Spectrum of the Cesàro operator in ℓ p

We present a novel proof of the fact that the spectrum of the Cesàro operator acting in ? ^p , for 1 < p < ∞, consists of the closed disc centered at q/2 and with radius q/2, where q is the conjugate index of p.     

Boundedness of the Cesàro Operator in Hardy Spaces

The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$.     

Factorization of Cesàro and Hilbert matrices based on generalized Cesàro matrix

ABSTRACT

In this paper, we introduce factorizations for the Hilbert and Cesàro matrices based on generalized Cesàro matrix and as the application, we generalize the Hardy's inequality versus Hilbert's.     

The Dyadic Cesàro Operator on R+

The dyadic Cesàro operator C is introduced for functions in the space L ¹ := L ¹(R ₊) by means of the Walsh-Fourier transform defined by

Mean ergodicity and spectrum of the Cesàro operator on weighted $$c_0$$ spaces

A detailed investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator \(\mathsf {C}\) acting on the weighted Banach sequence space \(c_0(w)\) for a bounded, strictly positive weight w. New features arise in the weighted setting (e.g. existence of eigenvalues, compactness, mean ergodicity) which are not present in the classical setting of \(c_0\).     

Composition Cesàro Operator on the Normal Weight Zygmund Space in High Dimensions

Let n>1 and B be the unit ball in n dimensions complex space Cⁿ.Suppose thatφis a holomorphic self-map of B andψ∈H(B)withψ(0)=0.A kind of integral operator,composition Cesàro operator,is defined by T_φψ(f)(z)=∫¹0f[φ(tz)]Rψ(tz)dt/t,f∈(B)z∈B.In this paper,the authors characterize the conditions that the composition Cesàro operator T_φ,ψis bounded or compact on the normal weight Zygmund space Z_μ(B).At the same time,the sufficient and necessary conditions for all cases are given.     

A note on the powers of Cesàro bounded operators

In this note we give a negative answer to Zemánek’s question (1994) of whether it always holds that a Cesàro bounded operator T on a Hilbert space with a single spectrum satisfies $ \mathop {\lim }\limits_{n \to \infty } $ \mathop {\lim }\limits_{n \to \infty } ∥T ⁿ⁺¹ − T ⁿ ∥ = 0.     

The Cesàro Operator in Growth Banach Spaces of Analytic Functions

The Cesàro operator C, when acting in the classical growth Banach spaces \({A^{-\gamma}}\) and \({A_0^{-\gamma}}\), for \({\gamma} > 0\), of analytic functions on \({\mathbb{D}}\), is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of C acting in these spaces. In addition, we determine the largest Banach space of analytic functions on \({\mathbb{D}}\) which C maps into \({A^{-\gamma}}\) (resp. into \({A_0^{-\gamma}}\)); this optimal domain space always contains \({A^{-\gamma}}\) (resp. \({A_0^{-\gamma}}\)) as a proper subspace.     

On the Cesàro summability of the ultraspherical series

Summary The author has obtained theorems for Cesàro summability of the ultraspherical series which are analogous to those of Izumi and Sunouchi (2) in Fourier series.     

On the Cesàro summability of orthogonal sequences

(C, ). , . 0<<1. 1) - ( _k ), _k =a _k , (C, ), . 2) , , (C, ) ; _k = =¦a _k ¦.     

Point spectra of Cesàro matrices

For all positive a the point spectrum of the (C, α) matrix is determined, where the matrix is regarded as an operator on certain Banach sequence spaces. In particular the point spectrum is obtained in the spaces I_p(X), with 1<p≤∞, where X is a Banach space.     

On Subspaces of Cesàro Spaces

We obtain a characterization of subspaces of L _p , with 1 < p < ∞, on which the L _p -norm is equivalent to the norm of the Cesàro space Ces _p . Also, we show that Ces _p has a complemented copy of the Cesàro sequence space ces _p .     

A theorem on the absolute Cesàro summability of factored fourier series

Summary In this paper, the author proves a theorem on the absoluteCesàro summability of factoredFourier series. His theorem extends a theorem ofMatsumoto and generalizes a theorem ofPrasad andBhatt.     

Statistical Extensions of Some Classical Tauberian Theorems for Cesàro Summability of Triple Sequences

Some growth conditions of the resolvent function of a Banach space operator are investigated using higher order Cesàro means. More precisely, Abel and Nevanlinna estimates are obtained under the condition of boundedness of some weighted Cesàro averages. Also, certains results related to the (strong or uniform) convergence of Cesàro means are mentioned.     

Cesàro summability of one- and two-dimensional Walsh-Fourier series

ВВОДьтсьp-кВАжИлОкАл ьНыЕ ОпЕРАтОРы И ОДНО МЕРНыЕ ДИАДИЧЕскИЕ МАРтИНг АльНыЕ пРОстРАНстВА хАРДИH _p . ДОкАжАНО, ЧтО ЕслИ сУБлИНЕИНыИ ОпЕРАтО РT p-кВАжИлОкАлЕН И ОгРА НИЧЕН ИжL _∞ ВL _∞, тО ОН ьВльЕтсь тАкжЕ ОгРАН ИЧЕННыМ ИжH _p ВL _p , (0<p<1). В кАЧЕстВЕ пРИ лОжЕНИь ДОкАжАНО, ЧтО МАксИМАльНыИ ОпЕРАт ОР ОДНОгО ЧЕжАРОВскОгО пАРАМЕтРА И МОДИФИцИ РОВАННых ЧЕжАРОВскИх сРЕДНИх МАРтИНгАлА ьВльЕтсь ОгРАНИЧЕННыМ ИжH _p ВL _p И ИМЕЕт слАБыИ тИп (L ₁,L ₁). Мы ВВОДИМ ДВУМЕРНыИ ДИА ДИЧЕскИИ гИБРИД пРОс тРАНстВ хАРДИH ₁ И пОкАжыВАЕМ, Ч тО МАксИМАльНыИ ОпЕРАт ОР сРЕДНИх ЧЕжАРО ДВУ МЕРНОИ ФУНкцИИ ИМЕЕт слАБыИ тИп (H ₁ ^# ,L ₁). тАк Мы пОлУЧАЕМ, Ч тО ДВУпАРАМЕтРИЧЕск ИЕ сРЕДНИЕ ЧЕжАРО ФУНкц ИИf ?H ₁ ^# ?L logL схОДьтсь пОЧтИ ВсУДУ к ИсхОДНОИ ФУНк цИИ.