Backward Φ-shifts and universality

In this paper we consider spaces of sequences which are valued in a topological space E and study generalized backward shifts associated to certain selfmappings of E. We characterize their universality in terms of dynamical properties of the underlying selfmappings. Applications to hypercyclicity theory are given. In particular, Rolewicz's theorem on hypercyclicity of scalar multiples of the classical backward shift is extended.     

Common hypercyclic vectors for multiples of backward shift

We prove that ℓ² contains vectors which are hypercyclic simultaneously for all multiples of the backward shift operator by constants of absolute value greater than 1. The set of such vectors is dense G_δ.     

Backward shifts on function algebras

J.R. Holub (1988) [10] introduced the concept of backward shift on Banach spaces. We show that an infinite-dimensional function algebra does not admit a backward shift. Moreover, we define a backward quasi-shift as a weak type of a backward shift, and show that a function algebra A does not admit it, under the assumption that the Choquet boundary of A has at most finitely many isolated points.     

Triangular extension spectrum of weighted shifts

A necessary and sufficient condition for a complex number to be in the triangular extension spectrum of a weighted backward shift is obtained. It is shown that the triangular extension spectrum of a weighted backward shift is always a closed annulus when it is not empty. Moreover, for any given closed annulus, there exists a weighted backward shift with the annulus as its triangular extension spectrum.

     

Backward shift invariant subspaces with applications to band preserving and phase retrieval problems

The band preserving and phase retrieval problems have long been interested and studied. In this paper, we, for the first time, give solutions to these problems in terms of backward shift invariant subspaces. The backward shift method among other methods seems to be direct and natural. We show that a function , with , that makes the band of fg to be within that of f if and only if g divided by an inner function related to f, belongs to some backward shift invariant subspace in relation to f. By the construction of backward shift invariant space, the solution g is further explicitly represented through the span of the rational function system whose zeros are those of the Laplace transform of f. As an application, we also use the backward shift method to give a characterization for the solutions of the phase retrieval problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Some remarks on analytic continuation in weighted backward shift invariant subspaces

By a famous result of Douglas, Shapiro, and Shields, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation outside the closed unit disk. More can be said when the spectrum of the associated inner function has holes on \mathbb T{{\mathbb T}}. Then the functions of the invariant subspaces even extend analytically through these holes. Here we will be interested in weighted backward shift invariant subspaces which appear naturally in the context of kernels of Toeplitz operators. Note that such kernels are special cases of so-called nearly invariant subspaces. In our setting a result by Aleksandrov allows to deduce analytic continuation properties which we will then apply to consider embeddings of weighted invariant subspaces into their unweighted companions. We hope that this connection might shed some new light on known results. We will also establish a link between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation in the spirit of results by Aleman, Richter, and Ross.     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

Backward Aluthge Iterates of a Hyponormal Operator Have Scalar Extensions

The backward Aluthge iterate (defined below) of a hyponormal operator was initiated in [11]. In this paper we characterize the backward Aluthge iterate of a weighted shift. Also we show that the backward Aluthge iterate of a hyponormal operator has an analogue of the single valued extension property for . Finally, we show that backward Aluthge iterates of a hyponormal operator have scalar extensions. As a corollary, we get that the backward Aluthge iterate of a hyponormal operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

Common supercyclic vectors for a path of operators

We examine common supercyclic vectors for a path of operators. In particular, we show that the path consisting of convex combinations of two arbitrary unilateral weighted backward shifts has a dense Gδ set of common supercyclic vectors. Moreover, we show there exists a path with a dense Gδ set of common supercyclic vectors between a unilateral weighted backward shift which satisfies the Supercyclicity Criterion, and an operator which does not. Lastly, we provide an example of a path of unilateral weighted backward shifts that fails to have a common supercyclic vector.     

The unilateral backward shift is w-quasisubscalar

In [Ko 2] we extend Putinar's theorem to every operator on a finite dimensional space by generalizing Putinar's techniques. Using these techniques, we construct a functional model for the unilateral backward shift. We show, in particular, that the unilateral backward shift is w-quasisubscalar. As a corollary we prove that every strict contraction is w-subscalar, i.e., is similar to the restriction to an invariant subspace of a w-scalar operator.Research partially supported by GARC.     

Distributional chaos for backward shifts

We provide sufficient conditions which give uniform distributional chaos for backward shift operators. We also compare distributional chaos with other well-studied notions of chaos for linear operators, like Devaney chaos and hypercyclicity, and show that Devaney chaos implies uniform distributional chaos for weighted backward shifts, but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic.     

Backward Extensions of Recursively Generated Weighted Shifts and Quadratic Hyponormality

Given the weight sequence for a subnormal recursively generated weighted shift on Hilbert space, one approach to the study of classes of operators weaker than subnormal has been to form a backward extension of the shift by prefixing weights to the sequence. We characterize positive quadratic hyponormality and revisit quadratic hyponormality of certain such backward extensions of arbitrary length, generalizing earlier results, and also show that a function apparently introduced as a matter of convenience for quadratic hyponormality actually captures considerable information about positive quadratic hyponormality.     

Algebras of bounded functions invariant under the restricted backward shift

Let T be the unit circle in the complex plane and let A be a vector space of bounded Lebesgue measurable functions on T. A is said to be invariant under the restricted backward shift if, whenever ? is in A and the 0-th Fourier coefficient of ? vanishes, then $\text{e}{}^{\mathrm{?i\theta }}\text{?(e}{}^{\mathrm{i\theta }}\text{)}$ is also in A. The theorems of this paper provide a characterization of the uniformly closed subalgebras of C(T) which contain the constants and which are invariant under the restricted backward shift and, a similar characterization of the weak-¹ closed subalgebras of L^∞(T, dθ) which contain the constants and which are invariant under the restricted backward shift.     

Trace class backward weighted shifts are quasisubscalar

Recently, the author generalized Putinar techniques. In this paper we use those recent techniques and results to show (Theorem 3.1) that every trace class backward weighted shift with a monotone decreasing weight sequence is quasisubscalar.

     

Inverse limit spaces arising from problems in economics

In this paper we use tools from topology and dynamical systems to analyze the structure of solutions to implicitly defined equations that arise in economic theory, specifically in the study of so-called “backward dynamics”. For this purpose we use inverse limit spaces and shift homeomorphisms to describe solutions which are typical in that they are likely to be observed in future time. These predicted solutions corresponds to attractors in an inverse limit space under the shift homeomorphism(s).     

Compressions of Stable Contractions

The stability of compressions of stable contractions is studied and a sufficient orbit condition is given. On the other hand, it is shown that there are non-stable compressions of the 1-dimensional backward shift and a complete characterization of weighted unilateral shifts with this property is provided. Dilations of bilateral weighted shifts to backward shifts are also considered.     

Non-archimedean shift operators

We introduce and study non-Archimedean analogs of the operators of unilateral shift and backward shift playing crucial roles in the classical theory of nonselfadjoint operators. In particular, we find various functional models of these operators having both common and different features compared to their classical counterparts.     

Hypercyclic algebras

We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators.     

Generalized Appell bases

In the context of Köthe spaces we study the bases related with the backward unilateral weighted shift operator, the so-called generalized derivation operator, extending known results for spaces of analytic functions. These bases are a subclass of Sheffer sequences called generalized Appell sequences and they are closely connected with the isomorphisms invariant by the weighted shift. We use methods of the non classical umbral calculi to give conditions for a generalized Appell sequence to be a basis.