Let $μ$ be a positive finite Borel measure on $[0,1).$ Cesàro-type operators $C_μ$ when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that $C_μ$ is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of $C_μ$ on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments $(μ_n)_{n\in\N_0}$. The continuity, compactness and spectrum of $C_μ$ acting on Fréchet and (LB) Korenblum type spaces are also considered.
Complete mapping criteria for generalized Cesàro operators between Hardy spaces
School of Mathematics and Statistics, Hunan University of Science and Technology（湖南科技大学数学与统计学院）；School of Financial Mathematics and Statistics, Guangdong University of Finance（广东金融学院金融数学与统计学院）
For $p>\frac{2λ}{2λ+1}$ with $λ>0$, the Hardy spaces $H_λ^{p}(\mathbb{R}^{2}_+)$ associated with the Dunkl transform $\mathscr{F}_λ$ and the Dunkl operator $D_x$ on the line, where $D_xf(x)=f'(x)+\fracλ{x}[f(x)-f(-x)]$, is the set of function $F=u+iv$ on the upper half plane $\mathbb{R}_+^2=\big\{(x, y): y>0\big\}$, satisfying the $λ$-Cauchy-Riemann equations: $D_xu-\partial_y v=0, \partial_y u +D_xv=0$, and $\sup_{y>0}\int_{\mathbb{R}}|F(x, y)||x|^{2λ}dx<0$. In this paper, we will study the boundedness of Cesàro operator on $H_λ^{p}(\mathbb{R}^{2}_+)$. We will prove the following inequality $…
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.
