Numerical ranges of weighted composition operators

Abstract

Abstract. The operator that takes the function f to ψf • φ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on 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.

Numerical Ranges of Normal Weighted Composition Operators on the Fock Space of CN

“…If there exists V ∈ { ∈ C | | | = ‖ , ‖} such that V ∈ ( , ), then by [1,Lemma 2.3] and Theorem 3(2), for any ∈ R, V ∈ ( , ), which implies that there is an uncountable collection of orthogonal vectors in F 2 since , is normal and eigenvalues of , that correspond to distinct eigenvalues are orthogonal, a contradiction. So…”

“…Inspired by results on numerical ranges of (weighted) composition operators on Hardy space [1], in this paper we give a complete characterization of numerical ranges of normal weighted composition operators on the Fock space of C ( ≥ 1), the -dimensional complex Euclidean space.…”

Weighted composition operators on the Fock space

“…The set W (T ) is convex, its closure contains σ(T ) and σ p (T ) ⊆ W (T ). There are some papers that the numerical range of composition operators and weighted composition operators on the Hardy space H 2 were investigated (see [1], [2], [9] and [12]).…”