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

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.Comment: 15 page

Dilations and constrained algebras

It is well known that unital contractive representations of the disk algebra are completely contractive. Let A denote the subalgebra of the disk algebra consisting of those functions f whose first derivative vanishes at 0. We prove that there are unital contractive representations of A which are not completely contractive, and furthermore provide a Kaiser and Varopoulos inspired example for A and present a characterization of those contractive representations of A which are completely contractive. In the positive direction, for the algebra of rational functions with poles off the distinguished variety V in the bidisk determined by (z-w)(z+w)=0, unital contractive representations are completely contractive.Comment: New to version 2 is a proof of rational dilation for the distinguished variety in the bidisk determined by (z-w)(z+w)=