Amemiya, Ichiro

Ando, Tadao

English

Amemiya Ichiro

Ando Tadao

University of Szeged

Convergence of random products of contractions in Hilbert space By I. AMEM1YA and T. ANDO in Sapporo (Japan) 1. Introduction Given two projections and P2 in. a Hilbert space, it is known that a product Tn-- T2Ti converges strongly as —°= where TJ=Pl or Tj = P2 at random. The problem in this paper is to observe the case of a finite number of projections. The result is that weak convergence is always valid, while strong convergence is still unsettled. After several comments on the convergence of the iterates of a single contraction, the convergence problem of random products will be discussed for a wider class of contractions, including all non-negative definite contractions. 2. Iterates Let T be a contraction in a Hilbert space, i. e., a linear operator with || T\\ S 1 , 1 " then the so-called mean ergodic theorem shows that the average — TJ converges « j = i strongly, as n -*- to the projection onto the subspace of all vectors invariant under T, i. e., the null space of I— T, and that the orthogonal complement of the null space of I—T coincides with the closure of its range (see [6], n° 143). When do the iterates T" themselves converge strongly or weakly? Since T operates as the identity on the null space of I — T, T" converges if and only if T"f converges to 0 for a l l / i n the range of /— T, so that T" converges strongly or weakly according as T"{I— T) converges to 0 strongly or weakly. Given / , | |T"/|| decreases monotonically with limit, say a ë O . If a = 0 , clearly T"(I— T)f—0 strongly, and if a > 0 , with g n = it follows llgj=l, Tn(I-T)f = ||T"f\\{I-T)g„, and \\TgJ This observation leads to the following criterion. T" converges strongly or weakly, if T has the following property (S) or (W), respectively: (S) \\f„\\ S1, || Tf„\\ -1 imply ( / - T)fn -0 strongly, (W) 11/,|| S I , H37J-1 imply (/-70/B-0 weakly. 240 I. Amemiya—T. Ando A non-negative definite contraction, in particular, a projection, has the prop-erty (S); in fact, if T is a non-negative definite contraction, \\(i-T)fnV = ( ( / - r)1/,,/») s ^ {(I-T)(I+T)fn,fn) = | | / J 2 - | | 7 7 „ | | 2 - 0 whenever ||/„|| s i and | | 7 / J - l . The product of two (hence a finite number of) contractions, each of which has (S) or (W), also has the same property; in fact, if T l , T2 are contractions with the property, say (S), and if | | / J S 1 and || T2TJn\\ - 1 , then || TJ„|| s 1 and \\TJJ - 1 , so that (/-T2Tx)fn = ( / - T,)fn + ( / - T2)TJ„ — 0 strongly. It should be mentioned that the statement about (S) was observed by H A L P E R I N [3] in proving the strong convergence of the iterates of a product of a finite number of projections. The condition (W) has a simpler equivalent form (W') : (W) \\Tf\\=\\f\\ implies T f = f . Only the implication (W')=>(W) needs a proof. Since 11/112-II7/112 = (<j-T*T)f,f) and I-T*T is non-negative definite, | | 7 / | | = | | / | | is equivalent to (I-T*T)f= 0 , so that (W') implies that the null space of I-T*T is contained in that of I - T . By taking the orthogonal complements, it follows that the closure of the range of I-T*T contains the range of I-T*. Now if \\fn\\^ \ and | | 7 / n | | - l , then \^\\T*Tfn\\^\\fn\\-\\T*Tfn\\^(T*Tfn,fn) = H O T - * ! , so that the property (S) for the non-negative definite contraction T*T shows (I-T*T)f„ — 0 strongly, which, in turn, implies ( / „ , / 0 — 0 for all h in the closure of the range of I—T*T. For an arbitrary g, ( ( I - T ) f n , g ) = ( / „ , ( / - r * ) g ) - 0 , because (I-T*)g is in the closure of the range of / - T*T. Thus (W') implies (W). Clearly (S) implies (W) and equivalently (W'). If a contraction T has (W'), •its adjoint T* has ( W ) too. In fact, || T*f\\ = | | / | | implies TT*f=f, so that l i r r * / | | = | | / | | = | | r * / l l , hence TT*f=T*f by (W') of T, and the assertion follows. A contraction T is called completely non-unitary if \\T"f\\=\\T*"f\\=\\f\\ for all « S 0 implies / = 0 . The decomposition theorem, proved idenpendently in [4] and [7], is quite useful in analysing an arbitrary contraction; it says that for a contraction T there is a uniquely determined closed linear subspace such that it reduces T and that T is unitary on it and is completely non-unitary on its orthogonal complement. Indeed, the subspace consists of all vectors / for which || T"f\\ = = ||T*"f\\ = LL/H for all n S l . Moreover S Z . - N A G Y and F O I A § [7] proved that the Random products of contractions 241 spectral measure of the minimum unitary dilation o f ' a completely non-unitary contraction is absolutely continuous with respect to the Lebesgue measure on the unit circle. This result can give the following improvement of the criterion (W') for the weak convergence of the iterates: T" converges weakly, if \\Tnf\\=\\T*nf\\=\\f\\ for all «Si implies T f = f . Here is an alternative proof, not using spectral representation (cf. [2]). The hypothesis means that the unitary part of T in the decomposition mentioned above acts as the identity, so that there is no loss of generality in assuming the complete non-unitarity of T, which is, as in the proof of the implication (W')=>-(W), equivalent to the statement that the intersection of all null spaces of A„ (n = ± 1 , ± 2 , ...) consists of 0 only, where An = I - T*"T" and A_n = I - T"T*" for n SO. By taking orthogonal complement, the linear span of the union of all the ranges of A„ (n = ± 1 , ± 2 , ...) is dense, so that to prove the weak convergence, it suffices to show that for all non-zero integer n and vectors / , g (TJf A„g)-> 0 as j Since HT j f \ \ decreases as j increases, it results for « S i ( J J f A J J f ) = | | 7 y | | 2 - | | r » + J / | | 2 ^ 0 as co, so that the generalized Schwarz's inequality ([6], n° 104) for the scalar product induced by A„ yields | ( P f , A„g)\2 ( T J f , A„TJf)-(g, A„g) (TJf AnTJf)-\\g\\2^. Since A_„TJ — T"AnTi~n for / s « S l , the generalized Schwarz's inequality for the scalar product induced by A_ n yields |(;Pf, A.ng)|4 S (TJf, A-„TJf)2-(g, A_„g)2 ^ : ( P f , T"AnTJ-"f)2-\\g\\4 = (T*>Pf, A„TJ~nf)2 ||g||4 : ^(TJ-"fAnTJ-»f)-(T*nTJfAnT*»TJf)^\g\\*^ ^(TJ-f,AnTJ-"f).\\f\\2-\\g\\4^0. • 3. Random products Let Tj (j= i, 2, ..., N) be a finite set of contractions. A mapping r(-) f rom the-set of all positive integers to {1, ...,N} will be called a (random) selection. Given a random selection /•(•), construct the sequence of contractions {£„} by setting Sn = ^r(n)"" 'T r(2yT r ( i ) , then what can be said about the convergence of Sn or of 1 " the average — 2 S j l The random ergodic theorem (cf. [1]) shows that if each nj=i selection is considered as a point of the infinite product of the copies of the proba-bility space {1,2, . . . , N } (on which each point has the same probability TV-1), 1 " then the average — 2 Sj converges strongly for almost all selections. Without n j = i any further restriction on the Tf s this would be the best result. Suppose now that each Tj ( j = l , 2, ..., N) has (S). If a selection r(-) is periodicy i. e., r(k + m) = r(k) for some m and all k, then Smk=(Sm)k, and since Sm has (S), 242 I. Amemiya—T. Ando Smk converges strongly, as to the project ion onto the null space of I—Sm. F o r an index n, take k such that m{k — \)^n^mk, then — Sn-m(k- l)Sm(k- l) where it is assumed that S0 = I. If / is in the null space of I—Sm, I I / I I = I I Sm(k - l)/ll = I I SJ\\ S || Smkf\\ = ll/H, so tha t S„f=f because of ( W ) for Sn, and if / i s in the closure of the range of I I S „ f { \ S| |S'„, (k_!)/!! — 0 as « - c o , because k-*°= as n — Thus S„ converges strongly. W h e n every T j is a projection, P R A G E R [5] derived the weak convergence of S„ f r o m a quasi-periodicity assumption on the selection /'(•); Sn converges weakly, if there is m such that every m consecutive r(k),r(k+ 1), ..., r(k + m — 1) contains each j at least once ( j —1,2, . . . . N). On starting with an observation tha t in his p roof the hypothesis that T j is a projection is no t essential, but only the proper ty (W) is necessary, the goal of this paper is to derive the weak convergence f r o m ( W ) without any periodicity assumption on a selection. I t should be mentioned tha t when the Hilbert space is of finite dimension, P R A G E R attained the same goal in the case of projections; he proved the strong convergence, but strong conver-gence is equivalent to the weak one in the finite dimensional case. T h e o r e m . If Tj is a contraction with (W) or ecjuiucdeiilly ( W ) ( j = 1, 2, . . . , N), then for any random selection /•(•) the sequence Sn — Tr(„yTr(2)Tr( J) converges weakly as n The proof will be divided into several steps. (i) In what follows, by a weak neighborhood there is meant a convex symmetric neighborhood of 0 with respect to the weak topology. The condition (W) can be stated in the following f o r m : for any weak neighborhood S there is an e =-0 such tha t 11/11 s 1, || Tjf\\ S I - e imply ( / - Tj)f€ 8 . (ii) The intersection of the null spaces of I—Tk (k = 1, 2, . . . , / ) coincides with the null space of I-^Tj...T2Ti. In fact, the former is obviously contained in the latter. If / is iri the latter, 11/11 = II T, -T2TJ\\^ I I T J W * 11/11 so tha t TJ=f by (W') of T, and by induction TJ=f(k = 1, 2, ...,./). Let Qj stand fo r the projection onto the null space o f / - Tj...T2T1, then TkQj=Qj (k = 1, 2, ...,./') so tha t T k Q j = Q j because a vector invariant under a contraction is also invariant under its adjoint (cf. [6], n° 143). Thus f r o m TkQ} = Qj and T*kQj = Qj, the commu-tativity of Qj with Tk (k = 1 ,2 , . . . , / ) follows. (iii) Let P j = I— Q j , then for any weak neighborhood 35, there is another 2B such that | | / | | S l , ( f - T k ) f i % s (k = l, 2, . . . , / ) imply P / 6 S 3 . In fact, since (I—Tk)f= 0 (lc = l, 2, . . . , /') is equivalent to Pjf = 0 by (ii), the mapping which assigns to Pjf the ordered / t u p l e {(/— 7 \ ) / , . . . , ( / — T j ) f } is one-to-one. Since {I—Tk)f = (/— Tk)Pjf (k = 1, 2, .. . . j ) by (ii), i t is continuous f r o m the image of Random products of contractions 243 the unit ball under P} into the product of j copies of the Hilbert space, when they are provided with their respective weak topologies. Since the domain is compact, the mapping is bi-continuous and the assertion is just the statement that the inverse mapping is continuous at the origin. (iv) Let Sfty be the collection of contractions which are in a multiplicative semi-group with unit generated by j of the contractions {Tk}'f ( j = 1, 2, ..., N) and let dJt0 = {/}. Given a weak neighborhood 23 and S 6 2fty, there is a positive number e = e(93,/) depending only on 58 and j such that | | / | | S i , \\Sf\\ S i —e implies ( /—S) /693 . Proof proceeds by induction on j as follows. The assertion for j=0 is trivial. Suppose that the assertion is true up to j— 1. Only S in Wj but not in fflj-x needs consideration. There is no loss of generality in assuming that >S is in the multiplicative semi-group generated by T1} T2, Tj. For any index I s k ^ j , S can be written in the form S = Ri TkR2 = R3 TkRA where RX,RA are in 3JIj-i- Given a weak neighborhood 93, take SB which is in relation of (iii) to 93, and choose a weak neighborhood U such that 411 + 4 7 : 1 1 C S® ( ¿ = 1 , . . . , / ) , which is possible because of the weak continuity of T{. Now by the inductive assumption and (i) it is possible to take a positive number s, independent of S, such that | | g | | S l , \\Rg\\ S l - e with ReWj-r or R = Tk imply (I~R)g£ 11. Now if ll/H S i and \\Sf\\ S i —e, obviously 1 ̂ \\RJ\\ S l - e and | | T k R J \ \ s 1 - s , hence (I— R4)f£U and (I--Tk)RJiX\, so that ( / - Tk)f = (I-JUV+ ( / - Tk)RJ~ Tk(I-Rjfe2U + TkU g ±28 and quite similarly (I-Tk)Sf = (Ri-I)TkR2f+Tk(I-R1)TkR2f+Tk(.I-Tk)R2ftU+2Tk)l g ± 2 3 . Since the relation is valid for k = l, 2, ...J, (iii) guarantees 3 and 3, consequently P / / — S ) / ( : 9 3 . A S in (ii), I — P j is just the projection onto the null space of I—S, because S has Tk as a factor (k = 1 , 2 , . . . , / ) , so that ( / - s)f = ( / - s) Pjf = Pj(i- s)fe 93. (v) S„f converges weakly for all / In fact, if || S„f\\ -+0, the assertion is trivial. If inf„ II S,J\\ > 0 , given a weak neighborhood 93, take e = s(93, N) in (iv), then for sufficiently large n^m we have | |S„/| | S (1 — e)i|5m/ | | . Since S„ = S-S„, for some ^efflijv and [15̂ 11 = 4 1 ^ ^ 1 - 8 with g - S m f l |S m / l l ~ 6 I IS m / | | ' (iv) guarantees (I— S)g£iS, so that smf-sj=\\s,jKi-s)gz\\№-The weak convergence follows f rom the arbitrariness of 93. This completes the proof. * 244 I. Amemiya—T. Ando: Random products of contractions When every index j appears infinitely many times in a selection r(-), the limit of the sequence S„ is the projection onto the subspace of vectors invariant under all Tj (j= 1, 2, ..., N). In fact, with the notations in the proof of the Theorem, fo r sufficiently large m SmQN = QN, and if f=PNf and inf„ | |S„/ | | > 0 , for any weak neighborhood 33 and sufficiently large m there is n>m such that r(m +1), ..., r(n) contains every j at least once (y = 1, 2, ..., N) and ¡¡f"^!! S i — e where e = £(23, N), ll'-Wll so that as in (iv) and (v) PN ( ¡ 7 ^ 4 ) hence Ml SmJ 11/ smf=smpNf=pNsmfe II/II®. The arbitrariness of $ implies the weak convergence of SnPN to 0. Thus Sn con-verges weakly to QN. * C o r o l l a r y . I f T j is a contraction with (W) or equivalently ( W ) ( / = 1, 2 , . . . , N), then for any random selection >'(•) the sequence sn = TrWTr,2y..Tr(n) converges weakly as « — P r o o f . It is proved in § 2 that T j has (W) 0 = 1,2 , ...,N), so that by the Theorem, the product T f ^ . . . T i m T f w converges weakly, hence ( S „ f , g ) = - ( / , 7%...Tr\2)T*wg) converges for all / , g. This completes the proof. References [1] A. BECK, and J. T. SCHWARTZ, A vector valued random ergodic theorem, Proc. Amer. Math. Soc., 8 (1957), 1049-1059. [2] R. S. FOGUEL, Powers of a contraction in Hilbert space, Pacific J. Math., 13 (1963), 551 — 562. [3] I. HALPERIN, The product of projection operators, Acta Sci. Math., 23 (1962), 96 -99. [4] H. LANGER, Ein Zerspaltungssatz fur Operatoren im Hilbertraum, Acta Math. Acad. Hungar., 12 (1961), 441-445. [5] M. PRAGER, On a principle of convergence in a Hilbert space, Czech. Math. J., 10 (1960), 271-282. [6] F. RIESZ and B. SZ.-NAGY, Leçons d'analyse fonctionnelle (Budapest, 1952). [7] B. SZ.-NAGY and C. FOIAÇ, Sur les contractions de l'espace de Hilbert. IV, Acta Sci. Math., 21 (1960), 251-259. (Received November 6, ¡964) 

Convergence of random products of contractions in Hilbert space

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Convergence of random products of contractions in Hilbert space

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