We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

Let u and $\varphi $ be two analytic functions on the unit disk D such that $\varphi (D) \subset D$ . A weighted composition operator $uC_{\varphi }$ induced by u and $\varphi $ is defined on $A^2_{\alpha }$ , the weighted Bergman space of D, by $uC_{\varphi }f := u \cdot f \circ \varphi $ for every $f \in A^2_{\alpha }$ . We obtain sufficient conditions for the compactness of $uC_{\varphi }$ in terms of function-theoretic properties of u and $\varphi $ . We also characterize when $uC_{\varphi }$ on $A^2_{\alpha }$ is Hilbert–Schmidt. In particular, the characterization is independent of $\alpha $ when $\varphi $ is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on $A^2_{\alpha }$ .

We characterise bounded and compact generalised weighted composition operators acting from the weighted Bergman space $A^p_\omega $ , where $0<p<\infty $ and $\omega $ belongs to the class $\mathcal {D}$ of radial weights satisfying a two-sided doubling condition, to a Lebesgue space $L^q_\nu $ . On the way, we establish a new embedding theorem on weighted Bergman spaces $A^p_\omega $ which generalises the well-known characterisation of the boundedness of the differentiation operator $D^n(f)=f^{(n)}$ from the classical weighted Bergman space $A^p_\alpha $ to the Lebesgue space $L^q_\mu $ , induced by a positive Borel measure $\mu $ , to the setting of doubling weights.

We investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.

We provide complete characterisations for the compactness of weighted composition operators between two distinct $L^{p}$-spaces, where $1\leq p\leq \infty$. As a corollary, when the underlying measure space is nonatomic, the only compact weighted composition map between $L^{p}$-spaces is the zero operator.

Let $u$ and $\unicode[STIX]{x1D711}$ be two analytic functions on the unit disc $D$ such that $\unicode[STIX]{x1D711}(D)\subset D$. A weighted composition operator $uC_{\unicode[STIX]{x1D711}}$ induced by $u$ and $\unicode[STIX]{x1D711}$ is defined by $uC_{\unicode[STIX]{x1D711}}f:=u\cdot f\circ \unicode[STIX]{x1D711}$ for every $f$ in $H^{p}$, the Hardy space of $D$. We investigate compactness of $uC_{\unicode[STIX]{x1D711}}$ on $H^{p}$ in terms of function-theoretic properties of $u$ and $\unicode[STIX]{x1D711}$.

Let $\phi $ and $\psi $ be analytic maps on the open unit disk $D$ such that $\phi (D) \subset D$. Such maps induce a weighted composition operator $C_{\phi ,\psi }$ acting on weighted Banach spaces of type $H^{\infty }$or on weighted Bergman spaces, respectively. We study when such operators are order bounded.

We give derivative-free characterizations for bounded and compact generalized composition operators between (little) Zygmund type spaces. To obtain these results, we extend Pavlović’s corresponding result for bounded composition operators between analytic Lipschitz spaces.

Let ϕ:D→D and ψ:D→ℂ be analytic maps. These induce a weighted composition operator ψCϕ acting between weighted Bloch type spaces. Under some assumptions on the weights we give a necessary as well as a sufficient condition when such an operator is continuous.