Invertible weighted composition operators

Abstract

A weighted composition operator C ψ,ϕ takes an analytic map f on the open unit disc of the complex plane to the analytic map ψ · f • ϕ where ϕ is an analytic map of the open unit disc into itself and ψ is an analytic map on the open unit disc. This paper studies the invertibility of such operators. The two maps ψ and ϕ are characterized when C ψ,ϕ acts on the Hardy-Hilbert space of the unit disc H 2 (D). Depending upon the nature of the fixed points of ϕ spectra are then investigated.

Hermitian weighted composition operators on $H^{2}$

“…Although it will have little impact on our work, it is not necessary for f to be bounded in order for W f,ϕ to be bounded (see [11]). Weighted composition operators have been studied occasionally over the past few decades, but have usually arisen in answering other questions related to operators on spaces of analytic functions, such as questions about multiplication operators or composition operators.…”

“…Weighted composition operators also arise in the description of commutants of analytic Toeplitz operators (see for example [2,3]) and in the adjoints of composition operators (see for example [5,9,6]). Only recently have investigators begun to study the properties of weighted composition operators in general (see [11]). …”

Weighted composition operators on the Dirichlet space: boundedness and spectral properties

“…In a very recent work, Hyvärinen, Lindström, Nieminen and Saukko [13] have described the spectra of invertible weighted composition operators W u,ϕ acting on a large class of analytic function spaces including the weighted Bergman and the weighted Hardy spaces; generalizing previous results obtained in [11]. Nevertheless, as they also remark, their results do not apply directly to the Dirichlet space since they rely on the fact that the algebra of the multipliers of the spaces considered is H ∞ .…”

“…Nevertheless, {ϕ n (z 0 )}, z 0 ∈ D, is no longer interpolating in M(D) and therefore, our proof completely differs from the previous ones. Likewise, the work in [11] made use of inner functions, which are inappropriate in the context of D.…”

Invertible and normal composition operators on the Hilbert Hardy space of a half–plane

“…Thus, if C is a bounded, invertible operator, so is the weighted composition operator T ;' . According to [6], ' must be a disc automorphism and hence, must be one of the conformal automorphisms of … C inducing bounded composition operators. Also, according to [12], only the identity and parabolic half-plane automorphisms induce isometric composition operators.…”