Karl-G. Grosse-Erdmann, Dimitris Papathanasiou
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.
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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.
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.
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…
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.
Extension of m-isometric weighted composition operators on directed graphs
该研究针对含单回路、多分支顶点的有向图，探讨了k-拟-m-等距复合算子及加权复合算子相关内容，丰富了有向图上算子理论的研究范畴。
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.
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 …
Completion problem for extension of m- isometric weighted composition operators on directed graphs
本文研究了有向图上加权复合算子的k-准m-等距完成问题，探讨了单侧加权位移算子的扩展方法。
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 …
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…
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…
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。
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.
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。
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…
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。
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…
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 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.
On several dynamical properties of shifts acting on directed trees
本文研究了有向树上逆移位算子的F-transitivity和拓扑F-复发性，揭示了无根树上算子的复发性与超循环性的等价关系，并探讨了有根树中超循环性的条件。
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 …
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.
We systematically develop the multivariable counterpart of the theory of weighted shifts on rooted directed trees. Capitalizing on the theory of product of directed graphs, we introduce and study the notion of multishifts on directed Cartesian product of rooted directed trees. This framework unifies the theory of weighted shifts on rooted directed trees and that of classical unilateral multishifts. Moreover, this setup brings into picture some new phenomena such as the appearance of system of linear equations in the eigenvalue problem for the adjoint of a multishift. In the first half of the 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.
Wold-type decompositions for norm-increasing $m$-concave weighted shifts on directed trees
本文证明范数递增的 $3$-凹加权位移具有 Wold 型分解，分类了无根有向树上的 $4$-等距，并构造了无游荡子空间性质的严格 $4$-等距，否定了 Shimorin 对所有 $m\geq 4$ 的猜想。
Cesàro Operators on Rooted Directed Trees
本文研究了有根定向树上的Cesáro算子的性质，证明其在特定条件下为次正规算子，并指出经典定理在该上下文中失效。
Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}$ and let $S_λ \in B(l^2(V))$ be a left-invertible weighted shift on ${\mathscr T}$. We show that $S_λ$ can be modelled as a multiplication operator $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on a disc centered at the origin, where $E:=\ker S^*_λ$. The reproducing kernel associated with $\mathscr H$ is multi-diagonal and of bandwidth $k_{\mathscr T}.$ Moreover, $\mathscr H$ admits an orthonormal basis consisting of polynomials in $z$ with at most $k_{…
A rooted directed tree $\mathscr T=(V, E)$ with can be extended to a directed graph $\mathscr T_\infty=(V_\infty, E_\infty)$ by adding a vertex $\infty$ to $V$ and declaring each vertex in $V$ as a parent of $\infty.$ One may associate with the extended directed tree a family of semigroup structures $\sqcup_{b}$ with extreme ends being induced by the join operation $\sqcup$ and the meet operation $\sqcap$. Each semigroup structure among these leads to a family of densely defined linear operators $W^{b}_{λ_u}$ acting on $\ell^2(V),$ which we refer to as weighted join operators at a given base p…
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…
Chaotic weighted shifts on directed trees
本文研究有向树上加权后移位的混沌性，利用周期点、不动点及权重和广义连分数给出一般 Fréchet 空间与 ℓ^p、c_0 情形的刻画。
Wold-type decomposition for left-invertible weighted shifts on a rootless directed tree
本文研究无根有向树上左可逆加权位移的Wold型分解问题，通过级数收敛性条件完整刻画了具有该分解的所有有界左可逆加权位移。
In this paper, we study the boundedness, compactness and Schatten class membership of composition operators on the weighted $L^{p}$-space of a tree $L^{p}_λ(T)$ with $1\leq p <\infty$.
We study the weighted composition operators between the Lipschitz space and the space of bounded functions on the set of vertices of an infinite tree. We characterized the boundedness, the compactness, and the boundedness from below of weighted composition operators. We also determine the isometric weighted composition operators.
Dynamics of the translation semigroup on directed metric trees
研究有向度量树上的加权L^p空间中左翻译半群的动力学，推导出强连续性的必要充分条件，并通过树结构中权重的渐近衰减特性刻画超循环性和弱混属性。
The single equality $A^{*n}A^n = (A^*A)^n$ does not imply the quasinormality of weighted shifts on rootless directed trees
本文证明了该等式蕴含双边加权移位的拟正规性，但在无根有向树上构造了满足该等式的非拟规范加权移位反例，并指出该反例亦存在于 $L^2$-空间的复合算子类中。
In this paper, we introduce a discrete analogue of weighted Hardy spaces on rooted trees and study weighted composition operators between them in detail. In particular, we characterize bounded and compact weighted composition operators between discrete Hardy spaces. We also consider isometric weighted composition operators between these spaces.
In this paper, estimates for norms of weighted summation operators (discrete Hardy-type operators) on a tree are obtained for $1<p<q<\infty$ and for arbitrary weights and trees.
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