Similarity invariant subspaces and lie ideals in operator algebras

We study subspaces of algebra L(X) which are invariant with respect to all similarity transformations and give a full description of these subspaces for finite-dimensional space X, for separable Hilbert space X and for X=l(sub p) (1 <= p < (infinity)) or c0. The problem is equivalent to a study of Lie ideals in L(X).