Wang Gong-bao, Ma Ji-pu
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.
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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 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.
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…
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 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…
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.
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。
We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and weights are multiplication operators in $L^2$-spaces under the assumption of existence of a family of probability measures whose Radon-Nikodym derivatives behave regular along the trajectories of the symbol. We build the quasinormal extension which is a weighted composition operator induced by the same symbol. We give auxiliary results concerning commutativity of operator-weighted composition operators with multiplication operators.
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 $\mathcal{U}$.
When is a CPD weighted shift similar to a subnormal operator?
本文证明III型CPD单边加权移位是乘法算子的拟仿射变换，据此给出其与乘法算子相似的充要条件，并建立非次正规CPD移位相似于次正规算子的具体类。
The subnormal completion problem on a directed tree is to determine, given a collection of weights on a subtree, whether the weights may be completed to the weights of a subnormal weighted shift on the directed tree. We study this problem on a directed tree with a single branching point, $η$ branches and the trunk of length $1$ and its subtree which is the "truncation" of the full tree to vertices of generation not exceeding $2$. We provide necessary and sufficient conditions written in terms of two parameter sequences for the existence of a subnormal completion in which the resulting measures…
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…
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。
The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. \ We study LPCS within the class of commuting 2-variable weighted shifts $\mathbf{T} \equiv (T_1,T_2)$ with subnormal components $T_1$ and $T_2$, acting on the Hilbert space $\ell ^2(\mathbb{Z}^2_+)$ with canonical orthonormal basis $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ The \textit{core} of a commuting 2-variable weighted shift $\mathbf{T}$, $c(\mathbf{T})$, is the restriction of $\mathbf{T}$ to the invarian…
The Cauchy dual subnormality problem in two variables
研究解析左逆交换对环面 2 - 等距算子，满足联合核条件时与可逆权重算子值加权移位酉等价，初始权重交换时柯西对偶联合次正规，还构造反例说明初始权重不交换时柯西对偶非联合次正规。
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。
The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts
本文解决单边加权移位理论中两个未解问题：亚正交性问题和平方根问题，通过梅林变换和指数多项式理论证明两者等价当且仅当自然函数方程成立。
A criterion for subnormality of unbounded composition operators in L2-spaces, written in terms of measurable families of probability measures satisfying the so-called consistency condition, is established. It becomes a new characterization of subnormality in the case of bounded composition operators. Pseudo-moments of a measurable family of probability measures that satisfies the consistency condition are proved to be given by the Radon-Nikodym derivatives which appear in Lambert's characterization of bounded composition operators. A criterion for subnormality of composition operators induced …
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…
A Note on Spectral Mapping Theorems for Subnormal Operators
本文通过构造具体的紧集、测度与函数实例，证明了满足特定可逆性差异的$R^\infty(K,\mu)$空间存在，解决了1984年提出的次正规算子谱映射定理相关公开问题。
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。
The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{\prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if $T$ is a $2$-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then $T^{\prime}$ is subnormal if and only if $T$ satisfies the (unperturbed) kernel condition. The second counterexample arises from a $2$-isometric adjacency operator of a …
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.
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…
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.
Let H_0 (resp. H_\infty denote the class of commuting pairs of subnormal operators on Hilbert space (resp. subnormal pairs), and for an integer k>=1 let H_k denote the class of k-hyponormal pairs in H_0. We study the hyponormality and subnormality of powers of pairs in H_k. We first show that if (T_1,T_2) is in H_1, then the pair (T_1^2,T_2) may fail to be in H_1. Conversely, we find a pair (T_1,T_2) in H_0 such that (T_1^2,T_2) is in H_1 but (T_1,T_2) is not. Next, we show that there exists a pair (T_1,T_2) in H_1 such that T_1^mT_2^n is subnormal (all m,n >= 1), but (T_1,T_2) is not in H_\in…
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.
Moment infinitely divisible weighted shifts
研究了具有矩无限可分权重的位移的子正规性条件，证明了权重序列是log完全交替序列当且仅当位移是矩无限可分的，并给出了新的子正规性条件和无限可分Hankel矩阵的例子。
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。
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 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…
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。
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…
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。
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 …
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…
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…
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。
We present some results concerning the $l^p$ norms of weighted mean matrices. These results can be regarded as analogues to a result of Bennett concerning weighted Carleman's inequalities.
We give a proof of Cartlidge's result on the $l^{p}$ operator norms of weighted mean matrices for $p=2$ on interpreting the norms as eigenvalues of certain matrices.
We consider multiplication properties of elements in weighted Fourier Lebesgue and modulation spaces. Especially we extend some results by Pilipovic, Teofanov and Toft (2010).
When the backward shift operator on a weighted space $H^2_w=\{f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\}$ is an $n$-hypercontraction, we prove that the weights must satisfy the inequality $$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ As an application of this result, it is shown that such an operator cannot be subnormal. We also give an example to illustrate the important role that the $n$-hypercontractivity assumption plays in determining the similarity of Cowen-Douglas operators in terms of the curvatures of their eigenvector bundles.
We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with weighted sup-norms in terms of the symbol g. Thus we complement recent work by Bassallote, Contreras, Hernández-Mancera, Martín and Paul for spaces of holomorphic functions on the disc and by Constantin and Peláez for reflexive weighted Fock spaces.
In this paper we present sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces which complete the Osançliol result in 2014. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in $\mathbb{R}^n$.
This paper provides a class of complex symmetric weighted composition operators on $H^2(\mathbb{D})$ to includes the unitary subclass, the Hermitian subclass and the normal subclass obtained by Bourdon and Noor. A characterization of algebraic weighted composition operator with degree no more than two is provided to illustrate that the weight function of a complex symmetric weighted composition operator is not necessarily linear fractional. \end{abstract}
We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points. Using the technique of log majorization, we are able to characterize that the equality cases of said equalities occur if and only if the matrices commute, and hence are the same as in $\ell^p$.
A note on weighted composition operators on Dirichlet space
本文研究了狄里赫利空间上加权复合算子的紧性条件、数值范围刻画及零点包含准则，并引入了具有特定数值范围的算子类。
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.
In this note we consider weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work [4]. We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main Theorem 5.1 of [4]. In particular, we obtain that the multiplier space of the Gelfand-Shilov space $Σ^{r}_{s}(\mathbb{R}^{d})$ of Beurling type is ultrabornological, whereas the one of the Gelfand-Shilov space $\mathcal{S}^{r}_{s}(\mathbb{R}^{d})$ of Roumieu type …
For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $φ: \mathbb{R} \to \mathbb{R}$ for which the composition operator $C_φ: X \to Y, \, f \mapsto f \circ φ$ is well-defined and continuous. This problem has been recently considered for $X = Y$ being the space $\mathscr{S}$ of rapidly decreasing smooth functions [1] and the space $\mathscr{O}_M$ of slowly increasing smooth functions [2]. In particular, we recover both these results as well as obtain a characterization for $X =Y$ being the space $\mathscr{O}_C$ of very…
$\mathcal{MID}$ and safe quotients for GRWS
研究GRWS权重比产生$\mathcal{MID}$或次正规移位的条件，并推导乘积权重的结果。
The restricted invertibility theorem was originally introduced by Bourgain and Tzafriri in $1987$ and has been considered as one of the most celebrated theorems in geometry and analysis. In this note, we present weighted versions of this theorem with slightly better estimates. Particularly, we show that for any $A\in\mathbb{R}^{n\times m}$ and $k,r\in\mathbb{N}$ with $k\leq r\leq \mbox{rank}(A)$, there exists a subset $\mathcal{S}$ of size $k$ such that $σ_{\min}(A_{\mathcal{S}}W_{\mathcal{S}})^2\geq \frac{(\sqrt{r}-\sqrt{k-1})^2}{\|W^{-1}\|_F^{2}}\cdot\frac{r}{\sum_{i=1}^{r}σ_{i}(A)^{-2}}$, w…
In this note, we completely characterize complex symmetric weighted composition differentiation operator on the Hardy space $H^2$ with respect to the conjugation operator $C_{λ,α}$. Meanwhile, the normal and self-adjoint of the weighted composition differentiation operators on the Hardy space $H^2$ are also studied. This note could be considered as a continuation of the work initiated by Fatehi and Hammond.
We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.
The goal of this paper is to extend Nakai's generalized Morrey spaces to a wider function class, the one-sided Muckenhoupt weighted case. Morrey matching Muckenhoupt enables us to study the weak and strong type boundedness of one-sided sublinear operators satisfying certain size conditions on the one-sided weighted Morrey spaces. We also establish one-sided Fefferman-Stein inequalities on one-sided weighted Morrey spaces in this paper. Meanwhile, the boundedness and compactness of Riemann-Liouville integral operators on locally one-sided weighted Morrey space are considered. As applications, w…
Given a non-negative weight $v$, not necessarily bounded or strictly positive, defined on a domain $G$ in the complex plane, we consider the weighted space $H_v^\infty(G)$ of all holomorphic functions on $G$ such that the product $v|f|$ is bounded in $G$ and study the question of when is such a space complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight $v$, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.
In this note unbounded hyperexpansive weighted composition operators are investigated. AS a consequence unbounded hyperexpansive multiplication and composition operators are characterized.
In this note basic properties of unbounded weighted conditional expectation operators are investigated. A description of polar decomposition and quasinormality in this context are provided. Also, we study hyperexpan- sive weighted conditional expectation operators.
Sparse Operators and their boundedness on Morrey-type Spaces: An Expository Note
该说明性笔记概述调和分析中稀疏控制的核心思想，建立Komori-Shirai型加权Morrey空间上稀疏算子有界性的模型结果，为推导相关算子的Morrey空间估计提供简洁框架。
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