Weighted Shift Operators Research Articles

On the numerical radius of weighted shift operators with generalized geometric weights

In this paper, we give bounds on the numerical radius of the weighted shift operator T with generalized geometric weights(1,sq,q2,sq3,…,q2n−2,sq2n−1,…), where s>0 and 0<q<1. Also, we provide the entire function FT(z) whose minimal positive root gives the numerical radius of the weighted shift operator T. The purpose of this paper is to generalize the results of numerical radius for the weighted shift operator with geometric weights given in [5].

The fine spectra of a difference of weighted shift operators acting in some classical sequence spaces

In this paper we investigate the fine spectra, the ergodic properties and the linear dynamics of the operatorC(x)=(−nxn+n(n−1)xn−2)n∈N0,x=(xn)n∈N0∈CN0, and of its formal adjoint C′ acting in the classical sequence spaces s, s′, ω and φ. The operators C and C′ acting in s are conjugate to the one-dimensional Ornstein-Uhlenbeck differential operator Ou=u″−xu′ and to the differential operator O′u=u″+xu′+u acting in S(R), respectively.

Hyperinvariant subspaces for operators intertwined with weighted shift operators

Hyperinvariant subspaces for operators intertwined with weighted shift operators

Uncertainty principles in holomorphic function spaces on the unit ball

AbstractOn all Bergman–Besov Hilbert spaces on the unit disk, we find self-adjoint weighted shift operators that are differential operators of half-order whose commutators are the identity, thereby obtaining uncertainty relations in these spaces. We also obtain joint average uncertainty relations for pairs of commuting tuples of operators on the same spaces defined on the unit ball. We further identify functions that yield equality in some uncertainty inequalities.

Weighted translation semigroups: multivariable case

M. Embry and A. Lambert initiated the study of a weighted translation semigroup $$\{S_t\}$$ in $${\mathcal B}(L^2({{\mathbb R}_+})),$$ with a view to explore a continuous analogue of a weighted shift operator. We continued the work, characterized some special types of semigroups and developed an analytic model for the left invertible weighted translation semigroup. The present paper deals with the generalization of the weighted translation semigroup in multi-variable set up. We develop the toral analogue of the analytic model and also describe the spectral picture. We provide many examples of weighted translation semigroups in multi-variable case. Further, we replace the space $$L^2({{\mathbb R}_+})$$ by $$L^2({{\mathbb R}_+^d})$$ and explore the properties of weighted translation semigroup $$\{S_{\overline{t}}\}$$ in $${\mathcal B}(L^2({{\mathbb R}_+^d})),$$ in both one and multi variable cases.

Commutators on Fock spaces

Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.

Entropy, Pressure, Ground States and Calibrated Sub-actions for Linear Dynamics

Denote by $X$ a Banach space and by $T : X \to X$ a bounded linear operator with non-trivial kernel satisfying suitable conditions. We consider the concepts of entropy - for $T$-invariant probability measures - and pressure for H\"older continuous potentials. We also prove the existence of ground states (the limit when temperature goes to zero) associated with such class of potentials when the Banach space $X$ is equipped with a Schauder basis. We produce an example concerning weighted shift operators defined on the Banach spaces $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$, $1 \leq p < +\infty$, where our results do apply. In addition, we prove the existence of calibrated sub-actions when the potential satisfies certain regularity conditions using properties of the so-called Ma\~n\'e potential. We also exhibit examples of selection at zero temperature and explicit sub-actions in the class of H\"older continuous potentials.

The matricial subnormal completion problem

Given a truncated sequence of positive numbers a=(aj)j=0m, the subnormal completion problem (originally posed and geometrically solved by J. Stampfli) asks whether or not there exists a subnormal weighted shift operator on ℓ2 whose initial weights are given by a. Subsequently a concrete solution based on a solution of the truncated Stieltjes moment problem was discovered by Curto and Fialkow. In this paper we will consider a matricial analogue of the subnormal completion problem, where the truncated sequence of positive numbers (aj)j=0m is replaced by a truncated sequence of positive definite matrices. We will provide concrete conditions for a solution based on the parity of m and make a connection with a matricial truncated Stieltjes moment problem. We will also put forward a certain canonical subnormal completion which is minimal in the sense of the norm of the corresponding weighted shift operator and describe the support of the corresponding matricial Berger measure in terms of the zeros of a matrix polynomial which describes the initial positive rank preserving extension.

On the numerical range of some weighted shift operators

In this paper, we compute the numerical radius of a weighted shift operator with weights (h,k,a,b,a,b,…) where h,k,a,b>0. The purpose of this paper is to generalize the results of [1]. As an application of our main result, we derive a reduced lower bound of the numerical radius of some tridiagonal operators.

Behaviour of Birkhoff sums generated by rotations of the circle

For continuous functions $f$ with zero mean on the circle we consider the Birkhoff sums $f(n,x,h)$ generated by the rotations by $2\pi h$, where $h$ is an irrational number. The main result asserts that the growth rate of the sequence $\max_x f(n,x,h)$ as $n \to \infty$ depends only on the uniform convergence to zero of the Birkhoff means $\frac{1}{n}f(n,x,h)$. Namely, we show that for any sequence $\sigma_k \to 0$ and any irrational $h$ there exists a function $f$ such that the sequence $\max_x f(n,x,h)$ increases faster than $n\sigma_n$. We also show that for any function $f$ that is not a trigonometric polynomial there exist irrational $h$ for which some subsequence $\max_x f(n_k,x,h)$ increases faster than the corresponding subsequence $n_k\sigma_{n_k}$. We present applications to weighted shift operators generated by irrational rotations and to their resolvents. Namely, we show that the resolvent of such an operator can increase arbitrarily fast in approaching the spectrum. Bibliography: 46 titles.

Investigating Distributional Chaos for Operators on Fréchet Spaces

In this paper, various notions of chaos for continuous linear operators on Fréchet spaces are investigated. It is shown that an operator is Li–Yorke chaotic if and only if it is mean Li–Yorke chaotic in a sequence whose upper density equals one; that an operator is mean Li–Yorke chaotic if and only if it admits a mean Li–Yorke pair, if and only if it is distributionally chaotic of type 2, if and only if it has an absolutely mean irregular vector. As a consequence, mean Li–Yorke chaos is not conjugacy invariant for continuous self-maps acting on complete metric spaces. Moreover, the existence of invariant scrambled sets (with respect to certain Furstenberg families) of a class of weighted shift operators is proved.

Moment Infinite Divisibility of Weighted Shifts: Sequence Conditions

We consider weighted shift operators having the property of moment infinite divisibility; that is, for any \(p > 0\), the shift is subnormal when every weight (equivalently, every moment) is raised to the p-th power. By reconsidering sequence conditions for the weights or moments of the shift, we obtain a new characterization for such shifts, and we prove that such shifts are, under mild conditions, robust under a variety of operations and also rigid in certain senses. In particular, a weighted shift whose weight sequence has a limit is moment infinitely divisible if and only if its Aluthge transform is. As a consequence, we prove that the Aluthge transform maps the class of moment infinitely divisible weighted shifts bijectively onto itself. We also consider back-step extensions, subshifts, and completions.

Invariant probabilities for discrete time linear dynamics via thermodynamic formalism * *AOL partially supported by CNPq. AM partially supported by CNPq, project 311018/2018-1, and FAPESP, project 2013/24541-0. MS partially supported by CNPq, project 312632/2018-5. VV supported by INCTMat.

We show the existence of invariant ergodic σ-additive probability measures with full support on X for a class of linear operators L : X → X, where L is a weighted shift operator and X either is the Banach space or for 1 ⩽ p < ∞. In order to do so, we adapt ideas from thermodynamic formalism as follows. For a given bounded Hölder continuous potential , we define a transfer operator which acts on continuous functions on X and prove that this operator satisfies a Ruelle–Perron–Frobenius theorem. That is, we show the existence of an eigenfunction for which provides us with a normalised potential and an action of the dual operator on the one-Wasserstein space of probabilities on X with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of requires an a priori probability on the kernel of L. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces.

Topological Transitivity of Shift Similar Operators on Nonseparable Hilbert Spaces

In this paper, we investigate topological transitivity of operators on nonseparable Hilbert spaces which are similar to backward weighted shifts. In particular, we show that abstract differential operators and dual operators to operators of multiplication in graded Hilbert spaces are similar to backward weighted shift operators.

On C*-algebras of singular integral operators with PQC coefficients and shifts with fixed points

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift operators , acting on the space over the unit circle . Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms with finite sets of discontinuities for their derivatives , which acts topologically freely on , where is the interior of the nonempty closed set composed by all common fixed points for all shifts , with boundary of zero Lebesgue measure. A Fredholm symbol calculus for the -algebra is constructed and a Fredholm criterion for the operators is established.

Conditional positive definiteness as a bridge between k–hyponormality and n–contractivity

For sequences α≡{αn}n=0∞ of positive real numbers, called weights, we study the weighted shift operators Wα having the property of moment infinite divisibility (MID); that is, for any p>0, the Schur power Wαp is subnormal. We first prove that Wα is MID if and only if certain infinite matrices log⁡Mγ(0) and log⁡Mγ(1) are conditionally positive definite (CPD). Here γ is the sequence of moments associated with α, Mγ(0),Mγ(1) are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of Wα, and log is calculated entry-wise (i.e., in the sense of Schur or Hadamard). Next, we use conditional positive definiteness to establish a new bridge between k–hyponormality and n–contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift Wα is MID if and only if for all p>0, Mγp(0) and Mγp(1) are CPD.

Reducing subspaces of weighted shift operators on weighted Hardy space

Reducing subspaces of weighted shift operators on weighted Hardy space

On algebraic properties of fuzzy weighted shift operator

In this paper we introduce the concept of fuzzy weighted shift operators in a nonempty fuzzy set and investigate some of algebraic properties. The sufficient condition to satisfy that fuzzy weighted shift operators are fuzzy subsemigroub and fuzzy bi-ideal is given.

Complex symmetric monomial Toeplitz operators on the unit ball

Monomial Toeplitz operators Tzpz¯q on the weighted Bergman space of the unit ball in Cn are natural extension of the weighted shift operators. In higher dimensions, dim ker⁡Tzpz¯q=∞ may occur for non-trivial case, thus complex symmetric phenomena will appear. In this paper, we completely characterize when the monomial Toeplitz operator Tzpz¯q on the weighted Bergman space is JA-symmetric, where JAf(z)=f(Az‾)‾ with the symmetric unitary n×n matrix A. Moreover, we also determine all possible forms of A such that Tzpz¯q is JA-symmetric. As an application, we construct the weakly closed unital commutative algebra generated by complex symmetric Toeplitz operators.

The Commutant of Some Shift Operators

We obtain a characterization for the commutant of certain weighted shift operators of higher multiplicity. The characterization is based on the realization of the operator as multiplication by monomials on weighted Bergman spaces of the unit disk.