ANALYSIS International Journal of Analysis 2314-4998 2314-498X Hindawi Publishing Corporation 397262 10.1155/2013/397262 397262 Research Article The Near Subnormal Weighted Shift and Recursiveness Ben Taher R. Rachidi M. Qian Chuanxi 1 Group of DEFA, Department of Mathematics and Informatics Faculté des Sciences Université Moulay Ismail BP 11201, Zitoune Méknés Morocco umi.ac.ma 2013 27 3 2013 2013 19 10 2012 14 02 2013 2013 Copyright © 2013 R. Ben Taher and M. Rachidi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We aim at studying the near subnormality of the unilateral weighted shifts, whose moment sequences are defined by linear recursive relations of finite order. Using the basic properties of recursive sequences, we provide a natural necessary condition, that ensure the near subnormality of this important class of weighted shifs. Some related new results are established; moreover, applications and consequences are presented; notably the notion of near subnormal completion weighted shift is implanted and explored.

1. Introduction

Let α={αn}n be a bounded sequence of (or ) and a separable Hilbert space of basis {en}n0. The unilateral weighted shift with weight sequence α is defined by Wαen=αnen+1. The moments of the operator Wα are given by (1)γ0=1,γkγk(α):=α02α12αk-12if    k>0.

It is well known that the bounded operator Wα can never be normal, and it is hyponormal if and only if |αn||αn+1| (see ). For a given positive DB(), the space of bounded operators on , an operator TB(), is called a D-near subnormal operator if there exists a constant m>0 satisfying mDT*DT. An hyponormal operator TB() is called near subnormal if it is QT-near subnormal, where QT=T*T-TT*(QT0). Some necessary and sufficient conditions guaranteeing the near subnormality for unilateral weighted shifts have been established in [4, 5].

In this paper, we are interested in studying the near subnormal unilateral weighted shifts, when the sequence γ{γn}n of moments α satisfies the following linear recursive relation of order r2: (2)γn+1=a0γn+a1γn-1++ar-1γn-r+1for  every  nr, where γ0,γ1,,γr-1 are the initial data and a0,a1,,ar-1 are fixed real or complex numbers with ar-10. Such sequences are widely studied in the literature, generally called r-generalized Fibonacci sequences (see  and references therein). A weighted shift Wα such that γ={γn}n0 is a sequence (2) is called a recursive weighted shift. Our motivation in considering sequence (2) is inspired from the fact that every weighted shift is norm-limit of recursively generated weighted shifts (for further information we refer, to [1, 2, 7], e.g.,). On the other hand, these sequences play a central role in the characterization of the subnormality via the truncated moment problem (see [1, 2, 7, 8]). It turns out that following Curto-Fialkow's approach, the roots of the polynomial P(z)=zr-a0zr-1--ar-1z-ar=i=1s(z-λi)ki, called the characteristic polynomial of (2), play an important role, for establishing properties of subnormality, via Berger's Theorem. Moreover, in the process of construction of the generating measure, related to the truncated moment problem and subnormality of [8, 9], it reveals some significant obstructions when all roots λj(1js) are not simple; some additional conditions on the initial data γ0,γ1,,γr-1 are necessary for one thing, to guarantee the existence of the generating measure. On the other hand, it was established in  that every γn can be expressed as a moment of distribution of discrete support.

In this paper, we describe a deductive reasoning to prove that appending a mild hypotheses to the natural necessary condition for the existence of the hyponormal operators is sufficient, so that the unilateral weighted shifts, whose associate moment sequences satisfy (1), are near subnormal (Section 2). The main tool employed is the Binet formula of sequence (2) (see ). We then uprise to the near subnormal completion problem (NSCP) of order m, while at the same time evolving the subnormal completion problem (see ), the case m=2 is examined and solved. In the last section, we are interested in stretching our study for characterizing the near subnormality of a recursive weighted shift Wα such that the moment sequence γ={γn}n0 satisfies (2); we employ some results on the moment of distributions of discrete support (see [8, 10]). The construction of the representing distribution is derived from the Binet formula of sequence (2); we preclude to set any condition on the initial data γ0,γ1,,γr-1. The closed relation of the near subnormality and the subnormality of weighted shifts is discussed.

2. Recursive Sequences and Near Subnormality of Unilateral Weighted Shifts

Let TB() be a hyponormal operator and a sequence α={αn}n such that Ten=αnen+1, for all n1. It was established in  that if 0<α1<α2<α3<, then T is near subnormal if and only if supn1{αn((αn+12-αn2)/(αn2-αn-12))1/2}<+, where α0=0. This characterization of near subnormality for unilateral weighted shifts is more practical and adequate in this section. Suppose that the moment sequence γ={γn}n of the weighted shift α satisfies (2). Expression (1) shows that αn2=γn+1/γn, for n1; as a matter of fact, we formulate easily our first telling result as follows.

Proposition 1.

Let TB() be a hyponormal unilateral weighted shift with Ten=αnen+1 for all n and γ={γn}n the moment sequence (1) associated with α, satisfying |αn|<|αn+1|, for all n. Then, T is near subnormal if and only if the sequence {(γn-1/γn)1/2((γn+2γn-γn+12)/(γn+1γn-1-γn2))1/2}n1 is bounded.

The advantage of this result consists of its application to the former sequence (2), for establishing sufficient conditions on the near subnormality.

Theorem 2.

Let γ={γn}n be a positive sequence (2) and P(z)=i=1s(z-λi)ki(k1++ks=r) its characteristic polynomial with 0<|λ1||λ2|<|λs-1|<|λs|. Let α={αn}n be the sequence defined by αn2=γn+1/γn, satisfying |αn|<|αn+1|, for all n. Then, the hyponormal operator T associated with unilateral weighted shift α is near subnormal.

Proof.

Let γ={γn}n be a positive sequence (2) and P(z)=i=1s(z-λi)ki its characteristic polynomial with 0<λ1λ2<λs-1<λs. It follows from the Binet formula that γn=i=1sj=1kici,jnjλin, for all n (see ). A straightforward computation leads to have limn+{(γn-1/γn)1/2((γn+2γn-γn+12)/(γn+1γn-1-γn2))1/2}=λs-11/2. By the above Proposition 1, we obtain the desired result.

More generally, suppose that 0<|λ1||λ2||λs-1||λs|; it may occur in the Binet formula γn=i=0spi(n)λin, where pi(n)  =j=0kici,jnj, that there exist λs-d,,λs(d1) such that λiλj, |λi|=|λj| for s-dijs and |λj|<|λs| for 1js-d-1. If deg(ps(z))>deg(pj(z)), for every 0<js-1, then a straightforward computation allows us to establish that supn1{αn((αn+12-αn2)/(αn2-αn-12))1/2}<+. Therefore, under the preceding data, the conclusion of Theorem 2 is still valid. Now, let us study the general situation when there exist k1,,kr with s-dkikjs(1ijr) such that deg(pki(z))=deg(pkj(z)); we can also demonstrate that supn1{αn((αn+12-αn2)/(αn2-αn-12))1/2}<+. To establish this result, a long and direct computation is necessary; and the following lemma will be useful.

Lemma 3.

Let γ={γn}n be a positive sequence (2) and suppose that its characteristic polynomial is given by P(z)=(z-λ1)s(z-λ2)s, where λ1λ2 and |λ1|=|λ2|0. Let α={αn}n be the sequence defined by αn2=γn+1/γn, satisfying |αn|<|αn+1|, for all n. Then, we have supn1{αn((αn+12-αn2)/(αn2-αn-12))1/2}<+.

The proof of this lemma is more technical and for the reason of clarity we omit it. For the reason of simplicity, we study the general situation, when the Binet formula is given by γn=ps(n)λsn+ps-1(n)λs-1n+i=1s-2pi(n)λin, where deg(pi(z))<deg(ps(z))=r for 1is-2, ps(n)=anr+qs(n)(a0), and ps-1(n)=bnr+qs-1(n)(b0) with qs(n)=j=1r-1ajnj and qs-1(n)=j=1r-1bjnj. Since limn+pi(n)λin/ps(n)λsn=0, for every 1is-2, Lemma 3 implies the following.

Lemma 4.

Let γ={γn}n be a positive sequence (2) and suppose that its characteristic polynomial is P(z)=(z-λs)r(z-λs-1)rj=2s-3(z-λj)mj, where λsλs-1, |λs|=|λs-1|0 and |λj|<|λs|, for all 1js-2. Let α={αn}n be the sequence defined by αn2=γn+1/γn, satisfying |αn|<|αn+1| for all n. Then, we have supn1{αn((αn+12-αn2)/(αn2-αn-12))1/2}<+.

Lemmas 3 and 4 permit us to formulate the following extension of Theorem 2.

Theorem 5.

Let γ={γn}n be a positive sequence (2)   and P(z)=i=1s(z-λi)ki its characteristic polynomial with 0<|λ1||λ2||λs-1||λs|. Let α={αn}n be the sequence defined by αn2=γn+1/γn, such that |αn|<|αn+1| for all n. Then, the hyponormal operator T associated with unilateral weighted shift α is near subnormal.

We manage to reassemble the above results as follows.

Theorem 6.

Let TB(H) be a hyponormal operator such that Ten=αnen+1(n0). Suppose that the sequence of moment γ={γn}n is a positive sequence (2). Let α={αn}n be the sequence defined by αn2=γn+1/γn, satisfying |αn|<|αn+1|, for all n. Then, T is near subnormal.

In , an important class of subnormal weighted shifts is explored by considering measures μ with two atoms λ1 and λ2. A sequence {γn}n0 such that γ0=1, γ1 and γn+1=a0γn+a1γn-1(forn1) is a moment sequence if and only if P(γ1)0. As a matter of fact, we study the fallout of our approach by providing a connection between hyponormality, near subnormality and subnormality for this class of operators of weighted shifts.

Proposition 7.

Let γ={γn}n be a positive sequence (2) and P(z)=(z-λ1)(z-λ2) its characteristic polynomial with 0<λ1λ2. Let α={αn}n be the sequence defined by αn2=γn+1/γn and T the operator associated with unilateral weighted shift α. Suppose that αnαn+1, for all n. Then, the following five statements are equivalent: (i) α1>α0; (ii) T is hyponormal; (iii) T is k-hyponormal for all k; (iv) T is near subnormal; (v) T is subnormal.

Similar to the subnormal completion problem (see [1, 2]), the NSCP can be formulated as follows: “Let α(m)={αn}0m be a finite collection of positive numbers, find necessary and sufficient conditions on αm to guarantee the existence of a near subnormal weighted shift whose initial weights are given by α(m)". The first obstructions encountered for solving this problem are the natural necessary condition for the existence of the hyponormal completion. That is, once we know that α(m):α0,,α2m(m1) admits a k-hyponormal completion which is recursively generated, the condition αn<αn+1 is not easy to be satisfied in order to apply Theorem 6. When m2, the problem becomes highly nontrivial. For m=1, the strategy to solve NSCP is as follows. Given α(1):α0,α1,α2 such that α0<α1<α2, set γ0=1,y1=γ02,y2=α02α12, and γ3=α02α12α22. First, we use these terms to construct a recursive sequence {γn}n0 of order 2, by setting γn=ρ0γn-1+ρ1γn-2 for all n2. A straightforward calculation gives ρ0=α12(α22-α02)/(α12-α02),ρ1=α12α22(α12-α22)/(α12-α02) and αn+22=αn+12+(αn-12(αn+12-αn2)/αn+12(αn2-αn-12)), n1, where αn2=yn+1/yn. Thus, we obtain αn>αn-1, for every n0. And the completion α^={α}n0 of α(2) is hyponormal and recursively generated, and the condition αn<αn-1 is satisfied. It follows from Theorem 6 that Wα is near subnormal. As a matter of fact, we mange to have the following result.

Proposition 8.

Let α:α0<α1<α2 be an initial segment of positive weights. Then, α has a near subnormal completion.

3. The Moment of Distributions and the Near Subnormality of Unilateral Weighted Shifts

In this section, we are interested in formulating some results in the moment of distributions of discrete support, with a view to characterize the near subnormality and its closed relation with the subnormality of weighted shifts. Let H be a separable Hilbert space and {en}n0 its orthonormal basis. Let α={αn}n0 be a bounded sequence of nonnegative real numbers and Wα the bounded operator defined by Wαen=αnen+1. Let γ={γn}n0 be the sequence of moments associated with Wα. By Berger's theorem, Wα is a subnormal operator if and only if there exists a nonnegative Borelean measure μ, which is called a representing measure of γ={γn}n0, with supp(μ)[0,Wα2] such that γn=0Wα2tndμ(t),  (n0), where Wα2=supn0αn. Hence, the moment problem and subnormal weighted shifts are closely related. And it was shown in  that a sequence (2) γ={γn}n0 admits a generating measure (not necessary positive) if and only if its characteristic (minimal) polynomial Pγ has distinct roots, with supp(μ)=Z(Pγ), the set of zeros of Pγ (for more details, see Proposition 2.4 of ). It was pointed out in  that if Pγ(z)=i=0s(z-λi)pi, with pi2 for some i, then γ is a moment sequence for some distribution T=i=0pj=1piρi,jδλi(j), where δλi(j) is the jth derivative in the meaning of distribution of the Dirac measure δλi (see ). To find this distribution (supported by a compact K) interpolating γ, the Binet formula plays a primordial role. Indeed, let K be compact subset of (or ), U a neighborhood of K, and consider the function gK of class C satisfying the following three conditions: (i) gK(t)=1 for every tK; (ii) 0gK(t)1 for every t or ; and (iii) gK(t)=0 for every t-U (or -U). It was established in  that for a distribution T of compact support K, the real (or complex) number γn=T|gK(t)tn is independent of the function gK(t), for every n (see Lemma 1 of ). The number γn=T|gK(t)tn=T|tn(n0) is called the moment (or power moment) of order n of the distribution T.

Let δi be the Dirac measure at the point λi and δi(j) its jth derivative. It is well known that δi and δi(j) define two distributions on the space of polynomial functions on (or ). Moreover, every distribution of discrete support {λ1,,λs} can be written under the form T=i=0pj=1pici,jδλi(j) (see,  e.g.,). We denote by DK the -vector space of distributions of discrete support, contained in K. Hence, the preceding discussion and Theorem 6, allow us to derive the following result.

Theorem 9.

Let WB(H) be a hyponormal operator such that Wen=αnen+1(n0) and |αn|<|αn+1|, for all n. Suppose that the sequence of moments γ={γn}n is a positive sequence and there exists a distribution TDK such that γn=T|tn, for every n0. Then, W is near subnormal.

The natural extension of the K-moment problem can be formulated as follows. Let K be a closed subset of and {γn}n0 a sequence of . The associated distributional K-moment problem consists of finding a distribution T of support contained in K such that (3)γn=T|tn,foreveryn(n0).

A distribution μ solution of problem (3) is called a representing distribution of the sequence γ={γn}n0. Therefore, the equivalent form of Theorem 9 can be expressed as follows.

Proposition 10.

Let WB(H) be a hyponormal operator such that Wen=αnen+1(n0) with |αn|<|αn+1|, for all n. Suppose that the sequence of moments γ={γn}n is a positive sequence. If the distributional moment problem (3) owns a solution μDK, then W is near subnormal.

In light of Riesz theorem, it is well known that if T is a positive distribution then T|φ=Ktndμ(t), where μ is a measure, and we write T=Tμ=μ (see,  e.g.,). As a consequence, we have the following result.

Theorem 11.

Let WB(H) be a hyponormal operator given by Wen=αnen+1(n0) such that its sequence of moments γ={γn}n is a positive sequence. Let α={αn}n be the sequence defined by αn2=γn+1/γn, satisfying |αn|<|αn+1|, for all n. If the moment problem (3) owns a positive representing distribution μDK, then W is subnormal.

Acknowledgments

The authors would like to thank the anonymous referee for his (or her) useful remarks and suggestions that improved this paper. M. Rachidi is an Associate with “Group of DEFA.”

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