Backward Shift Operator Research Articles

Bounded power series on the real line

We investigate power series that converge to a bounded function on the real line. First, we establish relations between coefficients of a power series and boundedness of the resulting function; in particular, we show that boundedness can be prevented by certain Turán inequalities and, in the case of real coefficients, by certain sign patterns. Second, we show that the set of bounded power series naturally supports three topologies and that these topologies are inequivalent and incomplete. In each case, we determine the topological completion. Third, we study the algebra of bounded power series, revealing the key role of the backward shift operator.

Explicit formulas for a family of hypermaps beyond the one-face case

Enumeration of hypermaps (or Grothendieck's dessins d'enfants) is widely studied in many fields. In particular, enumerating hypermaps with a fixed edge-type according to the number of faces and genus is one topic of great interest. The first systematic study of hypermaps with one face and any fixed edge-type is the work of Jackson (1987) [23] using group characters. Stanley later (2011) obtained the genus distribution polynomial of one-face hypermaps of any fixed edge-type expressed in terms of the backward shift operator. There is also enormous amount of work on enumerating one-face hypermaps of specific edge-types. Hypermaps with more faces are generally much harder to enumerate and results are rare. Our main results here are formulas for the genus distribution polynomials for a family of typical two-face hypermaps including almost all edge-types, the purely imaginary zeros property of these polynomials, and the log-concavity of the coefficients.

Generalized 𝑞-Fock spaces and structural identities

Using q q -calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their q q -calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.

When are the norms of the Riesz projection and the backward shift operator equal to one?

The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection P on the Lebesgue space Lp lead to ‖P‖Lp→Lp=1/sin⁡(π/p) for every p∈(1,∞). Hence L2 is the only space among all Lebesgue spaces Lp for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces Lp. We prove that the norm of P is equal to one on the space X if and only if X coincides with L2 and there exists a constant C∈(0,∞) such that ‖f‖X=C‖f‖L2 for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.

Many-Dimensional Duhamel Product in the Space of Holomorphic Functions and Backward Shift Operators

Many-Dimensional Duhamel Product in the Space of Holomorphic Functions and Backward Shift Operators

Phillips symmetric operators and functional calculus of maximal symmetric operators

The aim of this work is to develop a functional calculus for simple maximal symmetric operators. The proposed approach is based on the properties of self-adjoint extensions of Phillips symmetric operators. The obtained results are applied to the description of non-cyclic vectors of backward shift operators.

Extended Eigenvalues and Eigenoperators of Some Weighted Shift Operators

A complex number is called an extended eigenvalue for an operator on a Hilbert space H if there exists a nonzero operator such that: such is called an extended eigenoperator corresponding to. The goal of this paper is to calculate extended eigenvalues and extended eigenoperators for the weighted unilateral (Forward and Backward) shift operators. We also find an extended eigenvalues for weighted bilateral shift operator. Moreover, the closedness of extended eigenvalues for the weighted unilateral (Forward and Backward) shift operators under multiplication is proven.

Произведение Дюамеля в пространствах целых функций экспоненциального типа

Let Ω be a polydomain in CN containing the point 0; H(Ω) be the space of all holomorphic functions on Ω with the compact open topology. In the dual H(Ω)' of H(Ω) a multiplication ⊛ is introduced and investigated. It is defined by the convolution rule with the help of shifts which are associated with the system of partial backward shift operators. The realizations of the algebra (H(Ω)',⊛) with the help of the Cauchy and the Laplace transformations and its representation in H(Ω) are obtained. By means of the Laplace transformation the introduced multiplication ⊛ is realized as the many-dimensional Duhamel product in the appropriate space of entire functions of exponential type.

Invariant subspaces of the generalized backward shift operator and rational functions

The paper is devoted to a complete characterization of proper closed invariant subspaces of the generalized backward shift operator (Pommiez operator) in the Fréchet space of all holomorphic functions in a simply connected domain Ω ∋ 0 \Omega \ni 0 in the complex plane. In the case when the function that generates this operator does not have zeros in Ω \Omega , all such subspaces are finite-dimensional. If in addition Ω \Omega coincides with the entire complex plane, then the generalized backward shift operator under consideration is unicellular. If this function has zeros in Ω \Omega , then the above family of invariant subspaces splits into two classes: the first consists of finite-dimensional subspaces, and the second of infinite-dimensional ones.

Reproducing Kernel Hilbert Spaces of Polyanalytic Functions of Infinite Order

In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by \(\displaystyle e^{z\overline{w}+\overline{z}w}\) which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal–Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel \(\displaystyle \frac{1}{(1-z\overline{w})(1-\overline{z}w)}\) and \(\displaystyle \frac{1}{1-2\mathrm{Re}\, z\overline{w}}\). The corresponding backward shift operators are introduced and investigated.

Construction of dense maximal-dimensional hypercyclic subspaces for Rolewicz operators

In this paper, weighted backward shift operators Tw associated to a Schauder basis of a Banach space are considered. These operators are emblematic in the setting of linear chaos in topological vector spaces. In a constructive way, it is shown the existence of a dense linear subspace having maximal dimension, all of whose nonzero members are simultaneously Tw-hypercyclic for every w belonging to a sequence of admissible weights. Our proof does not use any general result about algebraic or topological genericity.

On the essential norms of singular integral operators with constant coefficients and of the backward shift

Let X X be a rearrangement-invariant Banach function space on the unit circle T \mathbb {T} and let H [ X ] H[X] be the abstract Hardy space built upon X X . We prove that if the Cauchy singular integral operator ( H f ) ( t ) = 1 π i ∫ T f ( τ ) τ − t d τ (Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau )}{\tau -t}\,d\tau is bounded on the space X X , then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator a I + b H aI+bH with a , b ∈ C a,b\in \mathbb {C} , acting on the space X X , coincide. We also show that similar equalities hold for the backward shift operator ( S f ) ( t ) = ( f ( t ) − f ^ ( 0 ) ) / t (Sf)(t)=(f(t)-\widehat {f}(0))/t on the abstract Hardy space H [ X ] H[X] . Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priloz̆en. 9 (1975), pp. 73-74] for the operator a I + b H aI+bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S S .

LINEAR ODE’S WITH LAURENT POLYNOMIAL COEFFICIENTS

The forward and backward shift operators (and hence all Laurent polynomials in these operators) act on two-sided infinite sequences of continuous functions as well as the differentiation operator. One can define therefore linear ODEs with Laurent polynomial coefficients, where the unknowns are such sequences. It turns out that equations of this type can be treated easily, exactly as linear ODEs with constant coefficients. Our motivation for considering these ODEs, which seem to be quite natural on their own, has been the fact that the collection of all Bessel functions is characterized by a very simple first order equation of this kind.

Disjoint and simultaneous hypercyclic Rolewicz-type operators

We characterize disjoint hypercyclic and supercyclic tuples of unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$, which are a generalization of the unilateral backward shift operator. We show that disjoint hypercyclicity and disjoint supercyclicity are equivalent among a subfamily of these operators and disjoint hypercyclic unilateral Rolewicz-type operators always satisfy the Disjoint Hypercyclicity Criterion. We also characterize simultaneous hypercyclic unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$.

The Spectrum of Mapping Ideals of Type Variable Exponent Function Space of Complex Variables with Some Applications

The topological and geometric behaviors of the variable exponent formal power series space, as well as the prequasi-ideal construction by s -numbers and this function space of complex variables, are investigated in this article. Upper bounds for s -numbers of infinite series of the weighted n th power forward and backward shift operator on this function space are being investigated, with applications to some entire functions.

Common hypercyclic algebras for families of products of backward shifts

In this paper, we generalize to the context of algebras some recent results on the existence of common hypercyclic vectors for families of products of backward shift operators. We also give, in a multi-dimensional setting, a positive answer to a question raised by F. Bayart, D. Papathanasiou and the author about the existence of a common hypercyclic algebra on ℓ1(N) with the convolution product for the family of backward shifts (Bw(λ))λ>0 induced by the weights wn(λ)=1+λ/n.

Least squares estimation for the high-order uncertain moving average model with application to carbon dioxide emissions

Uncertain time series analysis aims to explore how the current observation is affected by the disturbance terms and past imprecise observations characterized as uncertain variables. For the case that the current observation is affected by a single past disturbance term, the 1-order uncertain moving average (UMA) model has been tentatively explored. While for the situation that the current observation is affected by multiple past disturbance terms, this paper initiates a high-order UMA model to more accurately describe this relationship. By transforming the high-order UMA model into an uncertain autoregressive model via a backward shift operator, the unknown parameters are calculated through the least squares method. Then, a tth residual is defined to describe the properties of disturbance terms. Furthermore, the forecast value and the confidence interval are derived from the fitted model. Finally, two examples are presented to demonstrate the effectiveness of this method.

Kernels of perturbed Toeplitz operators in vector-valued Hardy spaces

Recently, Liang and Partington (Integr Equ Oper Theory 92(4): 35, 2020) show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article, we provide a vectorial generalization of a result of Liang and Partington. As an immediate application, we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (see Chattopadhyay et al. in Integr Equ Oper Theory 92(6): 52, 2020, Theorem 3.5 and O’Loughlin in Complex Anal Oper Theory 14(8): 86, 2020, Theorem 3.4) in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space.

Dynamics of shift operators on non-metrizable sequence spaces

We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on Köthe coechelon sequence spaces k_p((v^{(m)})_{m\in\mathbb{N}}) in terms of the defining sequence of weights (v^{(m)})_{m\in\mathbb{N}} . We further discuss several examples and show that the annihilation operator from quantum mechanics is mixing, sequentially hypercyclic, chaotic, and topologically ergodic on \mathscr{S}'(\mathbb{R}) .

Almost Invariant Subspaces of the Shift Operator on Vector-Valued Hardy Spaces

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar–Gallardo–Partington. Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.