Quartically hyponormal weighted shifts need not be 3-hyponormal

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.

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。

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.

Quasinormal and hyponormal weighted composition operators on $H^2$ and $A^2_α$ with linear fractional compositional symbol

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.

When is hyponormality for 2-variable weighted shifts invariant under powers?

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…

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 $\mathcal{U}$.

Hyponormal Toeplitz Operators on Weighted Bergman Space

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.

The weighted mean matrix with weight sequence $w_n=2n+1$ is a hyponormal operator on $\ell^2$

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.

k-hyponormality of finite rank perturbations of unilateral weighted shifts

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…

Hyponormal Toeplitz Operators on Weighted Bergman Spaces

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.

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。

加权 Bergman 空间上具有非调和符号的次正规块 Toeplitz 算子

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。

关于一类λ-次正规算子

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。

Spectral pictures of 2-variable weighted shifts

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…

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 …

交换算子对的半亚正则性

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。

广义Cesàro矩阵的每个正整数阶的超正规性

Hyponormality of Generalized Cesàro Matrices of Every Positive Integer Order

本文通过计算机辅助证明，给出了广义Cesàro矩阵超正规的充要条件（参数非负），利用缺陷算子分解和离散Gram多项式，覆盖所有正整数阶。

A class of operators with normal Aluthge transformation

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…

条件正定性作为k-次超正规性与n-合同性之间的桥梁

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。

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。

带符号的表示测度（Berger型电荷）在次正常性及相关性质中的作用

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 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.

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。

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…

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 …

几何正则加权移位

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。

Weighted shifts on directed trees

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…