Lie Derivations Of Reflexive Algebras

In this thesis, we study the linear maps Lie derivable at some points of the nest algebra on Hilbert space. Also, we investigate the linear maps Lie derivable at some points of the reflexive algebra on a Banach space, whose invariant subspace lattice is J-subspace lattice, or has the nontrivial smallest element, or has the nontrivial greatest element. The thesis consists of four chapters.In Chapter 1, we introduce some terminology and notation, and summarize the background. Also, we state the main results of this thesis.Chapter 2 studies the linear map Lie derivable at zero points or idempotent on nest algebra. We prove that every linear map is sum of a derivation and and a linear map with image in the center vanishing on commutators.In Chapter 3, the linear map Lie derivable at zero point on standard sub-algebra of JSL algebra that contains unit operator is described. It turns out that such linear map is sum of a derivation and a linear map with image in the center vanishing on commutators.In Chapter 4, we consider the reflexive algebra whose invariant subspace lattice has the nontrivial smallest element, or has the nontrivial greatest element. We state every the linear maps which is Lie derivable at zero point is a sum of a derivation and and a linear map with image in the center vanishing on commutators.