Bourhim, A

Chidume, C E

Zerouali, E H

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Available at: http://www.ictp.trieste.it/~pub off IC/2000/122United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSBOUNDED POINT EVALUATIONS FOR CYCLIC OPERATORSAND LOCAL SPECTRAA. Bourhim1Departement de Mathematiques - Informatique, Faculte des Sciences,Universite Mohammed V, B.P. 1014, Rabat, MoroccoandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,C.E. Chidume2The Abdus Salam International Centre for Theoretical Physics, Trieste, ItalyandE.H. Zerouali3Departement de Mathematiques, Universite Mohamed V, B.P. 1014, Rabat, MoroccoandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.MIRAMARE { TRIESTEAugust 20001E-mail: bourhim@ictp.trieste.it2E-mail: chidume@ictp.trieste.it3Regular Associate of the Abdus Salam ICTP. E-mail: zerouali@fsr.ac.ma1AbstractWe study in this paper the concept of bounded point evaluations for cyclic operators. We givea negative answer to a question of L.R. Williams Dynamic Systems and Applications 3(1994)103-112. Furthermore, we generalize some results of Williams and give a simple proof of theorem2.5 of [11] that nonnormal hyponormal weighted shifts have fat local spectra.21. introductionThroughout this paper, L(H) will denote the algebra of all linear bounded operators on aHilbert space H. For an operator S 2 L(H), let Sdenote its adjoint, (S) its spectrum, p(S)its point spectrum, ap(S) its approximate spectrum, ,(S) its compression spectrum, r(S) itsspectral radius, m(S) its lower bound (i.e., infkSxk : kxk = 1	) and r1(S) = supnm(Sn)1nwhich equals limnm(Sn)1n. For F  C , we denote by F = fz : z 2 C g its conjugate set, int(F )its interior and cl(F ) its closure.Let T 2 L(H) be a cyclic operator on H with cyclic vector x that is the nite linear com-binations of the vectors x; Tx; T2x; : : : are dense. A complex number  2 C is said to be abounded point evaluation of T if there is a constant M > 0 such thatjp()j Mkp(T )xkfor every complex polynomial p. The set of all bounded point evaluations of T will be denotedby B(T ). Note that it follows from the Riesz Representation Theorem that  2 B(T ) if andonly if there is a unique vector denoted k() 2 H such that p() = hp(T )x ; k()i for everycomplex polynomial p. By setting q(z) = (z   )p(z), we obtain  2 p(T). Conversely if k()is an eigenvector of Tassociated to the eigenvalue , we get hp(T )x ; k()i = p()hx ; k()i.Hence hx ; k()i 6= 0 and  2 ,(T ). Which givesProposition 1.1. B(T ) = ,(T ) = p(T).An open subset O of C is said to be an analytic set for T if it is contained in B(T ) and thevector valued function k :  7 ! k() is analytic on O. The largest analytic set for T will bedenoted by Ba(T ) and every point of it will be called analytic bounded point evaluationfor T . Remark that Ba(T ) coincides with conjugate of the usual spectral residuum STof Tgiven by complex numbers  such that there exist an open set V 3  and f : V ! H an analyticfunction satisfying (T  z)f(z) = 0 z 2 V ( see [12] )Tavan Trent proved in [9] that for cyclic subnormal operators, we have,(T )nap(T ) = Ba(T ) ()L.R. Williams proved in [10] proposition 1.3 that for every arbitrary cyclic operator T 2 L(H),,(T )nap(T )  Ba(T ) and asked if () remain valid for arbitrary cyclic operators ( [10] QuestionA ).In this paper, section 2, we rst give necessary and sucient conditions for weighted shiftsto satisfy () and we exhibit an operator which provides a negative answer to the question ofWilliams. Then, in section 3, we devote attention to operators with fat local spectra. Further-more, we generalize theorem 2.1 of [10] to the case of operators satisfying Dunford ConditionC and with no eigenvalues. Finally, we give a simple proof of theorem 2.5 of [11] using Zero-principle for analytic functions.32. analytic bounded point evaluations for unilateral weighted shiftIn [7], A.L. Shields represented a weighted shift operator as ordinary shift operator (that is, as"multiplication by z") on a Hilbert space of formal power series (in the unilateral case) or formalLaurent series (in the bilateral case). He dened the concept of bounded point evaluations ofa weighted shift to examine which power series and Laurent series represent analytic functions.In fact, this concept of bounded point evalutions for injective unilateral weighted shift coincideswith the one dened above by Williams (see [10]).We now describe the set of bounded point evaluations and the set of analytic bounded pointevaluations for an arbitrary injective unilateral weighted shift. Let S be a unilateral weightedshift on a Hilbert space H with a positive weight sequence (!n)n0, that isSen= !nen+1:where (en)n0is an orthonormal basis of H. Let  be the following sequence given by:n=8<:!0:::!n 1if n > 01 if n = 0The unilateral weighted shift S is cyclic with cyclic vector e0. It follows from Corollary 2 of [7]and Proposition 1.4 of [10] that the sets B(S) and Ba(S) have a circular symmetry about theorigin. Also, it follows from Theorem 8 of [7] and Proposition 1.1 that B(S) = f0g if r2(S) = 0otherwise 2 C : jj < r2(S)	 B(S)  2 C : jj  r2(S)	;where r2(S) = lim infn!+1(n)1n:Theorem 2.1. If r2(S) > 0 then Ba(S) = 2 C : jj < r2(S)	.Proof. Let  2 B(S) then there is k() 2 H such that p() = hp(S)e0; k()i for everypolynomial p. For every n  0 we havehen; k()i = h1nSne0; k()i=nn:Hence, k() =Pn0nnenfor every  2 B(S). And so, the desired result holds.For the weighted shift S, the spectrum (S) is known to be the disk 2 C : jj  r(S)	,and the approximate point spectrum ap(S) is known to be the annulus 2 C : r1(S)  jj r(S)	where r1(S) = limn!1infk0n+kk1n(see [7]). Therefore,,(S)nap(S) = (S)nap(S) = 2 C : jj < r1(S)	:Hence we have the following theorem.Theorem 2.2. Let S be a weighted shift. Then, ,(S)nap(S) = Ba(S) if and only if r1(S) =r2(S).4A negative answer to the question A (see [10]) can be given by a unilateral weighted shift Sfor which r1(S) < r2(S).Let us consider an example of a such weighted shift. For s 2 N there are unique n; k 2 N suchthat s = n! + k with 0  k  (n+ 1)!  n!  1. We sets= n!+k= ek:And so, for every k 2 N, we havek+1k=8<:e if n!  k < (n+ 1)!  11e(n+1)! n! 1if k = (n+ 1)!  1Hence, the unilateral weighted shift S corresponding to the weight n+1nn0is bounded. Forevery n  2, set kn= n!   n. Clearly, we have (n   1)!  kn< n! and kn= e n! (n 1)! n.And so,infk0n+kkn+knkn=1e n! (n 1)! n:Hence,r1(S) = limn!1infk0n+kk1n= 0:On the other hand, it is clear thatr2(S) = lim infn!1n1n= 1:Therefore, Ba(S) = 2 C : jj < 1	and ,(S)nap(S) = ;:For the unilateral weighted shift S the set of its analytic bounded point evaluations is exactlythe interior of the set of its bounded point evaluations. This suggests the following question,which was posed by J.B. Conway in [2] page 65 for arbitrary cyclic subnormal operator.Question: Does the interior of B(T ) always coincide with Ba(T ) for an arbitrary cyclic operatorT 2 L(H)?3. local spectra of cyclic operators with dunford's condition cLet T be a bounded operator on a complex Hilbert space H. For an element x 2 H, let T(x)be its local spectrum and T(x) be its local resolvent (see [1]). For a closed subset F of thecomplex plane C , let HT(F ) be the set of elements x 2 H such that T(x)  F ; it is a linearsubspace of H. The operator T is said to have the single valued extension property (svep)if zero is the unique vector x 2 H such that T(x) = ;, which is equivalent to that for everyopen set U  C , the only analytic solution of the equation (T   I)f() = 0 for  2 U is thezero function f  0. The operator T is said to satisfy Dunford's Condition C (DCC) if forevery closed subset F of C , the linear subspace HT(F ) is closed. It is known that every operatorwhich satises DCC has the single valued extension property (see [1]). The operator T is said tobe hyponormal if kTxk  kTxk for every x 2 H; J.G. Stampi [8] and M. Radjabalipour [4]have shown that hyponormal operators satisfy DCC. Recall that T is said to be pure if the only5reducing subspace M of T such that TjMis normal isM = f0g. In [10] Theorem 2.1, it is shownthat for a pure cyclic hyponormal operator T 2 L(H) with cyclic vector x, T(Sx) = (T ) forevery operator S 2 L(H) such that ST = TS and ker(S) is nite dimensional. We generalizethis result as follows.Theorem 3.1. Let T 2 L(H) be a cyclic operator on H with cyclic vector x 2 H and letS 2 L(H) be an operator on H which commutes with T such that ker(S) is nite dimensional.If T satises DCC and p(T ) = ; then T(Sx) = (T ):We prove this theorem using several lemmas. The following two lemmas are given in [10].Lemma 3.2. Let T 2 L(H) be a cyclic operator on H with a cyclic vector x 2 H. If T satisesDCC then T(x) = (T ):Lemma 3.3. Suppose that H = H1 H2where H1and H2are two Hilbert spaces such thatH2is nite dimensional. If T has the single valued extension property and H1is an invariantsubspace for T then A = TjH1has the single valued extension property and A(x) = T(x) forevery x 2 H1.Lemma 3.4. Suppose that H = H1H2where H1and H2are two Hilbert spaces such that H2is nite dimensional. If H1is an invariant subspace for an operator T which satises DCC thenthe operator A = TjH1satises the DCC.Proof. It follows from Lemma 3.3 that for every closed subset F of C , H1A(F ) = HT(F )\H1.So, the desired result holds.Lemma 3.5. Suppose that H = H1 H2where H1and H2are two Hilbert spaces such thatH2is nite dimensional. If T 2 L(H) is an operator without point spectrum such that H1is aninvariant subspace for T then (T ) = (A) where A = TjH1.Proof. We rst writeT =A B0 Cwith respect to the decomposition H = H1 H2. Now, suppose that A is invertible in L(H1)then TH1= H1. And so, H1\ TH2= f0g since T one to one. Let x22 H2such that Cx2= 0then Tx2= Bx22 H1\ TH2, hence C is one to one and so, by the nite-dimensionality, C isinvertible. Therefore the operator T is invertible with inverse given by:A 1 A 1BC 10 C 1Thus (T )  (A). Conversely, It is known that (A) =Sx2H1A(x) (see [3] Theorem 1.9), itfollows from Lemma 3.3 that (A) =Sx2H1T(x)  (T ).Proof of Theorem 3.1. Let H1be the closed linear subspace generated byTnSx : n  0	.Then it is clear that H1is an invariant subspace for T . Since TS = ST and x is a cyclic vector for6T , then H1= cl Im(S). Therefore H = H1H2where H2= ker(S). It follows from Lemma3.3 that T(Sx) = A(Sx) where A = TjH1. Since A is a cyclic operator with cyclic vector Sx andsatises DCC (Lemma 3.4) then it follows from Lemma 3.2 that A(Sx) = (A): Since p(T ) = ;,it follows from Lemma 3.5 that (T ) = (A). And so, (T ) = (A) = A(Sx) = T(Sx): Theproof is complete.Remark 3.6. Let T 2 L(H) be a cyclic operator with cyclic vector x 2 H and satises DCCsuch that p(T ) = ;. For every non-zero polynomial p, T p(T )x= (T ). Therefore, T(y) =(T ) holds for all y in a dense subset of H. L.R. Williams proved in Theorem 2.5 of [11] that ifT is a nonnormal hyponormal (unilateral or bilateral) weighted shift operator then T(x) = (T )for every non-zero element x 2 H.We state and give a simple proof of Theorem 2.5 of [11] using the fact that a non-zero analyticfunction has isolated zeros.Theorem 3.7. Let T be a nonnormal hyponormal weighted shift on H. Then for every a non-zero element x 2 H, T(x) = (T ).Proof. First suppose that T is a nonnormal hyponormal unilateral weighted shift, thenr(T ) = r1(T ) = r2(T ) = kTk > 0 and by Theorem 2.1 we have Ba(T ) = int (T ). Now, letx 2 H such that there is  2 (T )nT(x). So, there is vector valued analytic function f on anopen neighbourhood V of  such that(T   I)f() = x for every  2 V:Since ; 6= V \ int (T ) Ba(T ), then for every  2 V \ int (T )we have,bx() := hx ; k()i= h(T   I)f() ; k()i= hf() ; (T   I)k()i= 0:Hence, bx  0. And so, x = 0.The case of a nonnormal hyponormal bilateral weighted shift is similar using the boundedpoint evaluations in the sense of A.L. Sheilds [7].Remark 3.8. Using the same proof of this Theorem, one can see that for every injective uni-lateral weighted shift T , cl Ba(T ) T(x) for every non-zero element x 2 H. The same holdsfor every bilateral weighted shift.Acknowledgments. This work has been partially supported by the Associate Scheme ofthe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The rst and lastauthors are grateful to the mathematics section of the Abdus Salam ICTP.7References[1] I. Colojoara and C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, NewYork 1968.[2] J.B. Conway, The Theory of Subnormal Operators, volume 36 of Mathematical Surveys and Mono-graphs. American Mathematical Society, Providence, R.I 1991.[3] I. Erdelyi and R. Lange, Spectral Decomposition on Banach Spaces, Lecture notes in Mathematics,Springer Verlag New York, V. 623 (1977).[4] M. Radjabalipour, Ranges of Hyponormal Operators, Illinois J. Math., 21(1977) 70-75.[5] M. Raphael, Quasisimilarity and Essential Spectra for Subnormal Operators, Indiana Univ. Math.J., Vol 31. No.[6] W.C. Ridge, Approximate Point Spectrum of a Weighted Shift, Trans. Amer. Math. Soc. 147(1970)349-356.[7] A.L. Shields, Weighted Shift Operators and Analytic Function Theory, in Topics in Operator Theory,Mathematical Surveys, N0 13 (ed. C. Pearcy), pp. 49-128. American Mathematical Society, Providence,Rhode Island 1974.[8] J.G. Stampi, A Local Spectral Theory for Operators. V: Spectral Subspaces for Hyponormal Oper-ators, Trans. Amer. Math. Soc., 21(1976) 285-296.[9] T.T. Trent, H2() Spaces and Bounded Point Evaluations, Pac. J. Math., 80(1979) 279-292.[10] L.R. Williams, Bounded Point Evaluations and Spectra of Cyclic Hyponormal Operators, DynamicSystems and Applications 3(1994) 103-112.[11] L.R. Williams, The Local Spectra of Pure Quasinormal Operators, J. Math Anal. Appl. 187(1994)842-850.[12] F.H. Vasilescu, Analytic Functional Calculus and Spectral Decomposition, Editura Academei 1982.

Bounded point evaluations for cyclic operators and local spectra

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