Chatterjee, Sourav

Yadav, Bhuri Singh

English

Yadav Bhuri Singh

Chatterjee Sourav

University of Szeged

Acta Sci. Math., 42 (1980), 211—215 On a partial solution of the transitive algebra problem B. S. YADAV and S. CHATTERJEE Let B(H) denote the Banach algebra of all bounded linear operators on an infinite-dimensional separable complex Hilbert space H. A subalgebra sd of B (H) is called transitive if it is weakly closed, contains the identity operator and its only invariant subspaces are {0} and H. B(H) is obviously transitive. Whether there exists any other transitive algebra is a well known open problem, the so-called 'transitive algebra problem'. The problem was first raised by KADISON [5] and it continues to be still unsolved. However, partial solutions of the problem have been obtained by many mathematicians; see, for example, ARVESON [1], BARNES [2], DOUGLAS and PEARCY [3], NORDGREN [8], NORDGREN, RADJAVI and ROSENTHAL [9], and RADJAVI and ROSENTHAL [10], [11]. The first such solution was given by ARVESON [1] who proved that if a transitive algebra s i contains a maximal abelian self-adjoint algebra, then ¿rf=B(H). In the same paper, he also proved that B(H) is the only transitive algebra containing a simple unilateral shift. By using Arveson's techniques, NORD-GREN, RADJAVI and ROSENTHAL [9] have shown that a transitive algebra containing a Donoghue operator (backward weighted shift with a monotone decreasing and square-summable weight sequence) equals B(H). The purpose of this note is to go a step further in this direction and show that every transitive algebra containing a certain type of weighted shift, more general than a Donoghue operator, coincides with B(H). Our result assumes significance in the light of the conjecture that every transitive algebra containing a weighted shift is equal to B(H). We shall denote by H(n) the direct sum of n copies of H, and by A(n) the operator on H(n) which is the direct sum of n copies of A. Let {wk}™=1 be a bounded sequence of non-zero complex numbers and let foKlo be an orthonormal basis of H. The operator T on H defined by the require-ment Te0 = 0 and Tek = wkek-l (k = 1, 2, ...) is called a weighted unilateral (backward) shift with the weight sequence Received February 20, 1979. 14* 212 B. S. Yadav and S. Chatteijee We may and shall assume, without any loss of generality, that the weights wk are positive real numbers [4]. In this case, {wt}~=1 is said to be of bounded />th-power variation if J" K - W t + i l " < » . k=1 (For p = 1, we simply say "bounded variation".) The following theorem is an important tool to obtain our results: Theorem A. [9, Corollary 1 ] If a transitive algebra si contains an operator A such that (i) every eigenspace of A is one-dimensional, and (ii) for every n, each non-trivial invariant subspace of A(n) contains an eigenvector of A<n>, then st=B{H). In the rest of this paper, si will denote a transitive algebra containing a weighted unilateral shift T with the weight sequence {w t}~=1. Our first result is Theorem 1. If { w ^ } ^ is of bounded variation and ( 1 ) 5 = d ( n ) = s h + 2 - t H 2 < ~ for all hS2, then st=B(H). Proof . We know that there is a disc of eigenvalues for a backward weighted shift, but they are all of multiplicity one. Thus T satisfies condition (i) of Theorem A. Next, let (xlt x2, ..., x„) be a non-zero element of a non-zero invariant subspace M of r ( n ) and let oo xJ= 2xuet, 1 — j — n. i = 0 If, for each j, the sequence has only finitely many non-zero terms, then the invariant subspace of T(n) generated by {x1,x2, ...,*„) is finite-dimensional and thus contains an eigenvector. We therefore assume, without loss of generality, that for every m^O, there is a number r=r(m)and a number s=s(m), 1 ^s(m)^n, such that (2) K s l = . max {\xu\} > 0. Now, for a given integer m, we have (T^)r(x 1,X2,...,Xn) ( Xr, I „ xr,2 „ Xr,n „1 . / n w = e°>ire<» ->^eo\ + (yr.i,yr.2, -,yr,n), On a partial solution of the transitive algebra problem 213 where „ _ -V Xk,jWf-Wk-r + l „ YR,j — ZJ ~ k r • Now || ||2 _ y + I xk,j _ y ( wk+2-~ wk+r+l ^ I xk + r + l,j *=f+l I Wr...Wl ) I XriS ~ )\ xr>s M \ W1...Wr ) W1fc=0v w2...wr ) wlk=0i=0 w2...wr .7 (by Abel's transformation [12]) & 00 2 H + r + i - w l + r + i \ , by ( l ) , Wx fc=o <5 00 = -TT 2'|wt+ r + i -w t + r + 2|(w f c + r + 1 + w k + r + 2 )S k=0 ^ 2 K -w k + 1 | , where p = sup {vvt}, Wl *=-r * and hence yrj-+0 as m — °°. Also, for each j (l^jsri), the sequence | Xr'J i is contained in the unit 1 xr,s Jm=1 disc, and hence admits a convergent subsequence converging to a number, say Zj. A routine check reveals that a number j0 lying between 1 and n will occur infinitely often as a value s=s(m) and corresponding to this j0, we have Zj=l. The upshot of the above deliberation is that M contains an eigenvector of Tw, viz. ( z ^ , z2e0, ..., zne0). Thus, T also satisfies condition (ii) of Theorem A and we are done. Theorem 2. If is of bounded pth-power variation and (3) 5 = S(n)= 2(2Wj+i'Wj+n)9 k=o \ j f o w2...w„ ) for all n^2, where l < p < » and q is the Holder conjugate of p, then s/=B(H). < oo 214 B. S. Yadav and S. Chatteijee Proof. Proceeding as in the proof of Theorem 1, we have - u . t e r f t e i T - ter) te ~ TT ^ —w—Z, w*+'+1 = ~ 2J\2J — — I lwJt+r+i —Wft+r+2l Wi *=o W2...Wr Wlk=0\j=o w2...wr ) (by Abel's transformation [12]) = [ i f i Wj+J ' WJ+r)T(2 (by Holder's inequality) (51/« = - ( 2 h - w i t . l f , by(3); and hence, JV,j—0 as w— The rest of the proof follows as that for Theorem 1. Let /", be the Banach space of all complex />th-power summable sequences x= {x0, xlt xz,...} with the norm HP Ml = ( ¿1**1') Then a weighted unilateral (backward) shift T on lp appears as T{x0, Xi, X2, . . . } = { ^ . X j , w2x2, ...}. We denote by i f a strongly closed subalgebra of B(lp) containing the identity opera-tor and with no non-trivial invariant subspaces. We have the following analogue of Theorem 1 for lp spaces, which we state without proof: Theorem 3. If contains T with ttoV w2... wn ) k then &=B(lp). On a partial solution of the transitive algebra problem 215 Remark. A subalgebra i f of B(H) is called strictly cyclic if there exists a vector x0(LH such that A££f}=H, and an operator A£B{H) is strictly cyclic if the algebra generated by A is strictly cyclic. LAMBERT [7] has shown that every transitive algebra which contains a strictly cyclic algebra equals B{H). It follows, in particular, that every transitive algebra containing a strictly cyclic opera-tor is equal to B(H). Every Donoghue operator is strictly cyclic [6]. Whether the weighted shifts T in our Theorems 1 and 2 are also strictly cyclic, is not known. In case they are, these theorems will follow as corollaries to LAMBERT'S theorem [7, Theorem 4.5]. In fact, we strongly feel that the following is true: Conjecture. Every weighted unilateral shift whose weight sequence is of bounded variation and square-summable is strictly cyclic. References [1] W . B. ARVESON, A density theorem for operator algebras, Duke Math. J., 3 4 (1967), 635— 647. [2] B. A . BARNES, Density theorems for algebras of operators and annihilator Banach algebras,. Michigan Math. J., 19 (1972), 149—155. [3] R . G. DOUGLAS and C. PEARCY, Hyperinvariant subspaces and transitive algebras, Michigan Math. X , 1 9 (1972), 1—12. [4] P. R . HALMOS, A Hiibert space problem book, Van Nostrand (Princeton, N . J., 1967). [5] R. V. KADISON, On the orthogonalization of operator representations, Amer. J. Math., IS (1955), 600—621. [6] A . LAMBERT, Strictly cyclic weighted shifts, Proc. Amer. Math. Soc., 29 (1971), 331—336. [7] A . LAMBERT, Strictly cyclic operator algebras, Pacific J. Math., 39 (1971), 717—726. [8] E. A. NORDGREN, Transitive operator algebras, J. Math. Anal. Appl., 32 (1970), 639—643. [9] E. A . NORDGREN, H . RADJAVI and P. ROSENTHAL, On density of transitive algebras, Acta Sci. Math., 30 (1969), 175—179. [10] H . RADJAVI and P. ROSENTHAL, On invariant subspaces and reflexive algebras, Amer. J. Math.r 9 1 (1969), 683—692. [11] H. RADJAVI and P. ROSENTHAL, Invariant subspaces, Springer-Verlag (Berlin—Heidelberg— New York, 1973). [12] A . ZYGMUND, Trigonometric series, vol. I, 2nd ed., Cambridge Univ. Press (New York, 1959). DEPARTMENT OF MATHEMATICS UNIVERSITY OF DELHI DELHI-110007, INDIA 

On a partial solution of the transitive algebra problem

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On a partial solution of the transitive algebra problem

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