tailieunhanh - Báo cáo toán học: "A weighted bilateral shift with no cyclic vector "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một sự thay đổi song phương trọng với không có vector tuần hoàn. | J. OPERATOR THEORY 4 1980 287-288 Copyright by INCREST 1980 A WEIGHTED BILATERAL SHIFT WITH NO CYCLIC VECTOR B. BEAUZAMY Let us denote by ez the canonical basis of 2 Z and by T the weighted shift defined by Te wnen 1 n e z with Ị w 1 rt 0 1 4 n 0. This operator can also of course be considered as an operator on 2 T which we denote also by T. We are interested in the invariant subspaces for this operator the following theorem proves that they are very numerous. In fact no vector is cyclic. Theorem. For all 0 ỢỈ e L2 T span Tf T2f . Tnf . does not contain f. Proof. Assume on the contrary that we can find f f Q f e span 7 . . This means that for all 0 there exists a finite sequence of scalars a such that I II - s 8. 1 -hoo Let X akc k0 be the Fourier decomposition. By computing the Fourier 00 coefficients of Tmf one can write 1 as ft uflk m 2 1 tn-k @ m k m 2 s a-t-k Q k-m .2 By dividing each term of the second sum by the corresponding l 4fc putting bk ak if k 0 bk ak if k 0 one obtains 2 s bk s 0tm6fc_ t 2 2. kGZ M 1 288 B. BEAUZAMY This means that if u is the usual bilateral shift and if g J btfiikf then fcez llể - L amơ gỊ i2 T . m 1 Since a similar result holds for all 0 it follows that g e span Ug u2g . . From Szego s theorem we deduce that log g 0 dớ oo. But by the definition of b g z bkz IceZ zỊ 1 Ị- from which we obtain 4 J - OO log g ỡ dớ oo J co and the proof is complete. Remarks. 1. This operator is a typical example of a C . contraction not satisfying the assumption of our paper 1 the iterates T nx n 0 grow too quickly . One can wonder if the conclusion X ị span Tx .Tnx . does not hold more generally in such a case. 2. On the contrary since its spectrum is J -Ị- z I ST 1Ị it satisfies the assump- 14 J tion of the paper of Brown-Chevreau-Pearcy 2 Since their method provides a point not belonging to the closed span of its iterates it shows that in their paper no other description of the invariant subspaces was possible. REFERENCES 1. Beauzamy B. Sous-espaces