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We give the characterization of the embeddings between weighted Tandori and Cesàro function spaces using the combination of duality arguments for weighted Lebesgue spaces and weighted Tandori spaces with estimates for the iterated integral operators.
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We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers to the question posted in \cite{AM09}. This is rather surprising since $Ces_{\infty}$ has no natural lattice predual similarly as the known Talagrand's example \cite{Ta81}. We prove that neither $ces_{\infty}$ is isomorphic to $l_{\infty}$ nor $Ces_{\infty}$ is isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}= \|\wide…
We study real and complex interpolation of abstract Cesàro, Copson and Tandori spaces, including the description of Calderón-Lozanovski{\v ı} construction for those spaces. The results may be regarded as generalizations of interpolation for Cesàro spaces $Ces_p(I)$ in the case of real method, but they are new even for $Ces_p(I)$ in the case of complex method. Some results for more general interpolation functors are also presented. The investigations show an interesting phenomenon that there is a big difference between interpolation of Cesàro function spaces in the cases of finite and infinite …
We study the lattice of closed ideals in the algebra of continuous linear operators acting on $p$th Tandori and $p'$th Cesàro sequence spaces, $1\leqslant p<\infty$, which we show are isomorphic to the classical sequence spaces $(\oplus_{n=1}^\infty\ell_\infty^n)_p$ and $(\oplus_{n=1}^\infty\ell_1^n)_{p'}$, respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality.
We investigate structure of the optimal domains for the Hardy-type operators including, for example, the classical Cesàro, Copson and Volterra operators as well as for some of their generalizations. We prove that, in some sense, the abstract Cesàro and Copson function spaces are closely related to the space $L^1$, namely, they contain "in the middle" a complemented copy of $L^1[0,1]$, asymptotically isometric copy of $\ell^1$ and also can be renormed to contain an isometric copy of $L^1[0,1]$. Moreover, the generalized Tandori function spaces are quite similar to $L^\infty$ because they contai…
We study weighted altered Cesàro and Copson spaces, which is non-ideal enlargement of the usual spaces. We give full characterization of dual spaces for the spaces.
We study weigted altered Cesàro space Ch$_{\infty,w}(I)$, which is non-ideal enlargement of the usual Cesàro space. We prove the connection of the space with one weighted Sobolev space of first order on real line and give characterization of associate spaces for the space.
In this paper, we give the characterization of the embeddings between weighted Cesàro function spaces. The proof is based on the duality technique, which reduces this problem to the characterizations of some direct and reverse Hardy-type inequalities and iterated Hardy-type inequalities.
In this paper the boundedness of the weighted iterated Hardy-type operators $T_{u,b}$ and $T_{u,b}^*$ involving suprema from weighted Lebesgue space $L_p(v)$ into weighted Cesàro function spaces ${\operatorname{Ces}}_{q}(w,a)$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $R_u$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator $P_{u,b }$ from $L^p(v)$ into ${\oper…
In this paper the solution of the pointwise multiplier problem between weighted Copson function spaces $\operatorname{Cop}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces $\operatorname{Ces}_{p_2,q_2}(u_2,v_2)$ is presented, where $p_1,\,p_2,\,q_1,\,q_2 \in (0,\infty)$, $p_2 \le q_2$ and $u_1,\,u_2,\,v_1,\,v_2$ are weights on $(0,\infty)$.
Spectral Properties of the Compact Rhaly and Compact Generalised Ces{à}ro Operators on Weighted $c_0$ Spaces
本文全面研究了加权零序列空间上 Rhaly 算子与广义 Cesàro 算子的连续性、紧致性及谱性质，确定了其点谱、连续谱和剩余谱，并探讨了该算子的 Goldberg 分类。
Unlike for $\ell_p$, $1<p\leq\infty$, the discrete Cesàro operator $C$ does not map $\ell_1$ into itself. We identify precisely those weights $w$ such that $C$ does map $\ell_1(w)$ continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of $C$ are presented. It is also possible to identify all $w$ such that $C$ is a compact operator in $\ell_1(w)$. The final section investigates the mean ergodic properties of $C$ in $\ell_1(w)$. Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.
In this paper embeddings between weighted Copson function spaces ${\operatorname{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\operatorname{Ces}}_{p_2,q_2}(u_2,v_2)$ are characterized. In particular, two-sided estimates of the optimal constant $c$ in the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_0^t f(τ)^{p_2}v_2(τ)\,dτ\bigg)^{\frac{q_2}{p_2}} u_2(t)\,dt\bigg)^{\frac{1}{q_2}} \le c \bigg( \int_0^{\infty} \bigg( \int_t^{\infty} f(τ)^{p_1} v_1(τ)\,dτ\bigg)^{\frac{q_1}{p_1}} u_1(t)\,dt\bigg)^{\frac{1}{q_1}}, \end{equation*} where $p_1,\,p_2,\,q_1,\,q_2 \in…
The spectrum of the Cesàro operator $\mathsf{C}$ is determined on the spaces which arises as intersections $A^p_{α+}$ (resp. unions $A^p_{α-}$) of Bergman spaces $A_α^p$ of order $1<p<\infty$ induced by standard radial weights $(1-|z|)^α$, for $0<α<\infty$. We treat them as reduced projective limits (resp. inductive limits) of weighted Bergman spaces $A^p_α$, with respect to $α$. Proving that these spaces admit the monomials as a Schauder basis paves the way for using Grothendieck-Pietsch criterion to deduce that we end up with a non-nuclear Fréchet-Schwartz space (resp. a non-nuclear (DFS)-sp…
Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms
本文研究了广义 Cesàro 算子 $C_t$ 作用于全纯函数空间及赋权 Banach 空间时的连续性、紧性、谱、点谱、线性动力学与平均遍历性。
Cesàro summability of Taylor series in higher order weighted Dirichlet type spaces
研究了高阶加权Dirichlet型空间的Hadamard乘子，证明了当α>1/2时广义Cesàro和序列收敛，而当α=1/2时对于Dirac delta测度该结论不成立。
End-point Norm Estimates for Cesàro and Copson Operators
本文给出加权ℓ^∞空间上一大类算子及其在非负序列等锥上限制的范数精确公式，将其应用于Cesàro与Copson算子，得到范数、与恒等算子距离及幂权情形简化式，并给出相关不等式的最佳常数。
On absolutely Cesàro bounded operators
本文研究绝对Cesàro有界算子，证明了正算子的自改进性，刻画了加权移位的有界性，并给出了动力学障碍及泛性失败定理。
Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norm
本文研究由正有限Borel测度定义的Cesàro型算子在全纯函数加权空间上的连续性、紧性与谱性质，并以矩序列给出完整刻画。
Consider two continuous linear operators $T\colon X_1(μ)\to Y_1(ν)$ and $S\colon X_2(μ)\to Y_2(ν)$ between Banach function spaces related to different $σ$-finite measures $μ$ and $ν$. We characterize by means of weighted norm inequalities when $T$ can be strongly factored through $S$, that is, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all $f\in X_1(μ)$. For the case of spaces with Schauder basis our characterization can be improved, as we show when $S$ is for instance the Fourier operator, or the Cesàro operator. Our aim is to study the case when the map $T$ is besides…
For the weight function $\prod_{i=1}^{d+1}|x_i|^{2\k_i}$ on the unit sphere, sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the Cesàro $(C,δ)$ means in the weighted $L^p$ space for $δ$ below the critical index. Similar results are also proved for corresponding weight functions on the unit ball and on the simplex.
We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,α)$-summable to the function in the norm of the space, provided that $α>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.
Cesàro operators on the space of analytic functions with logarithmic growth
本文研究复平面开单位圆盘上对数增长解析函数空间$VH(\u2139)$上Cesàro算子的连续性、紧性、谱与遍历性质，该空间是带紧连接映射的加权解析函数Banach空间的可数归纳极限，由Taskinen和Jasiczak提出并研究。
We characterize those non-negative, measurable functions $ψ$ on $[0,1]$ and positive, continuous functions $ω_1$ and $ω_2$ on $\mathbb R^+$ for which the generalized Hardy-Cesàro operator $$(U_ψf)(x)=\int_0^1 f(tx)ψ(t)\,dt$$ defines a bounded operator $U_ψ:L^1(ω_1)\to L^1(ω_2)$. Furthermore, we extend $U_ψ$ to a bounded operator on $M(ω_1)$ with range in $L^1(ω_2)\oplus\mathbb Cδ_0$. Finally, we show that the zero operator is the only weakly compact generalized Hardy-Cesàro operator from $L^1(ω_1)$ to $L^1(ω_2)$.
Dynamics of Weighted Backward Shifts on Cesàro Spaces of Rooted Trees
本文研究无叶局部有限根树上Cesàro空间加权后移算子的动力学，刻画其有界性、传递性、超循环性及混沌性，并给出反例说明各性质间关系。
Hyponormality of Generalized Cesàro Matrices of Every Positive Integer Order
本文通过计算机辅助证明，给出了广义Cesàro矩阵超正规的充要条件（参数非负），利用缺陷算子分解和离散Gram多项式，覆盖所有正整数阶。
The generalized Cesàro operators $\mathcal{C}_t$, for $t\in[0,1)$, introduced in the 1980's by Rhaly, are natural analogues of the classical Cesàro averaging operator $\mathcal{C}_1$ and act in various Banach sequence spaces $X\subseteq {\mathbb C}^{{\mathbb N}_0}$. In this paper we concentrate on a certain class of Banach lattices for the coordinate-wise order, which includes all separable, rearrangement invariant sequence spaces, various weighted $c_0$ and $\ell^p$ spaces and many others. In such Banach lattices $X$ the operators $\mathcal{C}_t$, for $t\in[0,1)$, are always compact (unlike $…
On weighted Cesàro function spaces
本文利用离散化技术研究加权Cesàro和Copson函数空间的结构与嵌入关系，解决了长期开放问题，并为点态乘子和关联空间等提供表征。
Cesàro-Hardy operators on $L^p[0,1]$: fine spectrum, weighted Koopman semigroups and invariant subspaces
本文研究Cesàro-Hardy算子及其泛化在$L^p[0,1]$上的有界性和精细谱性质，利用$C_0$半群理论，通过子ordination表达Cesàro-Hardy算子及其对偶算子，探讨其谱性质及不变子空间问题。
Weighted Cesàro type operators on the weighted Bergman spaces in the unit ball
本文研究单位球加权Bergman空间上由[0,1)测度诱导的加权Cesàro型算子，在0<p,q≤∞全范围给出有界性刻画，部分结果在一维或单位圆盘情形下也为新结论。
For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk we prove that the spectrum of a generalized Cesàro operator $T_g$ is unchanged if the symbol $g$ is perturbed to $g+h$ by an analytic function $h$ inducing a quasi-nilpotent operator $T_h$, i.e. spectrum of $T_h$ equals $\{0\}$. We also show that any $T_g$ operator which can be approximated in the operator norm by an operator $T_h$ with bounded symbol $h$ is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function $g \in$ BMOA to be in the BMOA-norm closure of $H^{\infty…
We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesàro bounded and strong Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing absolutely Cesàro bounded operators on $\ell^p(\mathbb{N})$, $1\le p < \infty$, which are not power bounded, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesàro bounded. These results complement very limited number of known examples (see \cite{Shi} and \cite{AS}). In \ci…
We study associate and double associate spaces of two-weighted Sobolev spaces of the first order on real half-line and we show that unlike the notion of duality the associativity is divided into two cases which we call "strong" and "weak" ones with the division of the second associativity into four cases. On the way we prove that the Sobolev space of compactly supported functions possess weak associated reflexivity and the double weak-strong associate space is vacuous. The case of power weights was recently characterized by reduction to Cesàro or Copson type spaces [18].
Dynamics for affine composition operators on weighted Bergman space of a half plane
本文研究了半平面加权伯格曼空间中由仿射自映射诱导的复合算子的动力学特性，包括正 expansive 和绝对 Cesàro 算子的性质以及正 shadowing 性质的刻画。
Mean ergodicity of multiplication operators in weighted Dirichlet spaces
本文刻画了加权 Dirichlet 空间中乘法算子及其伴随算子的有界性条件，证明了均值遍历性与 Cesàro 有界性的等价性，并构造出一致均值遍历但非幂有界的算子特例。
Cesàro-Type Operators Acting on the Drury-Arveson Space
本文提出并研究了作用于多复变量Drury-Arveson空间的Cesàro型算子，证明了其有界性、紧性及谱性质。
Porosity and supercyclic operators on solid Banach function spaces
本文研究了固Banach函数空间中超循环加权复合算子的特性，引入了Cesáro超转移概念，并通过实例展示了相关结果。
Polynomial Approximation in Higher-Order Weighted Dirichlet Spaces
本文研究了高阶加权狄利克雷空间中Taylor部分和的收敛性问题，通过引入修改多项式和系数对应原理，证明了在特定条件下修改后的求和在范数下收敛，并给出了有限Dirac点质量权重下的显式公式。
Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Cesàro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.
In this work, we give well-known results related to some properties, dual spaces and matrix transformations of the sequence space bv and introduce the matrix domain of space bv with arbitrary triangle matrix A. Afterward, we choose the matrix A as Cesàro mean of order one, generalized weighted mean and Riesz mean and compute alpha-; beta- gamma-duals of these spaces. And also, we characterize the matrix classes of the new spaces.
Cesàro operator induced by a Bergman kernel
本文研究由加权Bergman空间再生核诱导的Cesàro型算子在Hardy空间和加权Bergman空间上的有界性，利用尾积分或矩给出完全刻画，结果在标准权及Cauchy核情形下亦为新。
Denote by $ B_X $ the unit ball of an infinite-dimensional complex Hilbert space $ X. $ Let $ψ\in H(B_X),$ the space of all holomorphic functions on the unit ball $B_X,$ $φ\in S(B_X)$ the set of holomorphic self-maps of $B_X. $ Let $C_{ψ, φ}: \mathcal B_ν(B_X)$ (and $ \mathcal B_{ν,0}(B_X)$) $\to \mathcal B_μ(B_X) $ (and $ \mathcal B_{μ,0}(B_X)$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $ C_φ$ and integral operator $T_ψ.$ In this paper, we characterize the boundedness and compactness of $ C_{ψ,φ} $ via the estimates for the restrictions of $ ψ$ …
Norm of the Cesàro operator between some spaces of analytic functions
本文研究 Cesàro 算子在 Korenblum 空间、对数加权空间、α-Bloch 空间及 Hardy 空间等解析函数空间上的范数，给出若干精确值及上下界估计。
Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms
本文研究了广义 Hilbert 算子在全纯函数加权 Banach 空间上的良定义、有界与紧致性，发现其有界性与紧致性等价且由测度矩可和性刻画，并分析了其核性。
A closed subspace is invariant under the Cesàro operator $\mathcal{C}$ on the classical Hardy space $H^2(\mathbb D)$ if and only if its orthogonal complement is invariant under the $C_0$-semigroup of composition operators induced by the affine maps $φ_t(z)= e^{-t}z + 1 - e^{-t}$ for $t\geq 0$ and $z\in \mathbb D$. The corresponding result also holds in the Hardy spaces $H^p(\mathbb D)$ for $1<p<\infty$. Moreover, in the Hilbert space setting, by linking the invariant subspaces of $\mathcal{C}$ to the lattice of the closed invariant subspaces of the standard right-shift semigroup acting on a pa…
On Weighted Local Morrey-type and Complementary Local Morrey-Type Spaces
该研究针对加权局部Morrey型与互补局部Morrey型空间，通过一维归约方法结合Cesàro和Copson空间嵌入刻画的最新结果，给出两类空间间连续嵌入的充要条件，覆盖所有有限参数情形。
We investigate (uniform) mean ergodicity of weighted composition operators on the space of smooth functions and the space of distributions, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic with period 2. Our results are based on a characterization of mean ergodicity in terms of Cesàro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest.
Optimal domain of Volterra operators in classes of Banach spaces of analytic functions
研究Volterra算子、Cesàro算子等在不同解析函数Banach空间中的最佳定义域，探讨其在Hardy空间、Korenblum增长空间等中的表现及新空间的生成。
Li-Yorke chaotic weighted composition operators on Hardy and Bergman spaces over the unit disk
研究加权复合算子在单位盘上的Hardy和Bergman空间中的李-约尔克混沌及均值李-约尔克混沌，证明算子混沌当且仅当非幂有界，均值混沌当且仅当非绝对Cesàro有界。
On the Dynamics of Weighted Composition Operators
本文研究了加权复合算子在不同空间上的动力学性质，并通过具体应用展示了其在序列空间和函数空间中的表现。
We characterize a four-weight inequality involving the Hardy operator and the Copson operator. More precisely, given $p_1, p_2, q_1, q_2 \in (0, \infty)$, we find necessary and sufficient conditions on nonnegative measurable functions $u_1, u_2, v_1, v_2$ on $(0,\infty)$ for which there exists a positive constant $c$ such that the inequality \begin{align*} &\bigg(\int_0^{\infty} \bigg(\int_0^t f(s)^{p_2} v_2(s)^{p_2} ds \bigg)^{\frac{q_2}{p_2}} u_2(t)^{q_2} dt \bigg)^{\frac{1}{q_2}} \notag \\ & \hspace{3cm} \leq c \bigg(\int_0^{\infty} \bigg(\int_t^{\infty} f(s)^{p_1} v_1(s)^{p_1} ds \bigg)^{\…
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