Weighted Shift Operators Research Articles

Weighted shift operators, orthogonal polynomials and chain sequences

The main purpose of this paper is to use chain sequences to study spectral properties of weighted shift operators A and of tridiagonal operators Re A. Characterizations of chain sequences and relations to Haar sequences are derived. We use these results to compare the spectral radius, the numerical radius and the norm of A and Re A. As an example we study orthogonal polynomials defined by a recursion formula with almost constant coefficients.

Matrix powers with circular numerical range

Let K2=[0200], Kn be the n×n weighted shift matrix with weights 2,1,…,1︸n−3,2 for all n≥3, and K∞ be the weighted shift operator with weights 2,1,1,1,…. In this paper, we show that if an n×n nonzero matrix A satisfies W(Ak)=W(A) for all 1≤k≤n, then W(A) cannot be a (nondegenerate) circular disc. Moreover, we also show that W(A)=W(An−1)={z∈C:|z|≤1} if and only if A is unitarily similar to Kn. Finally, we prove that if T is a numerical contraction on an infinite-dimensional Hilbert space H, then limn→∞⁡‖Tnx‖=2 for some unit vector x∈H if and only if T is unitarily similar to an operator of the form K∞⊕T′ with w(T′)≤1.

On m-complex symmetric weighted shift operators on [formula omitted

In this paper we study m-complex symmetric weighted shift operators on Cn. Let T be the (backward) weighted shift on Cn for some n≥2. We consider when T and Ta (the matrix of entries the moduli of those of T) are both m-complex symmetric with the (same) standard conjugation C, give as well some unitary operators useful in the study, and generalize to upper triangular matrices. Also, we show that if T is 2k-complex symmetric with the standard conjugation C for some k∈N with k<n, then T is (2k−1)-complex symmetric with the conjugation C.

Compact operators in the C$$^*$$-algebra generated by a matrix weighted shift

Complex symmetry operators have notable applications in extension and dilation results, rank one perturbations of Jordan operators, matrix-valued inner functions and free interpolation theory in the disk and so on. While in the study of the complex symmetric operators, one of the problems one always encounters is when a C $$^*$$ -algebra singly generated contains no nonzero compact operators. In this paper, we answer this question recur to matrix weighted shift operators.

On the generalized mean transforms of complex symmetric operators

In this paper, we prove that if $$T\in {\mathcal {L}({\mathcal {H}})}$$ is complex symmetric, then its generalized mean transform $${\widehat{T}}(t)~ (t\not =0)$$ of T is also complex symmetric. Next, we consider complex symmetry property of the mean transform $${\widehat{T}}(0)$$ of truncated weighted shift operators. Finally, we study properties of the generalized mean transform of skew complex symmetric operators.

Hyperexpansive weighted translation semigroups

Weighted shift operators turn out to be extremely useful in supplying interesting examples of operators on Hilbert spaces. With a view to studying continuous analogues of weighted shifts, M. Embry and A. Lambert initiated the study of operator semigroups

T-Entropy formulae for concrete classes of transfer operators

t-Entropy is a principal object of the spectral theory of operators, generated by dynamical systems, namely, weighted shift operators and transfer operators. In essence t-entropy is the Fenchel – Legendre transform of the spectral potential of an operator in question and derivation of explicit formulae for its calculation is a rather nontrivial problem. In the article explicit formulae for t-entropy for two the most exploited in applications classes of transfer operators are obtained. Namely, we consider transfer operators generated by reversible mappings (i. e. weighted shift operators) and transfer operators generated by local homeomorphisms (i. e. Perron – Frobenius operators). In the first case t-entropy is computed by means of integrals with respect to invariant measures, while in the second case it is computed in terms of integrals with respect to invariant measures and Kolmogorov – Sinai entropy.

On higher order selfadjoint operators

For a positive integer m and a Hilbert space H an operator T in B(H), the space of all bounded linear operators on H, is called m-selfadjoint if ∑k=0m(−1)k(mk)T⁎kTm−k=0. In this paper, we show that if T∈B(H) and the spectrum of T consists of a finite number of points then it is m-selfadjoint if and only if it is an n-Jordan operator for some integer n. Moreover, we prove that if T is m-selfadjoint then T is nilpotent when it is quasinilpotent. Then we characterize m-selfadjoint weighted shift operators. Also, we show that if T is m-selfadjoint then so is p(T) when p(z) is a polynomial with real coefficients. After that, we investigate an elementary operator τ and a generalized derivation operator δ on the Hilbert-Schmidt class C2(H) which are m-selfadjoint. Finally, we prove that no m-selfadjoint operator on an infinite-dimensional Hilbert space, can be N-supercyclic, for any N≥1.

An analytic model for left invertible weighted translation semigroups

M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2(\mathbb R_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space $\cal H$ of vector-valued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with $\cal H$ is a diagonal operator. As it turns out that every hyperexpansive weighted translation semigroup is left invertile, the model applies to these semigroups. We also describe the spectral picture for the left invertible weighted translation semigroup. In the process, we point out the similarities and differences between a weighted shift operator and an operator $S_t.$

On $$(m,\infty )$$-isometries: Examples

An operator T on a Banach space X is said to be an $$(m,\infty )$$ -isometry, if $$\begin{aligned} \max _{\begin{array}{c} 0 \le k \le m\\ k \;\; \mathrm{even}\\ \end{array}} \Vert T^kx\Vert =\max _{\begin{array}{c} 0 \le k \le m\\ k \;\; \mathrm{odd} \end{array}} \Vert T^kx\Vert , \end{aligned}$$ for all $$x\in X$$ . In this paper, we study unilateral weighted shift operators which are $$(m,\infty )$$ -isometries for some integers m. In particular, we show that any power of an $$(m,\infty )$$ -isometry is not necessarily an $$(m,\infty )$$ -isometry. We also study strict $$(3,\infty )$$ -isometries on $${{\mathbb {R}}}^2$$ and give an example of a strict $$(2n-1, \infty )$$ -isometry on $${\mathbb {C}}^2$$ , for any odd integer n.

Entropy statistic theorem and variational principle for t-entropy are equivalent

Equivalence of entropy statistic theorem and variational principle for t-entropy is established. It is obtained by means of new direct proofs of variational principles for t-entropy, spectral radius of weighted shift operators, and entropy statistic theorem.

S-Numbers of Weighted Shift Operators on P-Summable Formal Entire Functions of M-Variables

The idea of multiplying a formal Taylor power series by z to make a right shift operator on the space of all square summable sequences of real numbers was due to A.L. Shield. In this work, we consider Taylor power series in m-variables and we give upper and lower estimations of s-numbers for multiplication of m- right weighted shift operators. This allowed us to estimate upper bounds for s-numbers of infinite series of m-right weighted shift operators and give some applications.

A NOTE ON POSITIVE OPERATORS

We show that positive absolutely norm attaining operators can be characterised by a simple property of their spectra. This result clarifies and simplifies a result of Ramesh. As an application we characterise weighted shift operators which are absolutely norm attaining.

Growth orders and ergodicity for absolutely Cesàro bounded operators

In this paper, we extend the concept of absolutely Cesàro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if T is an absolutely Cesàro bounded operator of order α with 0<α≤1, then ‖Tn‖=o(nα), generalizing the result obtained for α=1. Moreover, if α>1, then ‖Tn‖=O(n). We apply such results to get stability properties for the Cesàro means of bounded operators.

C*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ϕ on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l2(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.

Subspace-hypercyclic weighted shifts

Our aim in this paper is to obtain necessary and sufficient conditions for weighted shift operators on the Hilbert spaces l(Z) and l(N) to be subspace-transitive, consequently, we show that the Herrero question [5] 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 weighted shift operators to be subspace-hypercyclic.

Lyapunov-maximizing measures for pairs of weighted shift operators

Motivated by recent investigations of ergodic optimization for matrix cocycles, we study the measures of the maximum top Lyapunov exponent for pairs of bounded weighted shift operators on a separable Hilbert space. We prove that, for generic pairs of weighted shift operators, the Lyapunov-maximizing measure is unique, and show that there exist pairs of operators whose unique Lyapunov-maximizing measure takes any prescribed value less than $\log 2$ for its metric entropy. We also show that, in contrast to the matrix case, the Lyapunov-maximizing measures of pairs of bounded operators are, in general, not characterized by their supports: we construct explicitly a pair of operators and a pair of ergodic measures on the 2-shift with identical supports, such that one of the two measures is Lyapunov-maximizing for the pair of operators and the other measure is not. Our proofs make use of the Ornstein $\overline{d}$-metric to estimate differences in the top Lyapunov exponent of a pair of weighted shift operators as the underlying measure is varied.

Distributionally n-chaotic dynamics for linear operators

This paper investigates distributionally n-chaotic dynamics of linear operators on Frechet spaces. It is shown that an uncountable distributionally scrambled sets under a linear operator may not be distributionally n-scrambled for any \(n \ge 3\). In addition, the existence of invariant distributionally n-scrambled linear manifolds for a composition operator and for a bilateral weighted shift operator are proved by explicit construction.

Distributionally n-Scrambled Set for Weighted Shift Operators

The aim of the present paper is to investigate distributionally n-scrambled sets for weighted shift operators. We prove that the unilateral weighted shift operator admits densely invariant distributionally n-e-scrambled linear manifolds for any e ∈ (0, 1) and any integer n ⩾ 2, showing that this operator can exhibit maximal distributional n-chaos on a dense invariant linear manifold. Analogous results for the bilateral weighted shift operator are also obtained.

On linear operators and bases on Köthe spaces

Abstract We make a survey of results published by the authors about the backward and forward unilateral weighted shift operators in Kóthe spaces, the so-called generalized derivation and integration operators, extending well-known results for spaces of analytic functions.