Wang Gongbao, Ma Jipu
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.
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We characterize $k$-hyponormality and quadratic hyponormality of powers of weighted shifts using Schur product techniques.
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.
In this paper, we study hyponormal weighed composition operators on the Hardy and weighted Bergman spaces. For functions $ψ\in A(\mathbb{D})$ which are not the zero function, we characterize all hyponormal compact weighted composition operators $C_{ψ,φ}$ on $H^{2}$ and $A^{2}_α$. Next, we show that for $φ\in \mbox{LFT}(\mathbb{D})$, if $C_φ$ is hyponormal on $H^{2}$ or $A^{2}_α$, then $φ(z)=λz$, where $|λ| \leq 1$ or $φ$ is a hyperbolic non-automorphism with $φ(0)=0$ and such that $φ$ has another fixed point in $\partial \mathbb{D}$. After that, we find the essential spectral radius of $C_φ$ o…
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.
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.
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。
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.
For 2-variable weighted shifts W_{(α,β)}(T_1, T_2) we study the invariance of (joint) k- hyponormality under the action (h,\ell) -> W_{(α,β)}^{(h,\ell)}(T_1, T_2):=(T_1^k,T_2^{\ell}) (h,\ell >=1). We show that for every k >= 1 there exists W_{(α,β)}(T_1, T_2) such that W_{(α,β)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but W_{(α,β)}(T_1, T_2) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor core we find a computable necessary condition for invariance. Next, we exhibit a large nontrivial class for which hyponormality is indeed invar…
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 $\mathcal{U}$.
We give the first example of a quartically hyponormal unilateral weighted shift which is not 3-hyponormal.
A weighted mean matrix whose weight sequence is linear with positive coefficients is shown to be a posinormal operator on $\ell^2$. This operator is also shown to be coposinormal, so it and its adjoint have the same null space and the same range. The posinormality result leads to a proof that the weighted mean matrix associated with the sequence of odd positive integers is hyponormal, as well as a conjecture regarding a more general linear case.
In this paper we explore finite rank perturbations of unilateral weighted shifts $W_α$. First, we prove that the subnormality of $W_α$ is never stable under nonzero finite rank pertrubations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients of $D_n(s):=\text{det} P_n [(W_α+sW_α^2)^*, W_α+s W_α^2] P_n$ are nonnegative, for every $n\ge 0$, where $P_n$ denotes the orthogonal projection onto the basis vectors $\{e_0,...,e_n\}$. Finally, for $α$ strictly increasing and…
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。
Hyponormal block Toeplitz operators with non-harmonic symbols on the weighted Bergman space
研究向量值加权 Bergman 空间中符号为 $Az^p\bar{z}^q+Bz^s\bar{z}^t$ 的块 Toeplitz 算子，给出其次正规性的两个条件。
On a class of lambda-hyponormal operators
本文定义了无限维希尔伯特空间上的λ-次正规算子，证明了一类此类算子不能超循环，并研究了加权复合算子的值域闭性与λ-次正规性，给出了相关例子。
In this paper, we study quasinormal and hyponormal composition operators \W with linear fractional compositional symbol $\ph$ on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on $H^{2}$ and $A_α^{2}$ by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on $H^2$ which are not quasinormal.
We consider the Toeplitz operator with symbol z^n+C|z|^s acting on certain weighted Bergman spaces and determine for what values of the constant C this operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix.
We study the class of hyponormal 2-variable weighted shifts with two consecutive equal weights in the weight sequence of one of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of consecutive equal weights leads to horizontal or vertical flatness, in a way that resembles the situation for 1-variable weighted shifts. In 1-variable, it is well known that flat weighted shifts are necessarily subnormal (with finitely atomic Berger measures). By contrast, we exhibit a large collection of flat (i.e., horizontally and vertically flat) 2-variab…
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 …
We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix. We apply this result to the Toeplitz operator with specific algebraic symbols acting on certain weighted Bergman spaces of the unit disk and determine when such operators are hyponormal.
Semi-hyponormality of commuting pairs of Hilbert space operators
本文通过构造正2×2算子矩阵的平方根公式，研究了交换算子对的半亚正则性，分析了3参数家族W_(α,β)(a,x,y)在单位立方体内的亚正则性、亚正则性、半亚正则性和弱亚正则性区域，并指出弱亚正则性与半亚正则性不重合的子区域。
Hyponormal contractions and analytic shifts
本文研究亚正交算子的两种有限类型，揭示其与解析位移和解析模型的关系，并提出基本模型在分析中的关键作用。
In this paper, some various partial normality classes of weighted conditional expectation type operators on L2() are investigated. Also, some applications of weak hyponormal weighted conditional type operators are pre- sented.
In this paper, we show that the generalized Aluthge transforma- tions of a large class of operators (weighted conditional type operators) are normal. As a consequence, the operator MwEMu is p-hyponormal if and only if it is normal, and under a weak condition, if MwEMu is normal, then the Holder inequality turn into equality for w, u. Also, we conclude that, invertible weighted conditional type operators are normal. In the end we give some appli- cations of p-hyponormal weighted conditional type operators. In the end, some examples are provided to illustrate concrete application of the main res…
Conditional positive definiteness as a bridge between k-hyponormality and n-contractivity
研究权重序列诱导的加权移位算子的条件正定性，建立k-次超正规性与n-合同性之间的联系，揭示两者交互机制。
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。
We study the spectral pictures of (jointly) hyponormal 2-variable weighted shifts with commuting subnormal components. By contrast with all known results in the theory of subnormal single and 2-variable weighted shifts, we show that the Taylor essential spectrum can be disconnected. We do this by obtaining a simple sufficient condition that guarantees disconnectedness, based on the norms of the horizontal slices of the shift. We also show that for every k >= 1 there exists a k-hyponormal 2-variable weighted shift whose horizontal and vertical slices have 1- or 2-atomic Berger measures, and who…
Signed representing measures (Berger-type charges) in subnormality and related properties of weighted shifts
本文研究了带符号的表示测度在次正常性及相关性质中的作用，证明了k-次正则性单独可以提供测度/电荷相关信息，并展示了某些完全超扩张加权位移的Berger型电荷表示。
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.
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。
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…
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 …
Geometrically regular weighted shifts
本文研究了一类加权移位，其权重由特定公式给出，证明了这类移位在不同参数区域下表现出多种性质，如无限可分矩 moment、亚正则、k-但非(k+1)-次次正则等，并提供了新的移位家族用于探索新定义。
We study in this paper the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams {\it Dynamic Systems and Apllications} 3(1994) 103-112. Furthermore, we generalize some results of Williams and give a simple proof of theorem 2.5 of L.R. Williams (The Local Spectra of Pure Quasinormal Operators J. Math. anal. Appl. 187(1994) 842-850) that non normal hyponormal weighted shifts have fat local spectra.
We show that an invertible bilateral weighted shift is strongly structurally stable if and only if it has the shadowing property. We also exhibit a K{ö}the sequence space supporting a frequently hypercyclic weighted shift, but no chaotic weighted shifts.
Let $I$ be a countably infinite index set, and let $X$ be a Banach sequence space over $I.$ In this article, we characterize disjoint hypercyclic and supercyclic weighted pseudo-shift operators on $X$ in terms of the weights, the OP-basis, and the shift mappings on $I.$ Also, the shifts on weighted $L^p$ spaces of a directed tree and the operator weighted shifts on $\ell^2(\mathbb{Z,\mathcal{K}})$ are investigated as special cases.
We study bounded weighted shifts on directed trees. We show that the set of multiplication operators associated with an injective weighted shift on a rooted directed tree coincides with the WOT/SOT closure of the set of polynomials of the weighted shift. From this fact we deduce reflexivity of those weighted shifts on rooted directed trees whose all path-induced spectral-like radii are positive. We show that weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition. We prove that the pairwise orthogonality of the factors in the decomposition is equivalent t…
In this paper, we study the local spectral properties of unilateral operator weighted shifts.
Criteria for subnormality of unbounded injective weighted shifts on leafless directed trees with one branching vertex are proposed. The case of classical weighted shifts is discussed. The relevance of an inductive limit approach to subnormality of weighted shifts on directed trees is revealed.
In this paper, we study the local spectral properties for both unilateral and bilateral weighted shift operators.
A new class of (not necessarily bounded) operators related to (mainly infinite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency operators, on the other; they are called weighted shifts on directed trees. The basic properties of such operators, including closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees are discussed. The relationships between domains of a weighted shift on a d…
A formally normal weighted shift on a directed tree is shown to be a bounded normal operator. The question of whether a normal extension of a subnormal weighted shift on a directed tree can be modeled as a weighted shift on some, possible different, directed tree is answered.
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。
This paper provides a description of the spectrum of diagonal perturbation of weighted shift operator acting on a separable Hilbert space.
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.
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.
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.
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.
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 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.
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.
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, …
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。
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.
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.
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 $S$.
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…
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。
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.
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.
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