Continuity and Holomorphicity of Symbols of Weighted Composition Operators

Abstract

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.

Multiplier algebras of normed spaces of continuous functions

“…First, similarly to the continuous case, if X is a domain in C n , F is a 1-independent NSHF over X, and E is a NSHF over X, then Mult (F, E) ⊂ H (X). For the multiplier algebras an even stronger fact is true (see [8,Proposition 4.3]). Proposition 4.7.…”

Which multiplication operators are surjective isometries?

“…It is easy to see that MO's from a 1-independent NSCF determine their symbols, and WCO's from a 2-independent NSCF also determine their symbols (see [7,Proposition 2.8]). Moreover, some properties of the symbols of WCO can indeed be recovered (see [7,Corollary 3.3…”

“…Hence, if ω (y) = 0, then y H = 0, which leads to a contradiction. If in this case Φ has a dense image, then W Φ,ω is an injection (see [7,Proposition 2.6]).…”

“…In this article we continue our investigation (see [7]) of the general framework which allows to consider any Banach space that consists of continuous (scalar-valued) functions, such that the point evaluations are continuous linear functionals, and of WCO's on these spaces.…”

Maximum Modulus Principle For Multipliers and Mean Ergodic Multiplication Operators

“…One of the most prominent classes of linear operators on the spaces of functions is the class of multiplication operators (MO). In this article we continue our investigation (see [2,3]) of the general framework which allows to consider any Banach space that consists of continuous (scalar-valued) functions, such that the point evaluations are continuous linear functionals, and of MO's on these spaces.…”