Numerical ranges of weighted composition operators

The operator that takes the function f   to ψf°φ$\psi f°\phi$ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on H²$H^2$, the Hardy–Hilbert space of the unit disc. We consider the case where φ is a rotation of the unit disc and identify a class of convexoid operators. In the case of isometric weighted composition operators we give a complete classification of their numerical ranges. We also consider the inclusion of zero in the interior of the numerical range.     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Spectrum of a compact weighted composition operator

For analytic in the open unit disk and an analytic map from the unit disk into itself, the weighted composition operator is the operator on the weighted Hardy space given by This paper discusses the spectrum of when it is compact on a certain class of weighted Hardy spaces and when the composition map has a fixed point inside the open unit disk.

     

Sequences of Weighted Composition Operators on the Hardy–Hilbert Space

Let \(\varphi \) be an analytic self map of the open unit disc \(\mathbb {D}\). Assume that \(\psi \) is an analytic map of \(\mathbb {D}\). Suppose that f is in the Hardy–Hilbert space of the open unit disc \(H^2\). The operator that takes f into \(\psi \cdot f \circ \varphi \) is a weighted composition operator, and is denoted by \(C_{\psi ,\varphi }\). In this paper we relate the convergence of the sequence \(\{ C_{\psi _n,\varphi _n}\}\) in different operator topologies to the convergence of the two sequences of maps \(\{\varphi _n \}\) and \(\{ \psi _n\}\).     

Compact weighted composition operators on the Hardy space

A weighted composition operator takes an analytic map on the open unit disk of the complex plane to the analytic map , where is an analytic map of the open unit disk into itself and is an analytic map on the open unit disk. This paper studies how the compactness of depends on the interaction between the two maps and .

     

Synthesis of cleavable multi-functional mikto-arm star polymer by RAFT polymerization: example of an anti-cancer drug 7-ethyl-10-hydroxycamptothecin (SN-38) as functional moiety

Multi-functional mikto-arm star polymers containing three different arms [hydrophilic, SN-38-P(OEGMA_8–9)₁₁, cationizable, SN-38-P(DMAEMA)₃₈ and hydrophobic, SN-38-P(BMA)₂₆] were prepared by RAFT polymerization via an arm-first approach using a cleavable cross-linker. The star polymers were cleaved to the linear arms with tributylphosphine as a reducing agent. The decrease in molecular weight observed is consistent with the initial stars having approximately five arms. Blue fluorescence was observed when a solution of mikto-arm star was irradiated under a 365 nm light proving the retention of the SN-38 moiety during star formation by RAFT polymerization. Thus these polymer-drug conjugates can be considered as potential delivery vehicles for cancer therapy. The P(DMAEMA) arms can be quaternized using iodomethane, allowing star polymers to bind negatively charged small interfering RNA (siRNA) and potentially be used as a carrier for that material.     

Invertible weighted composition operators

A weighted composition operator Cψ,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 Cψ,φ acts on the Hardy-Hilbert space of the unit disc H²(D). Depending upon the nature of the fixed points of φ spectra are then investigated.