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We introduce and study the Bourgain index of an operator between two Banachspaces. In particular, we study the Bourgain $\ell_p$ and $c_0$ indices of anoperator. Several estimates for finite and infinite direct sums areestablished. We define classes determined by these indices and show that someof these classes form operator ideals. We characterize the ordinals which occuras the index of an operator and establish exactly when the defined classes areclosed. We study associated indices for non-preservation of $\ell_p^\xi$ and$c_0^\xi$ spreading models and indices characterizing weak compactness ofoperators between separable Banach spaces. We also show that some of theseclasses are operator ideals and discuss closedness and distinctness of theseclasses.

Classes of operators determined by ordinal indices