Curvature calculations for a class of homogeneous operators

Abstract

For an operator $T$ in the class ${\mathrm B}_n(\Omega)$, introduced in \cite{CD}, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle $E_T$ determine the unitary equivalence class of the operator $T$. In the paper \cite{CD2}, the authors ask if there exists some pair of inequivalent oprators $T_1$ and $T_2$ for which the simultaneous unitary equivalence class of the curvature along with all the covariant derivatives coincide except for the derivative of the highest order. Here we show that some of the covariant derivatives are necessary to determine the unitary equivalence class of the operators in ${\mathrm B}_n(\Omega)$. Our examples consist of homogeneous operators. For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives are determined from the simultaneous unitary equivalence class of these at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples.