Spectral Characterization of Hyponormal Weighted Shifts

The complete characterizations of the spectra and their various parts of hyponormal unilateral and bilateral weighted shifts are presented respectively in this paper. The results obtained here generalize the corresponding work of the references.

Some Notes on M-hyponormal Weighted Shifts and Hyponormalizable Weighted Shifts

Let {an}∞n=0be a weight sequence and let W denote the associated unilateral weighted shift on H. In this paper, we consider the connection between the M-hyponormal and hyponormalizable weighted shift operator. Main results are Theorems 4.1 and Theorems4.2. Theorem 4.1 gives the sufficient condition that a weighted shifts M-hyponormal operator is hyponormalizable. Theorem 4.2 gives the sufficient condition that a hyponormalizable weighted shift operator is M-hyponormal. Finally, invariant subspaces of such operators are discussed.

Near Subnormality of Weighted Shifts and the Answer to the Hilbert Space Problem 160

In this paper, necessary and sufficient conditions are obtained for unilateral weighted shifts to be near subnormal. As an application of the main results, many answers to the Hilbert space problem 160 are presented at the end of the paper.

Reducing subspaces of tensor products of weighted shifts

A unilateral weighted shift A is said to be simple if its weight sequence {α_n} satisfies ▽~3(α_n^2)≠0for all n≥2.We prove that if A and B are two simple unilateral weighted shifts,then AI+IB is reducible if and only if A and B are unitarily equivalent.We also study the reducing subspaces of A^kI+IB^l and give some examples.As an application,we study the reducing subspaces of multiplication operators Mzk+αωl on function spaces.

LOCAL SPECTRA OF UNILATERAL OPERATOR WEIGHTED SHIFTS

In this note, the local spectral properties of unilateral operator weighted shifts are studied.

Classification of multiple unilateral weighted shifts by Aχ0

Abstract In this paper, it is characterized when a multiple unilateral weighted shift belongs to the classes An （1 ≤ n ≤ x0）. As a result, we perfect and generalize the previous conclusions given by H. Bercovici, C. Foias, and C. Pearcy. Moreover, we remark that Question 21 posed by Shields has been negatively answered.

The Essential Spectrum and Banach Reducibility of Operator Weighted Shifts

For an operator weighted shift S,the essential spectrum σ_e(S) and the indices associated with holes in σ_e(S) are described.Moreover,Banach reducibility of S is investigated and a condition for S~* to be a Cowen-Douglas operator is characterized.

K_0-group and similarity of operator weighted shifts

By using simultaneous triangularization technique, similarity for operator weighted shifts with finite multiplicity is characterized in terms of K0-group of commutant algebra. The result supports the conjecture posed by Cao et al. in 1999. Moreover, we discuss the relations between similarity and quasi-similarity for operator weighted shifts.

ON REFLEXIVITY OF HYPONORMAL AND WEIGHTED SHIFT OPERATORS

By an elementary proof, we use a result of Conway and Dudziak to show that if A is a hyponormal operator with spectral radius r（A） such that its spectrum is the closed disc {z：｜z｜ ≤ r（A）} then A is reflexive. Using this result, we give a simple proof of a result of Bercovici, Foias, and Pearcy on reflexivity of shift operators. Also, it is shown that every power of an invertible bilateral weighted shift is reflexive.

Cesàro Bounded Weighted Backward Shift

In this paper, we give a criterion of the absolutely Cesàro bounded weighted backward shift in spirit of the comparison method. Our approach is to construct the proper product of weight functions <img src="Edit_7232e0dc-07ab-41c5-8657-a49f0463b47c.bmp" alt="" />by the fraction of two monomials of the indexes, then we apply proper scaling to give Cesàro boundedness. In particular, we present a new example of non Cesàro bounded weighted backward shift on <img src="Edit_799dadb7-40ab-48f9-bae3-191378f96164.bmp" alt="" />.

Strongly Cyclic and Strongly Irreducible Decomposition

Let W be an injective unilateral weighted shift, and let W(n) be the orthogonal direct sum of n copies of W. In this paper, we prove that, if the commutant of W is strictly cyclic, then W(n) has a unique (SI) decomposition with respect to similarity for every natural number n.

Chaotic Unilateral Weighted Backward Shifts on Sequentially Complete Topological Sequence Spaces

Extending previous results of Grosse-Erdmann and Peris we obtain a characterization of chaotic unilateral weighted backward shifts on sequentially complete topological sequence spaces in which the canonical unit vectors(e_(n))_(n=1)^(∞) form an unconditional basis.

About Operator Weighted Shift

Abstract Let H be a complex seperable Hilbert space and ^（Jt^） denote the collection of bounded linear operators on H. For an operator in L（H）, rad{A}＇ denotes the Jacobson radical of the commutant of A. This paper characterizes the similarity of strongly irreducible operator weighted shift in terms of {A}＇/rad{A}＇. Moreover, we suggest some ways to determine when an operator weighted shift is strongly irreducible and when its commutant is commutative.

Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators

High order discretization schemes playmore important role in fractional operators than classical ones.This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones;but for fractional operators the stencils for high order schemes and low order ones are the same.Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved.Using the fractional linear multistep methods,Lubich obtains the n-th order(n≤6)approximations of the a-th derivative(a>0)or integral(a<0)[Lubich,SIAM J.Math.Anal.,17,704-719,1986],because of the stability issue the obtained scheme can not be directly applied to the space fractional operator with a∈(1,2)for time dependent problem.By weighting and shifting Lubich’s 2nd order discretization scheme,in[Chen&Deng,SINUM,arXiv:1304.7425]we derive a series of effective high order discretizations for space fractional derivative,called WSLD operators there.As the sequel of the previous work,we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich’s 3rd and 4th order discretizations.In particular,we prove that the obtained 4th order approximations are effective for space fractional derivatives.And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.

Spectral Decomposition of Weighted Shift Operators

The reseearch on the relation between weighted shift operators on Hilbert spaces and other important class of operators attracted the attention of some mathematicians. For example, the relation between weighted shift operators and subnormal operators has been thoroughly studied by J. Stampfli, R. Gellar and D. A. Herrero, etc. (see reference [1]) But the decomposability of weighted shift operators has not yet attracted enough attention up to now. We made initial research

On Hyponormal Weighted Unilateral Shift of Multiplicity K(0

It is shown in[6]and[7]that a necessary and sufficient condition for a hyponormal weightedunilateral shift to be unitarily equivalent to a Toeplitz operator is that its weights satisfy(1-|an|2)=(1-|a0|2)(1-|an-1|2)n≥1,where ansatisfiesan=1.In[8],the author obtainedsimilar result for the hyponormal weighted unilateral shift of multiplicity 2.The aim of this

A Comprehensive Evaluation Method for Sensing Characteristics of Multi-Peak Terahertz Metamaterials Sensor in Polarization Mode

In this study,the multi-peak terahertz metamaterials sensors are designed and fabricated,whose structures are the asymmetrical single split ring(SSR)and three split rings(TSR).The resonant formation and sensing mechanism of the two structures are investigated by using the finite-difference time-domain(FDTD)method.Vitamin B6(VB6)and its reactants with bovine serum protein(BSA)are tested as the medium,and the sensing experiments of the SSR and TSR are carried out.The experimental and simulation results indicate the consistent law,which is the sensitivity of the resonance in the transverse magnetic(TM)mode is much greater than that in the transverse electric(TE)mode.According to the weighted average method and the law for unequal precision measuring,the quality factor of the resonance is used as the weighting coefficient to calculate the comprehensive evaluation parameter(CEP)of the multi-peak metamaterials sensors in the TE and TM modes based on the experimental data.When the CEP and frequency shifts are as the evaluation parameter in experiments,the law’s variation of the CEP is consistent with that of the frequency shift,indicating that it is feasible to characterize the sensing characteristics of metamaterials with the CEP,which presents simplified characteristics of multi-peak metamaterials at different polarization modes.The method implies that the different influencing factors may be integrated into the CEP with the idea of weight,which promotes the practical application of the metamaterials sensor.The revelation of the sensing law also provides a method for the design of the terahertz metamaterials sensor with the high sensitivity.

A conservative numerical method for the fractional nonlinear Schrodinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.