The self-commutator of a subnormal operator

Author

Nathan Stephen Feldman

Date of Award

5-1997

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

John B. Conway

Committee Members

Michael Guidry, Stefan Richter, Kenneth R. Stephenson, Carl Sundberg

Abstract

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.

Recommended Citation

Feldman, Nathan Stephen, "The self-commutator of a subnormal operator. " PhD diss., University of Tennessee, 1997.
https://trace.tennessee.edu/utk_graddiss/9487