Mathematics Faculty Publications

Isolation and component structure in spaces of composition operators

Christopher Hammond et al. — Tue, 05 Feb 2013 11:43:30 PST

We establish a condition that guarantees isolation in the space of composition operators acting between H ^p(B _N) and H ^q(B _N), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.

The norm of a composition operator with linear symbol acting on the Dirichlet space

Christopher Hammond — Tue, 05 Feb 2013 10:16:33 PST

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 ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.

Adjoints of composition operators with rational symbol

Christopher Hammond et al. — Tue, 05 Feb 2013 09:30:20 PST

Building on techniques developed by C. C. Cowen and E. A. Gallardo-Gutiérrez [J. Funct. Anal. 238 (2006), no. 2, 447–462;MR2253727 (2007e:47033)], 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.

Norms of linear-fractional composition operators

Paul S. Bourdon et al. — Tue, 05 Feb 2013 09:05:36 PST

Composition operators with maximal norm on weighted Bergman spaces

Brent J. Carswell et al. — Mon, 04 Feb 2013 12:07:18 PST

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space H², where every inner function induces a composition operator with maximal norm.