Imbedded operators with finite Blaschke product symbol

Abstract

LetF=(f _ij )be ann×n matrix ofH ^∞entries, and define

$$S_F = (T_{2 \otimes ln}^* \oplus T_{2 \otimes ln}^* )\left| {_{Graph T_F^* } } \right..$$

This type of operator plays an important role in Cowen-Douglas theory. We call it the imbedded operator with symbolF. In the paper [L], we have already shown thatS _F is a compact pertubation ofT ^* _z⊗ I _n by BDF Theorem. In this paper, we begin to investigate the compact part ofS _F . We give a practical method to calculate this compact part whenn=1 andF is any finite Blaschke product.