ.-NAGY—C. FOIAÇ, Complements à l'étude des opérateurs de classe C0, Acta Sci. Math.,. 31 (1970) 287—296. [2] H. BERCOVICI—C. FOIAS—B. SZ.-NAGY, Compléments à l'étude des opérateurs de classe CV III, Acta Sci. Math., 37 (1975), 313—322. 13] H. BERCOVICI, On the Jordan model of C0 operators, Studia Math, (to appear). [4] P. Y. Wu, Commutants of C0(N) contractions, Acta Sci. Math., 38 (1976), 193—202. B. SZ.-NAGY C. FOIAS BOLYAI INSTITUTE FACULTY OF MATH., UNIV. BUCHAREST ARADI VÉRTANÜK TERE 1 STR. ACADEMIEI 14 6720 SZEGED, HUNGARY BUCHAREST, ROMANIA 

Commutants and bicommutants of operators of class C0

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Commutants and bicommutants of operators of class C0

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