The self-commutator of a subnormal operator

This work is directed toward a study of the self-commutator of a subnormal operator. This can lead one in several directions. One place the self-commutator plays an important role is in the C*-algebra generated by a subnormal operator. We shall do a systematic study of the C*-algebra generated by a subnormal operator. We shall see the importance of having compact self-commutator and shall use C*-algebra techniques to get estimates on the essential norm of self-commutators. We also use these methods to show certain operators have diagonalizable self-commutators and to show that compactness of the self- commutator is preserved under similarity for (essentially) subnormal operators. These methods naturally lead into a new class of operators, namely essentially subnormal operators. We shall characterize these as those operators that have an essentially normal extension.

We give a sharp form of the Berger-Shaw Theorem for cyclic subnormal operators S and characterize those operators in the commutant of S that have trace class self-commutator.

We shall also study subnormal operators that have zero as an eigenvalue for there self- commutators. These are natural generalizations of operators with finite rank self- commutator and shall lead into generalized quadrature domains and some interesting approximation questions. In short we study questions involving the self-commutator.