Characterization of λ-Self-commuting operators and a class of 2-hyponormal Toeplitz operators

Toeplitz operators are of great interest in the study of Operator Theory. They have unique spectral properties, and classifying those that are subnormal remains an open question. Subnormality is a topic of primary interest in Operator Theory. Paul Halmos posed the question “Is every subnormal Toeplitz operator normal or analytic?” Carl Cowen and John Long showed that the story is more complicated by constructing an explicit example.

In this dissertation, we look at the operator of Cowen and Long through the lens of k-hyponormality. We provide a new proof that gives greater insight into the nature of this operator and why it is subnormal. This investigation led to the classification of a new class of operators, of which the Cowen-Long operator is a chief example. We call these operators λ-self-commuting. We determine the exact form of such operators for all relevant values of λ. In addition, we classify some 2-hyponormal Toeplitz operators.