semisimple operator

A semisimple operator is one that splits as fully as the field allows: every invariant subspace it has comes with an invariant complement. Nothing gets trapped without an escape route — wherever T preserves a subspace, you can peel off a matching invariant piece and reduce. This is the property called complete reducibility.

The convenient test over the complex numbers: semisimple is exactly the same as diagonalizable. Over an algebraically closed field the two notions coincide. Over a general field they part ways — semisimple is the right, field-honest concept, because an operator can be unsplittable-down-to-blocks (semisimple) without those blocks being one-dimensional. Equivalently, T is semisimple iff its minimal polynomial is squarefree (a product of distinct irreducibles).

Semisimplicity is exactly the obstruction that the Jordan-Chevalley decomposition isolates: T = S + N where S is the semisimple part and N measures how far T strays from semisimple. T is semisimple precisely when N = 0. A real example: the 90-degree rotation [0, -1; 1, 0] is semisimple over R (minimal polynomial x^2 + 1 is irreducible, hence squarefree) even though it is not diagonalizable over R.