Thomas Tonev

Uniform algebra isomorphisms and peripheral multiplicativity

Let φ: A→ B be a surjective operator between two uniform alge-(1) =1 we Show that if φ satisfies the peripheral multiplicativity bras with φ(1) = 1. We show that if if satisfies the peripheral multiplicativity condition σ π (φ(f)φ(g)) = σ π (fg) for all f,g ∈ A, where σ π (f) is the peripheral spectrum of f, then φis an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.

Spectral conditions for almost composition operators between algebras of functions

In this article we develop a unifying theory of many years of work by a number of researchers. Especially, we establish general sufficient conditions for maps between algebras of bounded continuous functions on locally compact Hausdorff spaces to be almost composition or almost weighted composition operators, which extend the main results of many previous works on this subject. The following are some typical results. Let T : A → B be a surjective map between two function algebras on locally compact Hausdorff spaces X and Y with Choquet boundaries δA ⊂ X and δB ⊂ Y . If ‖Tf Tg‖ = ‖fg‖ and there is an ε, 0 ≤ ε < 2/3, so that the peripheral spectrum σπ(Tf Tg) is contained in an ε ‖fg‖-neighborhood of σπ(fg) for all f ∈ A and all g ∈ A with ‖g‖ = 1, then there is a continuous function α : δB → {±1} and a homeomorphism ψ : δB → δA so that |(Tf)(y)−α(y) f(ψ(y))| ≤ 2ε |f(ψ(y))| for each f ∈ A and every y ∈ δB. Therefore, T is an almost weighted composition operator on δB. If ‖Tf Tg‖ = ‖fg‖, there are ε, 0 ≤ ε < 1, and η, 0 ≤ η < 1, so that d(σπ(Tf Tg), σπ(fg)) ≤ ε ‖fg‖, while σπ(Tf) is contained in an ηneighborhood of σπ(f) for all f ∈ A and all g ∈ A with ‖g‖ = 1. Then |(Tf)(y) − f(ψ(y))| ≤ (ε + η) |f(ψ(y))| for each y ∈ δB and every f ∈ A; i.e. T is an almost composition operator on δB. If σπ(Tf Tg) ⊂ σπ(fg) (or σπ(fg) ⊂ σπ(Tf Tg)), and d(σπ(Tg), σπ(g)) ≤ η for some η, 0 ≤ η < 1, for all f ∈ A and all g ∈ A with ‖g‖ = 1, then (Tf)(y) = f(ψ(y)) for each f ∈ A and every y ∈ δB. Consequently, T is a composition operator on δB and, therefore, an algebra isomorphism.

Automatic Differentiability and Characterization of Cocycles of Holomorphic Flows

In this paper we prove that cocycles of holomorphic flows on domains in the complex plane are automatically differentiable with respect to the flow parameter, and their derivatives are holomorphic functions. We use this result to show that, on simply connected domains, an additive cocycle is a coboundary if and only if this cocycle vanishes at the fixed point of the flow.

A New Proof for the Description of Holomorphic Flows on Multiply Connected Domains

We provide a new proof for the description of holomorphic and biholomorphic flows on multiply connected domains in the complex plane. In contrast to the original proof of Heins (1941) we do this by the means of operator theory and by utilizing the techniques of universal coverings of the underlying domains of holomorphic flows and their liftings on the corresponding universal coverings.