Lie Maps And Reverse Elementary Operators On Nest Algebras

Usually, operator algebras can be divided into two classes, the one of which is called selfadjoint operator algebras and the other one is called non-selfadjoint operator algebras. Non-selfadjoint operator algebras which are younger in history, and richer in contents than selfadjoint operator algebras, are one active and fertile subject in the research of operator algebras. Nest algebras are the natural generalizations of upper triangular matrix algebras in infinite dimensional spaces and an important object of the class of non-selfadjoint operator algebras. Also, Lie maps and (reverse) elementary operators are two important transformations on algebras or rings.This thesis is dovoted to the investigation of the additivity of Lie maps and the form of reverse elementary operators on nest algebras. Chapter 1 introduces some basic concepts and summarizes theb- ackground of the problems. Chapter 2 shows that a bijective Lie map acting on a standard operator algebra on a Hilbert space with dimension greater than one is additive, where the considered algeb- ra does not contain the identity operator. In Chapter 3, we generalize the result above to a general ne- st algebra. Also, we construct two counterexamples which show that these two results are not true fo- r those algebras containing the identity operator. In Chapter 4, we study the reverse elementary operat- ors on a nest algebra on a Banach space, giving the concrete form of surjective reverse elementary oper- ators. Finally, we characterize the elementary operators on completely distributive and commutative subspace lattice algebras, which are the extension of nest algebras.