Maximum Modulus Principle For Multipliers and Mean Ergodic Multiplication Operators

Abstract

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.

Multiplier algebras of normed spaces of continuous functions

“…Note that since F (X) is a reflexive locally convex space, there is J * * F : F * * → F (X). It was shown in [10] that J * * F is injective if and only if span {x F |x ∈ X } = F * and if and only if the weak and pointwise topologies coincide on B F . In this case F * * is a BSF over X and J F * * = J * * F .…”

“…If F is complete, then Mult (F) is a unital Banach Algebra, according to Corollary 2.5. If F is a 1-independent NSF over X, then Mult (F) contractively embeds into F ∞ (X) (see [10,Proposition 2.2]). Proposition 2.9.…”

“…Then, there is λ ∈ C such that M ω − λId F is a compact operator on F. But the latter operator is equal to M ω−λ1 . Since there can be no nonzero compact multiplication operator on an infinite-dimensional 1-independent NSCF (see [10,Proposition 2.10]), it follows that ω ≡ λ.…”