Bansal, Ranjana

Yadav, Bhuri Singh

English

Yadav Bhuri Singh

Bansal Ranjana

University of Szeged

Acta Sci. Math., 47 (1984), 93—106 On systems of /V-variable weighted shifts B. S. YADAV and RANJANA BANSAL*) 1. Introduction. Let H be an infinite-dimensional Hilbert space with an ortho-normal basis and let {vi'1}Jl0 be a bounded sequence of complex numbers. An operator T on H defined by Tel = wlei+1, i - 0 , 1 ,2 , . . . is called a (forward) unilateral weighted shift with the weight sequence {wjjl0. JEWELL and LUBIN [2] have recently extended the theory of weighted shifts to the sys-tems of ^-variable weighted shifts. They concentrate in their paper mainly on exhibit-ing the interplay between such systems of operators and the analytic function theory of several variables on the lines of the survey article on (one-variable) shifts by SHIELDS [9] . The object of this note is to study the existence of cyclic vectors for the systems of iV-variable backward and forward weighted shifts. We also obtain an appli-cation of these ideas to the theory of transitive operator algebras. 2. Notations and terminology. We shall follow mostly the definitions and nota-tions given in [2]. To recapitulate, we denote by N an arbitrary but fixed positive integer, by I—(Ji, z2, •••, iN) a multi-index of non-negative integers, and by | / | the sum /1+/2 + . . .+/w . Ek denotes the multi-index / having ik = 8Jk for / = 1,2, ..., N, and O is the multi-index (0,0, ...,0). We shall write I±Ek for the multi-index (z-!, . . . , 4 _ 1 , 4 ± 1 , 4 + 1 , ...,zN) (where 4 > 0 in I-Ek). If T={TX,T2, ...,TN) is a system of N commuting operators on H, then, for a multi-index I=(i\, ..., iN), T1 means the operator Tl=T^T^...T'N". Let {<?,: 1^0} be an orthonormal basis of if and let { w / j — 1, 2, ..., N, ISO} be a bounded net of non-zero complex num-bers. A system of TV-variable backward weighted shifts with the weight net {vf/^} is defined as a family S= {5l5 ..., SN} of N operators on H such that Sje, equals Wjjej-E if z'y?i0 and 0 otherwise, for all / = 1 , 2 , ..., N. Received October 1, 1982. *) The second author gratefully acknowledges the support of U.G.C. grant No. F—25—9 (11633)/80 for this research work. 94 B. S. Yadav and R. Bansal Similarly, a system T= {Tx, T2, ..., TN} of TV-variable forward weighted shifts with the weight net {M'J,,} on H is defined by TjeI=wrje,+E, l s / s J V . We may and do assume w, /s to be positive real numbers [2; Corollary 2]. We also assume that wr / s satisfy either of the following two conditions: (I) W/.^Vz-Ej,» = Wl,kW,-EkJ, if ij, ik ^ 0, (II) + = wI,kW[ + E k J , for all / , 1 As observed in [2], condition (I) (respectively condition (II)) implies that the operators Sj (respectively Tj) commute. We set /?(/, k, m) equal to wj,kwJ-Ek,k---wJ-(m-v)Ek,k 'f M—JK,AN& 0 otherwise; <x(J,k,m) equal to WJ,KWJ+EK,K -WJ+(M-I)EK,K if m^O and 1 otherwise, where J=(j\,j2, ...,jN), and 1 ^k^N. It is easy to see that, for l^k^N, S%e3 is equal to P(J, k, m)ej^mE^ if )K = M and 0 othewise, and Tke} is equal to OL(J, k, M)eJ+mEk. 3. Cyclic vectors for ^-variable weighted shifts. We make the following defini-tions : D e f i n i t i o n 1. A net {vf/,,} of (non-zero) positive real numbers is said to be monotonically decreasing if for all I = (i\, i2, ..., /A-), /' = (/,', i2, ..., i'N) with we have w,j^wI,j for 1 S j ^ N . D e f i n i t i o n 2. A vector x in H is said to be a cyclic vector for a system A = = {AX,A2, ...,AN} of N commuting operators on H, if H=\J A'x (i.e.: H\s spanned by the vectors A'x). ' Theorem 1. Let S= {S1, S2, ..., SN} be a system of N-variable backward weighted shifts with the weight-net {w, which is monotonically decreasing and satisfies the condition (Y). If {w^f), is square-summable (i.e. for some j, \=j=N, and x=x(J)ej is any vector in H such that infinitely many elements of the j set {x(J): ji =j2 =... =jN S 0} differ from zero, then x is a cyclic vector for S. Proof . For the sake of simplicity, we shall prove the theorem in case N=2. However, our method works also in the general case. Assume that {wf x} is square-summable. For given e>0, there exists an /<,= = (/?, such that ( 1 ) 2 WJ,I < £ 3 K , O ) , I ) ! -M|s|/„| Choose M=(m1, m2) with m^i^, such that (2) \x(M)\ = sup{|x((y1 + /»,y2 + /2°))|: AJ2 0} > 0. Systems of A'-variable weighted shifts 95 It is easy to verify, by using condition (I), that (3) W<X.mù.lK(°> 2> m<i) = H i 1 ' m*>> 2> W2)W(1,0),1-Now Sux = {SpSr)(2 x(J)ej) = = 2 x{(ji + "h,k + '«2»P(O'l + mi,à + m2), 1, m1)P((j1>j2 + m2), 2, m2)e}. Hence, by making a successive use. of (2), (3) and the fact that {h>/ y} is mono-tonically decreasing, we have SMJ x(M)p(M, 1, mi)ß((0, m2\ 2, m2) — e0 2 x(Ui + m1,j2 + m2)) !(P(Ui + "h,k + m2). 1, »?))/?((./], j2 + m2), 2, ?n2)V^ l ß ((m,, »/a), 1, m,) p ((0, m2), 2, #h2) J ~ 2 P (CA + '»1, k + »'2), I , mi)ß (0"i, h + 2, w2) V = P (0«i > w2), 1, /Mj) ß ((0, m2), 2, m2) J X x(M) 2 ( _ ^ i 1VOi + mi,ja + m2),ll x { ß((mi, m2), 1, m1 -1)/?((1, m2), 2, W72) J ~ (L. i.WOV Wfi.nl 1 / £ . This shows that e0£L=\f S'x. Now for each /= ( / l 5 /,), we see that / j , = S'x-x(I)p(I, 1, )/?((0, /2), 2, z2)e0 is in L. Using this fact and making computations such as given above, we show that e(i,o) is in L. Let iV=(« l5 «,) be such that |x(iV+£i)l = sup{|x(0'i,y2))|: jx S «i + l , k «2} > 0. Proceeding as above, )>N c i N + E ö ß i N + E ^ 1, n i)/f((l , «2), 2, n2) x(U1 + n1,j2 + n2)) ' 2 O'l, (1,0) *(("i + 1, «2)) e(i,o) X ( P(Ul + >h,k + n2), 1, n1)P((j1,j2 + n2), 2, 772) f I ^((»i + l ,«2) ; 1,«i)/?((1,»2),2,»2) J 96 B. S. Yadav and R. Bansal ^ ( fl(C/i + Wi J2 + K2). 1, n^PjUiJz + n^, 2, n2) ? ~ UvAVO.d.0)V + 1 , «2), 1 , njp((l, «2), 2 , « 2 ) J ( ^((./1+ «1-1,72+ »2)1, Wi- 1)/?(C/i,J« + hQ, 2, wg) f ^ X l ^ ( ( « i + l , «2), 1 , « 1 - 1 ) 0 ( ( 2 , nj). 2, «2) J ~ 1 ^(2,0), 1 Uvjt)*0, (1,0) This implies that Similarity, we can show that (̂0, L)Ç• Continuing this process, we see that for all / ^ 0 , and hence x is a cyclic vector for S. Theorem 2. Let T={TX, T2, ...,TN) be a system of N-variable forward weight-ed shifts with the weight-net {«'/_;} which is monotonically decreasing and satisfies condition (II). If {vt'/j}/ is square-summable for some j, ISj^N, then any vector x=2î x(J)eJ H with x(0)^0 is a cyclic vector for T. j Proof . Again, we prove the theorem for N=2. Let the net {vv, ,}, be square-summable. It can be easily seen from condition (II) that (4) Now a(O, 2, /2)w(l i 2 ) 1 = w0 l a( ( l , 0), 2, i2). T'x= {T$T£)(2 x(J)ej) = = 2 x{Ui>J\))<*(Ui,h), 2, '2)a((7i,72 +'2), 1, h)e(jl+h,h+il), and hence, using successively (4) and the fact that is monotonically decreasing, we have T'x ei ~ = Z x(0)<x(0, 2, i2)a((0, i2), 1, x(UiJù) I" [ « {tii ' 2> h) « {Ui, h + h), 1, ii) Y { <x(0, 2, /2)a((0, i2), 1, /1) ) x(0) = Î ^ Î " U'o.i '1 u j , x{tii,U) f( "((./1,72), 2, i2)cc(U1,j2 + i2), 1, il) { "((7i ,À , 2,/ ot (. { «((1, 0), 2, /,)« 77)̂ 01 x(O) | l (( , ), , /2 a((l, i2), 1, h) x((A,A)) |2 ^ ( M )a A—î 2 x(0) Since {wf i}, is square-summable and {e,}, is an orthonormal basis of H, using the fact tha tthe vectors {TIx)t are linearly independent, it follows by the Paley— Wiener Theorem [8] that x is a cyclic vector for T. Systems of Ar-variable weighted shifts 97 4. A partial solution of the transitive algebra problem. Let 21 denote a weakly closed subalgebra of B(H), the Banach algebra of all bounded operators on H. We shall write Lat 91 for the lattice of all invariant subspaces of 91, and shall call 9i transitive if Lat 91={{0}, H}. It is easy to see that B(H) is a transitive algebra. Whe-ther there exist transitive algebras other than B(H) is an open question, known as the transitive algebra problem, raised by K A D I S O N [3] first in 1955. Since then, although the problem remains unsolved in general, a number of partial solutions of the pro-blem have been obtained by a number of mathematicians, based mainly on the first partial solution given by ARVESON [1]. LOMONOSOV [4] , however, used different techniques to obtain a partial solution of far reaching consequences. For an elegant account of these results we refer to RADJAVI and ROSENTHAL [7], see also PEARCY and SHIELDS [6]. We prove here the following theorem which generalizes a result due to N O R D G R E N , RADJAVI a n d ROSENTHAL [ 5 ] : Theorem 3. If a transitive algebra 91 contains a system of N-variable backward weighted shifts S with the monotonically decreasing weight-net {wtj} and {vt'j ;}/ is square-summable for some j, l^j^N, then 9 l = B ( H ) . In order to prove the theorem we shall need the following lemma which is a generalization of Corollary 1 of [5; p. 176]. Let us denote by B'n> the direct sum of n copies of an operator B on H. Lemma. 7/91 is a transitive algebra containing operators A andB on H such that (i) every common eigenspace of A and B is one-dimensional, (ii) for each 0, every non-zero common invariant subspace of A("> and B(n) contains a common eigenvector, then 91 is B(H). In the proof of the theorem we consider again only the case N=2. Thus we assume that 91 contains the family S= {S^, £2}. Assume that {wt J j is square-summable. In the light of above Lemma, it is sufficient to show that Sx and S2 satisfy conditions (i) and (ii). Let L be a common eigenspace of SL and S2. Then L={x£H: S1x = /.x and S2x=fix} for some complex numbers /., ¡i. For any vector x = ̂  x(J)e} in L, we have 2 x(UiJJ)*Ui.MAeUi-i.j2> = K2 x{Ui,k))eUl,j^}, j 1^0 2 x(Ui>j2))w(h.h),2e(h,h-D = K2 x((jL,j^)e(juh}), which implies that M ( A , / 2 ) ) = X ( ( J l + ^ A ) ) W 0 l + l , j 2 ) > l , H x { { h , j , y ) = X ( 0 \ , A + l ) ) " ' « ! , . 72+1) ,2 7 98 B. S. Yadav and R. Bansal if 0- This leads to x((.hJ,)) = ¿J*tiJ*[P(Ui,h), Ui)P(<PJJ, 2,y2)]_1A'((0, 0)) for (j1,j2)^0. Thus A'=X((0, 0)) v, where y = Z^'^lttUiJd, Ui)P((0J2), 2,A)']-Ie(jljl) is a fixed vector in H, and hence L is one-dimensional. Next, let M be a non-zero common invariant subspace of S^ and S2\ for n>0 . Let A^© x2 ©.. . ©x n be any non-zero element of M, where xk = 2 x(k, J)e}, 1 ̂ k^n. If for each k the net {x(k,J)}j has only finitely many non-zero elements, then the common invariant subspace My of S^0 and .St") generated by Â  © x2 ©. . . . . .©xn is finite-dimensional. Therefore, Mx, and consequently M, has a common eigenvector of S'"^ and S2\ We therefore assume that for every I=(i\, i2), we have max{\x(k, Ui+h,.h + >i))\'- h,k = 0, 1 ^ k s= n} > 0. Let I=(i\, >•,) be fixed. Then there is a multi-index R = R(I) = (r1, /•,), rxs=, r,,i2 and a number s=s(I) between 1 and n such that (5) | x(s, 7?)| = max {\x(k, (j\ + h, .h + *'2))|: k , h > 0, l s ^ //}. Now (S(n))R J © A J = © SpS[Kxk gives us that ( S ^ y c x x e ^ e ... ®xn) R)p(R, 2, r2)P((ri, 0), 1, i^)]"1 = = R))-1 © x(k, R)e0] + [ © y(K *)], where for each k y(k,R) = y ( x(k, (A + r,,./, + r2))p((.A -f ry,7a + /-•?), 1, i\) p((h,h-\-roX 2, r2)\ I x(s,R)P((r1,r2),l,r1) fi((0, r2), 2, rs) ) e<h'hV Making use of (5), (I) and the facts that the net {»fT is monotonically decreasing and {w/.i}/ ' s square-summable, we have y UCfc.C/i + r! Jg+rQ) |2 ( P(U\ + r1J2+r2), i,r1)p((j1>j2 + r2),2,r2) V 0 I x(s, (rj, r2\) | I P((>1, r2), 1, ri)P((0, r2), 2, / ,) ) ~ ^ y ( ( P ( U i + ' i - 1 ,k + '-2), 1, /1 ~ l ) l ( ( J i J 2 + r2), 2, r2)I2 ^ OW^O V. '̂(1,0), 1 = — 2 Wc/i+'i.A+r,).i - 0 as | / | - CO. ,V(1,0),1 UvJi)*0 Systems of TV-variable weighted shifts 99 Thus the vector [JC (,s, i?)]_1 © x(k, R)e0 is in M. Now for each k, the net k = 1 {x(k, R)/x(s, R):R = R(I), I = (/l5 /,)} is contained in the unit disc, and therefore contains a subnet convergent to a number zk (say). It can be easily seen that there is a number k0 (1 ̂ ¡k0~n) such that k0 will occur infinitely many times as a value of s=s(I). Corresponding to this k0, we have z^ = 1. Thus z1e0®... ©z„e0 is a common eigenvector of S^ and in M. Lastly we make the following remarks: 1. The condition of monotonicity of the net {wItJ} in all our theorems can be replaced by the following less stringent requirements: if j0 is the integer for which {WIJ}I is square-summable, then (0 i,),j = wu[,i:,...,rN)j f o r l^J^N if ik^i'k, l^k^N, (ii) ¡. . ¡. _ i f . . , j ^w, / . .<, . for l ^ j ^ N if ik=i'k, 1 =k=N, k^j'o, = 1 (ijo = 1 in Theorem 2, Theorem 3). 2. We observe that Theorem 2 and Theorem 3 can be proved in a more general form on the lines of Theorem 1 in [10] and Theorem 1 in [11] respectively. References [1] W. B. ARVESON, A density theorem for operator algebras, Duke Math. J., 34 (1967), 635—647. [2] N . P . JEWELL and A . R . LUBIN, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory, 1 (1979) , 2 0 7 — 2 3 3 . 13] R . V . KADISON, On orthogonalization of operator representations, Amer. J. Math., 7 8 (1955) , 600—621. [4] V. J. LOMONOSOV, Invariant subspaces for operators commuting with compact operators, Functional Anal., Appl. 7 (1973) , 2 1 3 — 2 1 4 . [5] E . NORDGREN, H . RADJAVI and P . ROSENTHAL, On density of transitive algebras, Acta Sci. Math., 30 (1969), 175—179. [6] C . PEARCY and A . L. SHIELDS, A Survey of Lomonosov technique in the theory of invariant subspaces, in: Topics in Operator Theory, Amer. Math. Soc. Surveys, No. 13 (Providence, R. I., 1974). [7] H. RADJAVI and P. ROSENTHAL, Invariant Subspaces, Springer-Verlag (New York, 1973). [8] F . RIESZ and B . SZ.-NAGY, Functional Analysis, Ungar (New York, 1955) . [9] A. L. SHIELDS, Weighted shift operators and analytic function theory, in: Topics in Operator Theory, Amer. Math. Soc. Surveys, No. 13 (Providence, R. L, 1974). [10] B . S . YADAV and S . CHATTERJEE, On a characterization of invariant subspace lattices of weighted shifts, Proc. Amer. Math. Soc., 84 (1982), 442—446. [11] B. S. YADAV and S. CHATTERJEE, A partial solution of transitive algebra problem, Acta Sci. Math., 42 (1980), 211—215. DEPARTMENT O F MATHEMATICS UNIVERSITY O F DELHI DELHI—110007, INDIA 7* 

On systems of N-variable weighted shifts

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On systems of N-variable weighted shifts

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