REMARKS CONCERNING SOME GENERALIZED CES ARO OPERATORS ON l2

Here we see that the p-Ces`aro operators, the generalized Ces`aro operators of order one, the discrete generalized Ces`aro operators, and their adjoints are all posinormal operators on l2, but many of these operators are not dominant, not normaloid, and not spectraloid. The question of dominance for Ck, the generalized Ces`aro operators of order one, remains unsettled when 1 2 k < 1, and that points to some general questions regarding terraced matrices. Sufficient conditions are given for a terraced matrix to be normaloid. Necessary conditions are given for terraced matrices to be dominant, spectraloid, and normaloid. A very brief new proof is given of the well-known result that Ck is hyponormal when k 1.