Ruishen Qian

Suppose $$(C_t)_{t\geqslant 0}$$ ( C t ) t ⩾ 0 is the composition semigroup induced by a one-parameter semigroup $$(\varphi _t)_{t\geqslant 0}$$ ( φ t ) t ⩾ 0 of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of $$(C_t)_{t\geqslant 0}$$ ( C t ) t ⩾ 0 acting on the weighted Bergman space induced by doubling weights, provided $$(\varphi _t)_{t\geqslant 0}$$ ( φ t ) t ⩾ 0 is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of $$(C_t)_{t\geqslant 0}$$ ( C t ) t ⩾ 0 also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of $$(C_t)_{t\geqslant 0}$$ ( C t ) t ⩾ 0 .

Let $0 \lt p,s \lt \infty $, $n$ be a nonnegative integer and $\mu $ be a positive Borel measure on $\mathbb {D}$. Let $\mathcal {T}_{s}^{p,n}{(\mu )}$ be the space of all analytic functions $f $ such that $$\sup _{I\subseteq \partial \mathbb D}\frac {1}{

In this paper, the boundedness and compactness of embedding from Dirichlet-Morrey spaces ${\mathcal{D}}_p^{\lambda }$ into tent spaces ${\mathcal{T}}_s(\mu )$ are investigated. As an application, we characterize the boundedness of the Volterra integral operator $J_g$ from ${\mathcal{D}}_p^{\lambda }$ to general function spaces $F(2,p(1-\lambda ),s)$. Meanwhile, the operator $I_g$ and the multiplication operator $M_g$ from ${\mathcal{D}}_p^{\lambda }$ to $F(2,p(1-\lambda ),s)$ are studied. Furthermore, the essential norm of $J_g$ and $I_g$ from ${\mathcal{D}}_p^{\lambda }$ to $F(2,p(1-\lambda ),s)$ are also considered.

In this paper, we study the boundedness and essential norm of Volterra integral operator $ V_g $ and integral operator $ S_g $ on Dirichlet type spaces $ {\mathcal{D}_{K, \alpha}} $.

Motivated by ^[11,12], under some conditions on weighted function $ K $, we investigated power bounded and power bounded below composition operators on Dirichlet Type spaces $ {\mathcal{D}_{K}} $.