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 invariant under all powers; moreover, for this class 2-hyponormality automatically implies subnormality. Our results partially depend on new formulas for the determinant of generalized Hilbert matrices and on criteria for their positive semi-definiteness.
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.
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…
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 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…
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…
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…
We show that an invertible bilateral weighted shift is strongly structurally stable if and only if it has the shadowing property. We also exhibit a K{ö}the sequence space supporting a frequently hypercyclic weighted shift, but no chaotic weighted shifts.
Let $I$ be a countably infinite index set, and let $X$ be a Banach sequence space over $I.$ In this article, we characterize disjoint hypercyclic and supercyclic weighted pseudo-shift operators on $X$ in terms of the weights, the OP-basis, and the shift mappings on $I.$ Also, the shifts on weighted $L^p$ spaces of a directed tree and the operator weighted shifts on $\ell^2(\mathbb{Z,\mathcal{K}})$ are investigated as special cases.
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…
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.
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.
It is shown that for a bounded weighted bilateral shift $T$ acting on $\ell_p(\Z)$ for $1\leq p\leq 2$ supercyclicity of $T$, weak supercyclicity of $T$, cyclicity of $T\oplus T$ and cyclicity of $T^2$ are equivalent. A new sufficient condition for cyclicity of a weighted bilateral shift is proved, which implies, in particular, that any compact weighted bilateral shift is cyclic.
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.
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…
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.
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.
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.
We introduce two natural notions of Aluthge transforms (toral and spherical) for 2-variable weighted shifts and study their basic properties. Next, we study the class of spherically quasinormal $2$-variable weighted shifts, which are the fixed points for the spherical Aluthge transform. Finally, we briefly discuss the relation between spherically quasinormal and spherically isometric 2-variable weighted shifts.
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.
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.
Bayart and Ruzsa [Ergodic Theory Dynam. Systems 35 (2015)] have recently shown that every frequently hypercyclic weighted shift on $\ell^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Proc. London Math. Soc. (3) 94 (2007)] who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. We first generalize the Bayart-Ruzsa theorem to all Banach sequence spaces in which the unit sequences are a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, …
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.
We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We also present an example of unitary equivalence between shifts defined on $\mathbb{C}^2$ which cannot be given by any unitary operator of diagonal form. The paper is concluded with investigation of unitary operators than can give unitary equivalence of bilateral operator valued weighted shifts.
We prove that an injective, not necessarily bounded weighted bilateral shift operator on $\ell^2(\mathbb{Z})$ is similar to a normal operator if and only if it is similar to a scalar multiple of the simple (i.e. unweighted) bilateral shift operator $S$.
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…
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 …
we study the hypercyclic and chaotic properties of the time varying weighted backward shift operator $(Tx)(t)=w(t)x(t+a)$ in $L_p(0,\infty)(1\leq p<\infty)$ and $C_0[0,\infty)$. And we also analyse the spectral structure of the operators if the spaces are complex.
We prove the equivalence of the frame property and the closedness for a weighted shift-invariant space. We also construct a sequence $Φ^{2k+1}$ and the sequence of spaces $V^p_μ(Φ^{2k+1})$, $k\in{\mathbb{N}}$, on $\mathbb{R},$ with the useful properties in sampling, approximations and stability.
In the following text for vector space $V$ over field $F$ we compute all eigenvalues of weighted generalized shift $σ_{φ,\mathfrak{w}}:V^Γ\to V^Γ$ (and generalized shift $σ_φ:V^Γ\to V^Γ$) for nonempty set $Γ$, weight vector $\mathfrak{w}\in F^Γ$ and self--map $φ:Γ\to Γ$.
Necessary and sufficient conditions are given for mean ergodicity, power boundedness, and topologizability for weighted backward shift and weighted forward shift operators, respectively, on Köthe echelon spaces in terms of the weight sequence and the Köthe matrix. These conditions are evaluated for the special case of power series spaces which allow for a characterization of said properties in many cases. In order to demonstrate the applicability of our conditions, we study the above properties for several classical operators on certain function spaces.
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 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…
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.
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.
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 study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function z is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries .
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 solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on $\ell^p(\mathbb Z)$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $\mathcal U$-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on $c_0$. The construction of these shifts lies on the constr…
Our aim in this paper is to obtain necessary and sufficient conditions for weighted shift operators on the Hilbert spaces $\ell^{2}(\mathbb Z)$ and $\ell^{2}(\mathbb N)$ to be subspace-transitive, consequently, we show that the Herrero question (D. A. Herrero. Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99 (1991)179-190) holds true even on a subspace of a Hilbert space, i.e. there exists an operator $T$ such that both $T$ and $T^*$ are subspace-hypercyclic operators for some subspaces. We display the conditions on the direct sum of two invertable bilateral forward weighte…
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 …
