Linear Maps On Nest Algebra Modules

This thesis is devoted to the investigation of some linear maps on weakly closed nest algebra modules.In Chapter 1, we summarize the background of this paper, and introduce some terminology and notation.In Chapter 2, we discuss rank preserving linear maps between weakly closed nest algebra modules on complex Hilbert space H . Structures of rank-1 preserving linear maps in various conditions are given. Especially, we give more concert characterizations on finite nest algebra modules.In Chapter 3, generalized module isomorphisms between weakly closed nest algebra modules are studied and the forms of such generalized module isomorphisms are described. As a corollary, we discuss module isomorphisms, and it turns out that module isomorphisms are necessarily trivial. Also, we prove that local module isomorphisms acting on weakly closed nest algebra modules are module isomorphisms.In Chapter 4, we first study local left (right) maps acting on weakly closed nest algebra modules, and prove that local left (right) maps acting on weakly closed nest algebra modules are left (right) maps. Furthermore, we define generalized left (right)α-maps acting on standard operator algebras, then such generalized left (right)α-maps are characterized.In Chapter 5, additive derivations of reflexive operator algebras are considered. Let X be a Banach space on real or complex number field and A be a reflexive operator algebra such that X -≠X in LatA .If X ( or X_-~⊥) is of infinite dimension, then every additive derivation of A is an inner derivation.