Sub-n-normal Operators

In this paper we introduce the class of sub-n-norrnal operators. By definition, such an operator is the restriction to an invariant subspace of an n-normal operator, and thus the sub-n-normal operators form a larger class than the subnormal operators. We obtain some modest structure theorems and contrast sub-n-normal operators with sub-Jordan operators. Finally we show that a sub-n-normal operator with rich spectrum has a nontrivial invariant subspace.