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We introduce and discuss the oscillation of an operator  T  mapping      L p ( S , Σ  , μ  ) {L^p}(S,\Sigma ,\mu )       into a Banach space. We establish results relating the oscillation, a “local norm", to the norm of the operator. Also using the oscillation we define a generalization of the Fredholm operators  T  with index      κ  ( T ) &gt; ∞  \kappa (T) &gt; \infty       and a corresponding perturbation class which contains the compact operators.

The oscillation of an operator on 𝐿^{𝑝}