We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form ϕ(z) = az + b. We compare this result to an upper bound for C ϕ that is valid whenever ϕ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez. 2004 Elsevier Inc. All rights reserved.
Recent work by several authors has revealed the existence of many unexpected classes of normal weighted composition operators. On the other hand, it is known that every normal operator is a complex symmetric operator. We therefore undertake the study of complex symmetric weighted composition operators, identifying several new classes of such operators.
Let
φ
\varphi
be a nonconstant analytic self-map of the open unit disk in
C
\mathbb {C}
, with
‖
φ
‖
∞
>
1
\|\varphi \|_{\infty }>1
. Consider the operator
D
φ
D_{\varphi }
, acting on the Hardy space
H
2
H^{2}
, given by differentiation followed by composition with
φ
\varphi
. We obtain results relating to the adjoint, norm, and spectrum of such an operator.
Building on techniques developed by Cowen and Gallardo-Guti\'{e}rrez, we find
a concrete formula for the adjoint of a composition operator with rational
symbol acting on the Hardy space $H^{2}$. We consider some specific examples,
comparing our formula with several results that were previously known.Comment: 14 page
Abstract. We obtain a representation for the norm of the composition operator C φ on the Hardy space H 2 whenever φ is a linear-fractional mapping of the form φ(z) = b/(cz + d). The representation shows that, for such mappings φ, the norm of C φ always exceeds the essential norm of C φ . Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form φ(z) = sz + t has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers s and t, Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of C 1/(2−z) is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator C φ , for which C φ > C φ e, an equation whose maximum (real) solution is C φ 2 . Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.