Operators with absolute continuity properties: an application to quasinormality

An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. Algebraic characterizations of some classes of operators that emerged in this context are invented. Various examples and counterexamples illustrating the concepts of the paper are constructed by means of weighted shifts on directed trees. Generalizations of these results that cover the case of q-quasinormal operators are established.