The main problem considered in this article is the following: if F, E are normed spaces of continuous functions over topological spaces X and Y respectively, and ω : Y → C and Φ : Y → X are such that the weighted composition operator W Φ,ω is continuous from F into E, when can we guarantee that both Φ and ω are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over complex manifolds. Additionally, we consider the most basic properties of the weighted composition operators, which only have been proven before for more concrete function spaces.

The main goal of this note is to show that (not necessarily holomorphic) multipliers of a wide class of normed spaces of continuous functions over a connected Hausdorff topological space cannot attain their multiplier norms, unless they are constants. As an application, a contractive multiplication operator is either a multiplication with a constant, or is completely non-unitary. Additionally, we explore possibilities for a multiplication operator to be (weakly) compact and (uniformly) mean ergodic.

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In this paper we study isometry-invariant Finsler metrics on inner product spaces over R or C, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most general concept of the Finsler metric is considered without any additional assumptions that are usually built into its definition. However, we present refined versions of the described results for more specific classes of metrics, including the class of Riemannian metrics. Our main result states that for an isometry-invariant Finsler metric the only possible linear maps under which the metric is invariant are scalar multiples of isometries. Furthermore, we characterise the metrics invariant with respect to all linear maps of this type.

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