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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 
$A^2_{\alpha }$.
$L^{p}$-SPACES
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.
$H^{p}$-SPACES
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 
$\unicode[STIX]{x1D711}$.
