Norms and spectral radii of linear fractional composition operators on the ball

TL;DR

This paper proves that linear fractional maps of the unit ball induce bounded composition operators on Hilbert function spaces, extends spectral radius formulas, and highlights their belonging to the Schur-Agler class.

Contribution

It provides a new proof of boundedness, norm bounds, and spectral radius extension for composition operators induced by linear fractional maps on the ball.

Findings

Linear fractional maps induce bounded composition operators.

Spectral radius formulas extend to these operators.

All such maps belong to the Schur-Agler class.

Abstract

We give a new proof that every linear fractional map of the unit ball induces a bounded composition operator on the standard scale of Hilbert function spaces on the ball, and obtain norm bounds analogous to the standard one-variable estimates. We also show that Cowen's one-variable spectral radius formula extends to these operators. The key observation underlying these results is that every linear fractional map of the ball belongs to the Schur-Agler class.