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Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. We define the generalized Ces\`aro operator as follows $$ C_{\omega}(f)(z)=\int_0^1 f(tz)\left(\frac{1}{z}\int_0^z B^{\omega}_t(u)\,du\right)\,\omega(t)dt,$$ where $\{B^{\omega}_\zeta\}_{\zeta\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_\omega$ induced by a radial weight $\omega$ in the unit disc $\mathbb{D}$. We study the action of the operator $C_{\omega}$ on weighted Hardy spaces of analytic functions $\mathcal{H}_{\gamma}$, $\gamma >0$ and on general weighted Bergman spaces $A^2_{\mu}$.

We consider the Hilbert-type operator defined by $$ H_{\omega}(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^{\omega}_t(u)\,du\right)\,\omega(t)dt,$$ where $\{B^{\omega}_\zeta\}_{\zeta\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_\omega$ induced by a radial weight $\omega$ in the unit disc $\mathbb{D}$. We prove that $H_{\omega}$ is bounded on the Hardy space $H^p$, $1

Analytic Morrey spaces belong to the class of function spaces which, like BMOA, are defined in terms of the degree of oscillation on the boundary of functions analytic in the unit disc. We consider semigroups of composition operators on these spaces and focus on the question of strong continuity. It is shown that these semigroups behave like on BMOA., Comment: 15 pages

In this paper we are concerned with two classes of conformally invariant spaces of analytic functions in the unit disc $\D$, the Besov spaces $B^p$ $(1\le p<\infty )$ and the $Q_s$ spaces $(0

If $\,\mu \,$ is a finite positive Borel measure on the interval $\,[0,1)$, we let $\,\mathcal H_\mu \,$ be the Hankel matrix $\,(\mu _{n, k})_{n,k\ge 0}\,$ with entries $\,\mu _{n, k}=\mu _{n+k}$, where, for $\,n\,=\,0, 1, 2, \dots $, $\mu_n\,$ denotes the moment of order $\,n\,$ of $\,\mu $. This matrix induces formally the operator $\,\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}^\infty a_kz^k\,$, in the unit disc $\,\mathbb D $. When $\,\mu \,$ is the Lebesgue measure on $\,[0,1)\,$ the operator $\,\mathcal H_\mu\,$ is the classical Hilbert operator $\,\mathcal H\,$ which is bounded on $\,H^p\,$ if $\,1

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of order $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb{D} $. This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators $\mathcal H_\mu $ on mean Lipschitz spaces of analytic functions., Comment: 11 pages, 0 figures

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of order $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb D $. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $H_\mu $ on Hardy spaces and on M\"obius invariant spaces., Comment: arXiv admin note: text overlap with arXiv:1612.08304

To each weighted Dirichlet space $\mathcal{D}_p$, $0

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu_{n, k})_{n,k\ge 0}$ with entries $\mu_{n, k}=\mu_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of orden $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\D $. This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu }$ on Hardy spaces has been recently studied. This paper is devoted to study the operators $H_\mu $ acting on certain conformally invariant spaces of analytic functions on the disc such as the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces., Comment: 24 pages