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 $W_α$ 2-hyponormal, we show that for a small finite-rank perturbation $α^\prime$ of $α$, the shift $W_{α^\prime}$ remains quadratically hyponormal.
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.
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…
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…
Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x \in \ell^2$ which are simultaneously hypercyclic for all $w$ in a fixed $W \subseteq \ell^\infty$ arbitrarily complicated by choosing $W$ appropriately.
We characterize disjoint and simultaneously hypercyclic tuples of unilateral pseudo-shift operators on $\ell^p(\mathbb{N})$. As a consequence, complementing the results of Bernal and Jung, we give a characterization for simultaneously hypercyclic tuples of unilateral weighted shifts. We also give characterizations for unilateral pseudo-shifts that satisfy the Disjoint and Simultaneous Hypercyclicity Criterions. Contrary to the disjoint hypercyclicity case, tuples of weighted shifts turn out to be simultaneously hypercyclic if and only if they satisfy the Simultaneous Hypercyclicity Criterion.
The aim of this paper is to study the Wold-type decomposition in the class of $m$-isometries. One of our main results establishes an equivalent condition for an analytic $m$-isometry to admit the Wold-type decomposition for $m\ge2$. In particular, we introduce the $k$-kernel condition which we use to characterize analytic $m$-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for $m\ge2$. As a result, we also show that $m$-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to…
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.
By analytic perturbations, we refer to shifts that are finite rank perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disc. Here shift refers to the multiplication operator $M_z$ on some analytic reproducing kernel Hilbert space. In this paper, we first isolate a natural class of finite rank operators for which the corresponding perturbations are analytic, and then we present a complete classification of invariant subspaces of those analytic perturbations. We also exhibit some instructive examples and…
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.
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 unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It relies on a model for such operators built on operator valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator valued unilateral weighted shifts in a fairly general context. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The members…
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.
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 …
We provide a characterization of $J$-class and $J^{mix}$-class unilateral weighted shifts on $l^{\infty}(\mathbb{N})$ in terms of their weight sequences. In contrast to the previously mentioned result we show that a bilateral weighted shift on $l^{\infty}(\mathbb{Z})$ cannot be a $J$-class operator.
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…
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…
We obtain a Disjoint Frequent Hypercyclicity Criterion and show that it characterizes disjoint frequent hypercyclicity for a family of unilateral pseudo-shifts on $c_0(\mathbb{N})$ and $\ell^p(\mathbb{N})$, $1\le p <\infty$. As an application, we characterize disjoint frequently hypercyclic weighted shifts. We give analogous results for the weaker notions of disjoint upper frequent and reiterative hypercyclicity. Finally, we provide counterexamples showing that, although the frequent hypercyclicity, upper frequent hypercyclicity, and reiterative hypercyclicity coincide for weighted shifts on $…
We generalize the concept of coarse hypercyclicity, introduced by Feldman in \cite{Fe1}, to that of coarse topological transitivity on open cones. We show that a bounded linear operator acting on an infinite dimensional Banach space with a coarsely dense orbit on an open cone is hypercyclic and a coarsely topologically transitive (mixing) operator on an open cone is topologically transitive (mixing resp.). We also "localize" these concepts by introducing two new classes of operators called coarsely $J$-class and coarsely $D$-class operators and we establish some results that may make these cla…
In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights $w_n=1$ in the classical unilateral shift $T$, defined as $Te_n=w_ne_{n+1}$, where $\{e_n\}_{n=1}^\infty$ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables $\{X_n\}_{n=1}^{\infty}$; that is, $w_n=X_n$. This paper answers basic questions concerning such a model. We propose that this model can be studied in comparison with the classical Hardy/Bergman/Dirichlet sp…
