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 $\mathcal{U}$.
A closed densely defined operator $ T $ on a Hilbert space $ \mathcal{H} $ is callled $M$-hyponormal if $\mathcal{D}(T) \subset \mathcal{D}(T^{*}) $ and there exists $ M > 0 $ for which $ \parallel(T-zI)^{*}x \parallel \leq M \parallel(T-zI)x \parallel $ for all $ z \in \mathbb{C}$ and for all $ x\in \mathcal{D}(T)$. In this paper, we prove that if bounded linear operator $ A : \mathcal{H} \rightarrow \mathcal{K}$ is such that $ AB^*\subseteq TA $, where $ B $ is a closed subnormal (resp. a closed $ M $-hyponormal) on $\mathcal{H}$, $ T $ is a closed $ M $-hyponormal (resp. a closed subnormal)…
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…
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 linear operator $S$ in a complex Hilbert space $\hh$ for which the set $\dzn{S}$ of its $C^\infty$-vectors is dense in $\hh$ and $\{\|S^n f\|^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every $f \in \dzn{S}$ is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator $S$ which generates Stieltjes moment sequences. What is more, $\dzn{S}$ is a core of any power $S^n$ of $S$. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of in…
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…
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…
UFR de Mathématiques, Université des Sciences et Technologies de Lille（里尔科学技术大学数学系）；Department of Mathematics, University of Iowa（爱荷华大学数学系）；Department of Mathematics, Bucknell University（巴克内尔大学数学系）
