On the local spectral properties of weighted shift operators

论复对称加权移位 II

On complex symmetric weighted shifts. II

研究有限有根有向树上的各类加权移位，刻画其复对称性，丰富了复对称算子相关理论。

The spectrum of diagonal perturbation of weighted shift operator

This paper provides a description of the spectrum of diagonal perturbation of weighted shift operator acting on a separable Hilbert space.

A weighted bilateral shift with cyclic square is supercyclic

It is shown that for a bounded weighted bilateral shift $T$ acting on $\ell_p(\Z)$ for $1\leq p\leq 2$ supercyclicity of $T$, weak supercyclicity of $T$, cyclicity of $T\oplus T$ and cyclicity of $T^2$ are equivalent. A new sufficient condition for cyclicity of a weighted bilateral shift is proved, which implies, in particular, that any compact weighted bilateral shift is cyclic.

Jumping flatness and Aluthge transform of recursive weighted shifts

We devote this paper to Hamburger type weighted shifts. We give in particular an affirmative answer to a problem concerning subnormality of the Aluthge transform of Hamburger moment measures with finite support. we also extend the notion flatness, jumping flatness property, introduced recently by Exner et all for Hamburger-type weighted shift and provide obtain several results related to the representing measure of such weighted shifts.

k-hyponormality of powers of weighted shifts via Schur products

We characterize $k$-hyponormality and quadratic hyponormality of powers of weighted shifts using Schur product techniques.

Joint backward extension property for weighted shifts on directed trees

Weighted shifts on directed trees are a decade old generalisation of classical shift operators in the sequence space $\ell^2$. In this paper we introduce the joint backward extension property (JBEP) for classes of weighted shifts on directed trees. If a class satisfies JBEP, the existence of a common backward extension within the class for a family of weighted shifts on rooted directed trees does not depend on the additional structure of the big tree (of fixed depth). We decide whether several classes of operators have JBEP. For subnormal or power hyponormal weighted shifts the property is sat…

算子值加权移位的传播现象研究

Propagation Phenomena for Operator-Valued Weighted Shifts

本文研究了2-次超norm、二次超norm和三次超norm算子值加权移位的传播现象，证明了二次超norm和三次超norm加权移位的平坦性，并提出局部平坦性概念，证明2-次超norm加权移位满足更强的传播现象，并给出结构分解定理。

Dynamics of weighted shifts on directed trees

We study the dynamical behaviour of weighted shifts defined on sequence spaces of a directed tree. In particular, we characterize their boundedness as well as when they are hypercyclic, weakly mixing and mixing.

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.

On the $N$-hypercontractions and similarity of multivariable weighted shifts

In \cite{SH}, A. L. Shields proved a well-known theorem for the similarity of unilateral weighted shift operators. By using the generalization of this theorem for multivariable weighted shifts and the curvature of holomorphic bundles, we give a necessary and sufficient condition for the similarity of $m$-tuples in Cowen-Douglas class. We also present a necessary condition for commuting $m$-tuples of backward weighted shift operators to be $n$-hypercontractive in terms of the weight sequences.

Toral and spherical Aluthge transforms of $2$-variable weighted shifts

We introduce two natural notions of Aluthge transforms (toral and spherical) for 2-variable weighted shifts and study their basic properties. Next, we study the class of spherically quasinormal $2$-variable weighted shifts, which are the fixed points for the spherical Aluthge transform. Finally, we briefly discuss the relation between spherically quasinormal and spherically isometric 2-variable weighted shifts.

A Note on Near Subnormal Weighted Shifts

A complete characterization of near subnormality for bilateral weighted shifts is obtained. As an application of the main results, many new answers to the Hilbert space problem 160 are presented at the end of the paper.

Note on Near Subnormal Weighted Shifts

In this note, 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.

Chaos and frequent hypercyclicity for weighted shifts

Bayart and Ruzsa [Ergodic Theory Dynam. Systems 35 (2015)] have recently shown that every frequently hypercyclic weighted shift on $\ell^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Proc. London Math. Soc. (3) 94 (2007)] who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. We first generalize the Bayart-Ruzsa theorem to all Banach sequence spaces in which the unit sequences are a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, …

带算子权重的m-等距加权移位

m-isometric weighted shifts with operator weights

本文研究带算子权重的m-等距加权移位，通过算子系数多项式给出刻画，解决了正权重构造、权重可交换性、完备化问题及伴随算子性质，并提供了相关示例。

Analytic structure of weighted shifts on directed trees

We show that a weighted shift on a directed tree is related to a multiplier algebra of coefficients of analytic functions. We use this relation to study spectral properties of the operators in question.

On unitary equivalence of bilateral operator valued weighted shifts

We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We also present an example of unitary equivalence between shifts defined on $\mathbb{C}^2$ which cannot be given by any unitary operator of diagonal form. The paper is concluded with investigation of unitary operators than can give unitary equivalence of bilateral operator valued weighted shifts.

Bilateral weighted shift operators similar to normal operators

We prove that an injective, not necessarily bounded weighted bilateral shift operator on $\ell^2(\mathbb{Z})$ is similar to a normal operator if and only if it is similar to a scalar multiple of the simple (i.e. unweighted) bilateral shift operator $S$.

Unbounded subnormal weighted shifts on directed trees

A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established. An approach to this issue via consistent systems of probability measures is invented. The role played by determinate Stieltjes moment sequences is elucidated. Lambert's characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi-analytic vectors, which is a new p…

Backward extensions of weighted shifts on directed trees

The weighted shifts are long known and important class of operators. One of known generalisation of this class are weighted shifts on directed trees, where we replace the linear order of coordinates in $\ell^2$ with a possibly more sophisticated graph structure. In this paper we focus on the question whether a weighted shift on a directed tree admits a subnormal or just power hyponormal (i.e. all powers of the operator are hyponormal) backward extension (a shift on larger directed tree). It comes out that in both cases the question whether we can obtain "joint extension" for a family of trees …

关于复对称加权移位算子

On complex symmetric weighted shifts

本文研究无界复对称加权移位算子，证明满足特定$C^\infty$向量条件的单侧复对称加权移位可分解为无穷多复自伴截断加权移位的正交和，推广了已有结果，还讨论了双侧情形并给出补充结果与开放问题。

Dynamic of Time Varying Weighted Backward Shift Operator

we study the hypercyclic and chaotic properties of the time varying weighted backward shift operator $(Tx)(t)=w(t)x(t+a)$ in $L_p(0,\infty)(1\leq p<\infty)$ and $C_0[0,\infty)$. And we also analyse the spectral structure of the operators if the spaces are complex.

Frames for weighted shift-invariant spaces

We prove the equivalence of the frame property and the closedness for a weighted shift-invariant space. We also construct a sequence $Φ^{2k+1}$ and the sequence of spaces $V^p_μ(Φ^{2k+1})$, $k\in{\mathbb{N}}$, on $\mathbb{R},$ with the useful properties in sampling, approximations and stability.

Eigenvalues of weighted generalized shifts over direct products of vector spaces

In the following text for vector space $V$ over field $F$ we compute all eigenvalues of weighted generalized shift $σ_{φ,\mathfrak{w}}:V^Γ\to V^Γ$ (and generalized shift $σ_φ:V^Γ\to V^Γ$) for nonempty set $Γ$, weight vector $\mathfrak{w}\in F^Γ$ and self--map $φ:Γ\to Γ$.

Mean ergodic weighted shifts on Köthe echelon spaces

Necessary and sufficient conditions are given for mean ergodicity, power boundedness, and topologizability for weighted backward shift and weighted forward shift operators, respectively, on Köthe echelon spaces in terms of the weight sequence and the Köthe matrix. These conditions are evaluated for the special case of power series spaces which allow for a characterization of said properties in many cases. In order to demonstrate the applicability of our conditions, we study the above properties for several classical operators on certain function spaces.

Asymptotic Behaviour and Cyclic Properties of Weighted Shifts on Directed Trees

In this paper we investigate a new class of operators called weighted shifts on directed trees introduced recently in [Z. J. Jablonski, I. B. Jung and J. Stochel, A Non-hyponormal Operator Generating Stieltjes Moment Sequences, J. Funct. Anal. 262 (2012), no. 9, 3946--3980.]. This class is a natural generalization of the so called weighted bilateral, unilateral and backward shift operators. In the first part of the paper we calculate the asymptotic limit and the isometric asymptote of a contractive weighted shift on a directed tree and that of the adjoint. Then we use the asymptotic behaviour …

有根有向树上平衡加权移位的酉等价性

Unitary equivalence of balanced weighted shifts on rooted directed trees

该研究在弱可逆假设下完全刻画了有根有向树上非周期平衡加权移位的酉等价性，推广了Bergman与Dirichlet型移位的相关结论，并通过反例说明该准则对最终周期移位不具必要性。

Conditionally positive definite unilateral weighted shifts

In a recent paper [15], Hilbert space operators $T$ with the property that each sequence of the form $\{\|T^n h\|^2\}_{n=0}^{\infty}$ is conditionally positive definite in a semigroup sense were introduced. In the present paper, this line of research is continued in depth in the case of unilateral weighted shifts. The conditional positive definiteness of weighted shifts is characterized in terms of formal moment sequences. The description of the representing triplet, the main object canonically associated with such operators, is provided. The backward extension problem for conditionally positi…

A hyponormal weighted shift on a directed tree whose square has trivial domain

It is proved that, up to isomorphism, there are only two directed trees that admit a hyponormal weighted shift with nonzero weights whose square has trivial domain. These are precisely those enumerable directed trees, one with root, the other without, whose every vertex has enumerable set of successors.

Subnormal weighted shifts on directed trees whose nth powers have trivial domain

It is shown that for any positive integer n there exists a subnormal weighted shift on a directed tree whose nth power is closed and densely defined while its (n + 1)th power has trivial domain. Similar result for composition operators in L2-spaces is established.

A note on weak positive matrices, finite mass measures and hyponormal weighted shifts

We study the class of Hankel matrices for which the $k\times k$-block-matrices are positive semi-definite. We prove that a $k\times k$-block-matrix has non zero determinant if and only if all $k\times k$-block matrices have non zero determinant. We use this result to extend the notion of propagation phenomena to $k$-hyponormal weighted shifts. Finally we give a study on invariance of $k$-hyponormal weighted shifts under one rank perturbation.

A Class of Sub-Hardy Hilbert Spaces Associated with Weighted Shifts

In this note we study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function z is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries .

有向森林上的加权移位与亚正规性

Weighted shifts on directed forests and hyponormality

本文介绍了有向森林上的加权移位算子，展示了图论与泛函分析的深层联系，并完整刻画了使每个亚正规有界加权移位均为幂亚正规的有向森林。

Hypercyclicity of Shifts on Weighted ${\mathbf L}^p$ Spaces of Directed Trees

In this paper, we study the hypercyclicity of forward and backward shifts on weighted $L^p$ spaces of a directed tree. In the forward case, only the trivial trees may support hypercyclic shifts, in which case the classical results of Salas apply. For the backward case, nontrivial trees may support hypercyclic shifts. We obtain necessary conditions and sufficient conditions for hypercyclicity of the backward shift and, in the case of a rooted tree on an unweighted space, we show that these conditions coincide.

Difference sets and frequently hypercyclic weighted shifts

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on $\ell^p(\mathbb Z)$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $\mathcal U$-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on $c_0$. The construction of these shifts lies on the constr…

von Neumann不等式对于一类双合同加权位移的不等式

von Neumann Inequality for a class of Doubly Contractive Weighted Shifts

本文研究了双合同加权位移的球版本von Neumann不等式，证明了平衡或满足适当权重条件的加权位移具有球单位正交扩张，从而满足欧几里得单位球上的von Neumann不等式，并进一步建立了一般双合同算子元的多项式不等式。

k-hyponormality of multivariable weighted shifts

We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k>=1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals.

单侧加权移位的多项式嵌入到2变量加权移位中

Polynomial embeddings of unilateral weighted shifts into $2$-variable weighted shifts

研究通过多项式嵌入将单侧加权移位表示为2变量加权移位，并探讨其子正规性及伯杰测度计算方法，回答了三个关键问题。

Semi-cubic hyponormality of weighted shifts with stampfli recursive tail

Let $α:\sqrt{x_{m}},\cdots ,\sqrt{x_{1}},(\sqrt{u},\sqrt{v},\sqrt{w}% )^{\wedge}$ be a backward $m$-step extension of a recursive weight sequence and let $% W_{α}$ be the weighted shift associated with $α$. In this paper we characterize the semi-cubic hyponormality of $W_{α}$ having the positive determinant coefficient property and discuss some related examples.

Subspace-hypercyclic weighted shifts

Our aim in this paper is to obtain necessary and sufficient conditions for weighted shift operators on the Hilbert spaces $\ell^{2}(\mathbb Z)$ and $\ell^{2}(\mathbb N)$ to be subspace-transitive, consequently, we show that the Herrero question (D. A. Herrero. Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99 (1991)179-190) holds true even on a subspace of a Hilbert space, i.e. there exists an operator $T$ such that both $T$ and $T^*$ are subspace-hypercyclic operators for some subspaces. We display the conditions on the direct sum of two invertable bilateral forward weighte…

算子权双边加权移位的酉等价性

Unitarily equivalent bilateral weighted shifts with operator weights

本文研究了算子权双边加权移位的酉等价性，在权重拟可逆假设下建立了酉等价的一般刻画，并证明定义在 $\mathbb{C}^{2}$ 上的移位其酉等价可由至多两条非零对角线的酉算子给出，同时给出定义在 $\mathbb{C}^{k}$ 上需至少 $k$ 条非零对角线的示例。

On analytic structure of weighted shifts on generalized directed semi-trees

Inspired by natural classes of examples, we define generalized directed semi-tree and construct weighted shifts on the generalized directed semi-trees. Given an $n$-tuple of directed directed semi-trees with certain properties, we associate an $n$-tuple of multiplication operators on a Hilbert space $\mathscr{H}^2(β)$ of formal power series. Under certain conditions, $\mathscr{H}^2(β)$ turns out to be a reproducing kernel Hilbert space consisting of holomorphic functions on some domain in $\mathbb C^n$ and the $n$-tuple of multiplication operators on $\mathscr{H}^2(β)$ is unitarily equivalent …

有向图上某些类绝对范数可达加权移位算子

Some Classes of Absolutely Norm Attaining Weighted Shift operators on Directed Graphs

研究有向图上绝对范数可达的拟$\ast$-超正规加权移位算子，通过举例揭示其相关性质，还对有向图上的加权复合算子和兰伯特算子展开研究。

双变量加斯赫变换

Aluthge transforms of 2-variable weighted shifts

本文研究了双变量加斯赫变换的性质，证明了其不保持双变量加权移位的联合次正规性，探讨了其在不同变换下的连续性和谱行为。

环面扩张的环面3-等距加权2-移位的柯西对偶次正规性问题的完整解

A complete solution to the Cauchy dual subnormality problem for torally expansive toral $3$-isometric weighted $2$-shifts

本文通过求解低双次数二变量多项式对应的一对Hausdorff矩问题，完整解决了环面扩张的环面3-等距加权2-移位的柯西对偶次正规性问题。

Semi-cubically hyponormal weighted shifts with Stampfi's subnormal completion

Let $α:1,(1,\sqrt{x},\sqrt{y})^{\wedge }$ be a weight sequence with Stampfli's subnormal completion and let $W_{α}$ be its associated weighted shift. In this paper we discuss some properties of the region $\mathcal{U}:\mathcal{=}\{(x,y):W_{α}$ is semi-cubically hyponormal$\}$ and describe the shape of the boundary of $\mathcal{U}$. In particular, we improve the results of \cite[Theorem 4.2]{LLB} with properties of $\mathcal{U}$.

Unbounded subnormal weighted shifts on directed trees I

A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established. An approach to this issue via consistent systems of probability measures is invented. The role played by determinate Stieltjes moment sequences is elucidated. Lambert's characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi-analytic vectors, which is a new p…

二变量加权移位的递推关系

Recursive relations for 2-variable weighted shifts

本文提出一种从表示测度确定次正规二变量加权移位递推关系的方法，证明原子密度不影响递推关系，并用于构造递推关系理想的 Gröbner 基。

有向树上加权移位的次正规与完全超膨胀完备化问题

Subnormal and completely hyperexpansive completion problem of weighted shifts on directed trees

本文研究了具有单个分支点的有向树上加权移位的次正规与完全超膨胀完备化问题，通过发展截断矩序列后向扩展的新结果，获得了此类完备化存在性的刻画。

Aluthge transforms of weighted shifts on directed trees

Aluthge transform of a bounded operator is generalized to the case of unbounded one. A formula for the Aluthge transform of a weighted shift on a directed tree is established and it is used to construct an example of a hyponormal operator whose Aluthge transform has trivial domain. It is proven that such an example can be also constructed in the class of composition operators. It is also shown that Aluthge transform of a closed, densely defined operator is not necessarily closable.

矩无限可分性与加权位移：序列条件

Moment Infinite Divisibility of Weighted Shifts: Sequence Conditions

研究加权位移运算的矩无限可分性，通过重新审视权重或矩的序列条件，获得新的特征化结果，并证明此类位移在一定条件下具有鲁棒性和刚性。

Berger measure for some transformations of subnormal weighted shifts

A subnormal weighted shift may be transformed to another shift in various ways, such as taking the p-th power of each weight or forming the Aluthge transform. \ We determine in a number of cases whether the resulting shift is subnormal, and, if it is, find a concrete representation of the associated Berger measure, directly for finitely atomic measures, and using both Laplace transform and Fourier transform methods for more complicated measures. \ Alternatively, the problem may be viewed in purely measure-theoretic terms as the attempt to solve moment matching equations such as $(\int t^n \, d…

关于 $\ell^{2}$ 上加权移位的数值半径作为 $\ell^{\infty}$ 上的范数及其与 Banach 极限的联系

On the numerical radius of weighted shifts on $\ell^{2}$ as a norm on $\ell^{\infty}$ and connections to Banach limits

本文研究了 $\ell^{2}$ 上加权移位的数值半径作为 $\ell^{\infty}$ 上范数的性质，证明了其与 Banach 极限、谱半径间的取值关系，并比较了不同算子范数的异同。

Subnormal weighted shifts on directed trees and composition operators in $L^2$ spaces with non-densely defined powers

It is shown that for every positive integer $n$ there exists a subnormal weighted shift on a directed tree (with or without root) whose $n$th power is densely defined while its $(n+1)$th power is not. As a consequence, for every positive integer $n$ there exists a non-symmetric subnormal composition operator $C$ in an $L^2$ space over a $σ$-finite measure space such that $C^n$ is densely defined and $C^{n+1}$ is not.