In this paper we characterize EP operators through the existence of different types of factorizations. Our results extend to EP operators existing characterizations for EP matrices and give new characterizations both for EP matrices and EP operators.2000 Mathematics Subject Classification. Primary 47A05, 15A09; Secondary 47B.

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A study of nonselfadjoint algebras of Hilbert space operators was begun by considering special types of such algebras, namely those determined by a commuting family of rank one operators. A first step in this direction was made by Erdos in [1] and is continued more extensively in [2].

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.