THE STRUCTURE OF QUASINILPOTENT OPERATORS

his thesis deals with structural results about the set of quasinilpotent operators in ;The second chapter is concerned with finite-dimensional results pertaining to nilpotent elements of matrix algebras. It is shown that if a two-dimensional subspace consisting of nilpotents contains rank-one operators then it has to lie in an algebra consisting of nilpotent operators. The fact that the nilpotents form a variety is then demonstrated. Linear maps which preserve both the nilpotents and adjoints are classified. Finally an example is given to show that the transpose map is not completely quasinilpotent preserving.;The third chapter deals with two-dimensional subspaces of ;The fourth chapter is concerned with structure results for linear maps on ;Finally the fifth chapter contains some assorted results on quasinilpotent operators. First is a result on nilpotent semigroups showing that, for infinite-dimensional Hilbert spaces, one can find a pair of operators which generate a semigroup consisting of quasinilpotent operators but whose sum is not quasinilpotent. Second are results on how much the spectral radius of an element of a Banach algebra can be perturbed by adding a nilpotent element to it. Lastly, it is shown that, for T...