In this paper, we characterize boundedness and compactness of weighted composition operators on the Dirichlet space and obtain the estimates for the essential norm.

We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.

Let $E^{\varphi }\left(\mu \right)$ be the subspace of finite elements of an Orlicz space endowed with the Luxemburg norm. The main theorem says that a compact linear operator $T:E^{\varphi }\left(\mu \right)\to C\left(\Omega \right)$ is extreme if and only if $T^{*}\omega \in ExtB\left({\left(E^{\varphi }\left(\mu \right)\right)}^{*}\right)$ on a dense subset of $\Omega$, where $\Omega$ is a compact Hausdorff topological space and $\langle T^{*}\omega ,x\rangle =\left(Tx\right)\left(\omega \right)$. This is done via the description of the extreme points of the space of continuous functions $C(\Omega ,L^{\varphi }\left(\mu \right))$, $L^{\varphi }\left(\mu \right)$ being an Orlicz space equipped with the Orlicz norm (conjugate to the Luxemburg one). There is also given a theorem on closedness of the set of extreme...