Hypercyclic operators on non-locally convex spaces

We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bès, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].

     

The finite rank perturbations of the product of Hankel and Toeplitz operators

In this paper we completely characterize when the product of a Hankel operator and a Toeplitz operator on the Hardy space is a finite rank perturbation of a Hankel operator, and when the commutator of a Hankel operator and a Toeplitz operators has finite rank.     

Spectral triples and manifolds with boundary

We investigate manifolds with boundary in noncommutative geometry. Spectral triples associated to a symmetric differential operator and a local boundary condition are constructed. We show that there is no tadpole for classical Dirac operators with a chiral boundary condition on spin manifolds.     

A Generalization of Tyuriemskih's Lethargy Theorem and Some Applications

Tyuriemskih's Lethargy Theorem is generalized to provide a useful tool for establishing when a sequence of (not necessarily) linear operators that converges point wise to the identity operator actually converges arbitrarily slowly. Then this generalization is used to answer affirmatively a 2010 conjecture of ours as well as establishing that all of the classical operators of Bernstein, Hermite-Fejer, Landau, Fejer, and Jackson converge arbitrarily slowly to the identity operator (and not just almost arbitrarily slowly as we established in 2010).     

Incompleteness of the linear span of the positive compact operators

We show that even in the case of a Banach lattice with an order continuous norm (or whose dual has an order continuous norm) the linear span of the positive compact operators on need not be complete under the regular norm.

     

Remarks on a result about hypercyclic non-convolution operators

In this paper we correct a proof by Aron and Markose in [R. Aron, D. Markose, On universal functions, J. Korean Math. Soc. 41 (2004) 65-76] for the hypercyclicity of the operator given by , in the case b≠0.     

On the sum of a narrow and a compact operators

Our main technical tool is a principally new property of compact narrow operators which works for a domain space without an absolutely continuous norm. It is proved that for every Köthe F-space X and for every locally convex F-space Y   the sum _T1+_T2$T_1+T_2$ of a narrow operator _T1:X→Y$T_1:X\to Y$ and a compact narrow operator _T2:X→Y$T_2:X\to Y$ is a narrow operator. This gives a positive answers to questions asked by M. Popov and B. Randrianantoanina [6, Problems 5.6 and 11.63].     

The effects of a new selection operator on the performance of a genetic algorithm

In this study, the new back controlled (BCSO) selection operator for genetic algorithm (GA) is presented. In the first stage, six existing operators and the BCSO were applied to the traveling salesman and space truss benchmark problems. An analysis of the results of the benchmark problems was made for each of the seven selection operators. In the second stage, the BCSO, the Roulette Wheel, dominant selection the hybrid II and operators were used to determine the fittest beam section has been used. In this stage, the cost of beam for a 18.3 m span developed by Mid-Atlantic States Pre-stressed Concrete Committee for Economic Fabrication and using a genetic algorithm the cost of a beam using BCSO were compared. It was determined that the beam produced using the proposed BCSO selection operator would be lower than those obtained with the existing selection operators. Also the determined beam cost is compares favourably with the average cost of a beam sold by the precast industry in the USA.     

Hyponormality of Toeplitz Operators with Polynomial Symbols: An Extremal Case

If T_φ is a hyponormal Toeplitz operator with polynomial symbol φ = ḡ + f (f, g ∈ H^∞ (𝕋 )) such that g divides f, and if ψ := then where μ is the leading coefficient of ψ and 𝒵(ψ) denotes the set of zeros of ψ. In this paper we present a necessary and sufficient condition for T_φ to be hyponormal when φ enjoys an extremal case in the above inequality, that is, equality holds in the above inequality.     

A short proof of existence of disjoint hypercyclic operators

We give a short proof of existence of disjoint hypercyclic tuples of operators of any given length on any separable infinite dimensional Fréchet space. Similar argument provides disjoint dual hypercyclic tuples of operators of any length on any infinite dimensional Banach space with separable dual.     

Characterizations of Disjointness preserving operators on vector-valued function spaces

We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator is compact (resp. completely continuous) if and only if

for all

where is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and is compact (resp.  is uniformly completely continuous).

     

复对称算子的一些等价性质

证明了在复对称算子的前提下,对数-亚正规算子与正规算子是等价的,并且给出了复对称算子的一些等价性质;最后通过给出例子来说明我们的结论.     

Universal elements for non-linear operators and their applications

We prove that under certain topological conditions on the set of universal elements of a continuous map T acting on a topological space X, that the direct sum T⊕Mg is universal, where Mg is multiplication by a generating element of a compact topological group. We use this result to characterize R₊-supercyclic operators and to show that whenever T is a supercyclic operator and z₁,…,zn are pairwise different non-zero complex numbers, then the operator z₁T⊕?⊕znT is cyclic. The latter answers affirmatively a question of Bayart and Matheron.     

C-半群的Lumer-Phillips定理与C-Hermitian算子

本文给出了稠定闭算子A(或A的扩张)生成压缩C-半群的充分条件,且在C是等距算子时,证明了该条件是必要的,推广了Lumer-Phillips定理.并用结果刻划了等距C-群的生成元.     

Cohen-Kwapień type theorems for multiple summing operators

Abstract

An old result of J.S. Cohen states that each p* -summing operator on a L _p -space has a p* -summing dual. Also, another old result of S. Kwapień states that each p* -summing operator on a Banach space X has a p* -summing dual if and only if X is isomorphic to a quotient of some L _p (µ). In this paper we prove some multilinear and polynomial variants of these results.     

Operators with connected spectrum + compact operators = strongly irreducible operators

Herrero’s conjecture that each operator with connected spectrum acting on complex, separable Hilbert space can be written as the sum of a strongly irreducible operator and a compact operator is proved. Jiang, C. L., Power, S., Wang, Z. Y., Biquasitriangular + small compact = strongly irreducible,J. London Math., to be published.     

关于某些次正规算子的表示

设T是完全非正规的次正规算子。给出了T＝N＋K的充要条件，其中N是正规算子，K是拟正规算子，且NK＝KN。     

Subelliptic Estimates for a Class of Degenerate Elliptic Integro-Differential Operators

In this paper we prove subelliptic estimates for operators of the form Δ_x + λ² (x)S in ?^N = ? × ?, where the operator S is an elliptic integro - differential operator in ?^N and λ is a nonnegative Lipschitz continuous function.     

On an extremal relation of Bernstein operators

In this note we present a new characterization of Bernstein operators by showing that they are the only solution of a certain extremal relation.