C. S. Kubrusly, B. P. Duggal
This paper provides conditions (i) to distinguish weak supercyclicity form supercyclicity for operators acting on normed and Banach spaces, and also (ii) to ensure when weak supercyclicity implies weak stability.
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Weak Supercyclicity -- An Expository Survey
本文是超循环性与弱超循环性领域的综述，梳理相关文献，探究赋范空间上多种超循环性的关系及与各类稳定性的关联，还给出弱l-序列超循环算子的谱描述，以推动该新兴领域发展。
This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is e…
On the dynamics of composition operators: supercyclicity, odometers and translations
本文研究可测函数Banach空间上复合算子的动力学性质，特别通过里程计诱导的复合算子给出新例子和反例，并得到L_p空间上复合算子的超循环性和频繁超循环性的一般结论。
We show that for every supercyclic strongly continuous operator semigroup ${T_t}_{t\geq 0}$ acting on a complex $\F$-space, every $T_t$ with $t>0$ is supercyclic. Moreover, the set of supercyclic vectors of each $T_t$ with $t>0$ is exactly the set of supercyclic vectors of the entire semigroup.
Weak l-sequential supercyclicity and weak quasistability
针对弱l-序列超循环性是否蕴含弱稳定性的开放问题，本文证明其蕴含弱拟稳定性，并给出非强超循环的弱l-序列超循环向量、弱l-序列超循环等距算子的刻画推论。
Weak Stability and Quasistability
本文研究弱l-序列超循环性与弱稳定性的关系，在界间隔子序列条件下证明前者蕴含后者，并给出一个幂有界拟稳定但不弱稳定的反例。
The purpose of this paper is to characterize weak supercyclicity for Hilbert-space contractions, which is shown to be equivalent to characterizing weak supercyclicity for unitary operators$.$ This is naturally motivated by an open question that asks whether every weakly supercyclic power bounded operator is weakly stable (which in turn is naturally motivated by a result that asserts that every supercyclic power bounded operator is strongly stable)$.$ Precisely, weakly supercyclicity is investigated in light of boundedly spaced subsequences as discussed in Lemma 3.1$.$ The main result in Theore…
We treat some questions related to supercyclicity of continuous linear operators when acting in locally convex spaces. We extend results of Ansari and Bourdon and consider doubly power bounded operators in this general setting. Some examples are given.
Various properties of the (continuous) Cesàro operator $\mathsf{C}$, acting on Banach and Fréchet spaces of continuous functions and $L^p$-spaces, are investigated. For instance, the spectrum and point spectrum of $\mathsf{C}$ are completely determined and a study of certain dynamics of $\mathsf{C}$ is undertaken (eg. hyper- and supercyclicity, chaotic behaviour). In addition, the mean (and uniform mean) ergodic nature of $\mathsf{C}$ acting in the various spaces is identified.
A bounded linear operator $T$ acting on a Banach space $\B$ is called weakly hypercyclic if there exists $x\in \B$ such that the orbit ${T^n x: n=0,1,...}$ is weakly dense in $\B$ and $T$ is called weakly supercyclic if there is $x\in \B$ for which the projective orbit ${λT^n x: λ\in \C, n=0,1,...}$ is weakly dense in $\B$. If weak density is replaced by weak sequential density, then $T$ is said to be weakly sequentially hypercyclic or supercyclic respectively. It is shown that on a separable Hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic. Th…
We call a bounded linear operator acting between Banach spaces weakly compactly generated ($\mathsf{WCG}$ for short) if its range is contained in a weakly compactly generated subspace of its codomain. This notion simultaneously generalises being weakly compact and having separable range. In a comprehensive study of the class of $\mathsf{WCG}$ operators, we prove that it forms a closed surjective operator ideal and investigate its relations to other classical operator ideals. By considering the $p$th long James space $\mathcal{J}_p(ω_1)$, we show how properties of the ideal of $\mathsf{WCG}$ op…
On several dynamical properties of shifts acting on directed trees
本文研究了有向树上逆移位算子的F-transitivity和拓扑F-复发性，揭示了无根树上算子的复发性与超循环性的等价关系，并探讨了有根树中超循环性的条件。
We consider operators T : M_0 -> Z and T : M -> Z, where Z is a Banach space and (M_0, M) is a pair of Banach spaces belonging to a general construction in which M is defined by a "big-O" condition and M_0 is given by the corresponding "little-o" condition. Prototype examples of such spaces M are given by $\ell^\infty$, weighted spaces of functions or their derivatives, bounded mean oscillation, Lipschitz-Hölder spaces, and many others. The main result characterizes the weakly compact operators T in terms of a certain norm naturally attached to M, weaker than the M-norm, and shows that weakly …
We present sharp interpolation theorems, including all limiting cases, for a class of quasilinear operators of joint weak type acting between Lorentz-Karamata spaces over $σ$-finite measure. This class contains many of the important integral operators. The optimality in the scale of Lorentz-Karamata spaces is also discussed. The proofs of our results rely on a characterization of Hardy-type inequalities restricted to monotone functions and with power-slowly varying weights. Some of the limiting cases of these inequalities have not been considered in the literature so far.
In this paper, we study the boundedness theory for maximal Calderón-Zygmund operators acting on noncommutative $L_p$-spaces. Our first result is a criterion for the weak type $(1,1)$ estimate of noncommutative maximal Calderón-Zygmund operators; as an application, we obtain the weak type $(1,1)$ estimates of operator-valued maximal singular integrals of convolution type under proper {regularity} conditions. These are the {\it first} noncommutative maximal inequalities for families of linear operators that can not be reduced to positive ones. For homogeneous singular integrals, the strong type …
In 2008, J. Parcet showed the $(1,1)$ weak-boundedness of Calderón-Zygmund operators acting on functions taking values in a von Neumann algebra. We propose a simplified version of his proof using the same tools : Cuculescu's projections and a pseudo-localisation theorem. This will unable us to recover the $L_p$-boundedness of Calderón-Zygmund operators with Hilbert valued kernels acting on operator valued functions for $1 < p < \infty$ and an $L_p$-pseudo-localisation result of P. Hytönen.
An Exposition on Weak Stability of Operators
本文综述赋范空间（尤其是希尔伯特空间）上有界线性算子的弱稳定性研究，梳理六十年文献并指出研究集中于希尔伯特空间算子及酉算子弱稳定性的核心地位。
We prove a weak version of Hardy's uncertainty principle using properties of the prolate spheroidal wave functions (PSWFs). We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on L2(R). A weak version of Hardy's uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.
A pre-adjoint approach on weighted composition operators between spaces of Lipschitz functions
本文利用加权Lipschitz算子这一预伴随视角，重新刻画Lipschitz函数空间间加权复合算子的有界性、紧性及映射性质，并改进已有范数估计。
In the present paper we establish a fixed point result of Krasnoselskii type for the sum $A+B$, where $A$ and $B$ are continuous maps acting on locally convex spaces. Our results extend previous ones. We apply such results to obtain strong solutions for some quasi-linear elliptic equations with lack of compactness. We also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a Banach space. In the last section we develop a sequentially weak continuity result for a class of operators acting on vector-valued Lebesgue spaces. Such …
We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with weighted sup-norms in terms of the symbol g. Thus we complement recent work by Bassallote, Contreras, Hernández-Mancera, Martín and Paul for spaces of holomorphic functions on the disc and by Constantin and Peláez for reflexive weighted Fock spaces.
In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and …
On an improved restricted reverse weak-type bound for the maximal operator
本文改进了Hardy-Littlewood最大算子的受限反弱型界，并证明了λ-中位最大算子与最大算子在Banach函数空间上的有界性等价。
We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplication-differentiation operators acting on $L^2(\Bbb{R}^n)$ which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behavior with the range of the sym…
The parabolic algebra was introduced by Katavolos and Power, in 1997, as the operator algebra acting on $L^2(R)$ that is weakly generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces of the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on $L^p(R)$, where $1<p<\infty$. It is also shown that the reflexive closures of the Fourier binests on $L^p(R)$…
In this paper we study the Bergman-Toeplitz operator $T_ψ$ induced by $ψ(w) = K_Ω^{-α}(w,w)d_Ω^β(w)$ with $α, β\geq 0$ acting from a weighted $L^p$-space $L_a^p(Ω)$ to another one $L_a^q(Ω)$ on a large class of pseudoconvex domains of finite type. In the case $1 < p \leq q < \infty$, the following results are established: \\ - Necessary and sufficient conditions for boundedness, which generalize the recent results obtained by Khanh, Liu and Thuc.\\ - Upper and lower estimates for essential norm, in particular, a criterion for compactness.\\ - A characterization of Schatten class membership of …
We address the problem of studying the boundedness, compactness and weak compactness of the integral operators $T_g(f)(z)=\int_0^z f(ζ)g'(ζ)\,dζ$ acting from a Banach space $X$ into $H^\infty$. We obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of $X$.
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