The Centralizers Of Nest Algebras

Nest algebra is an important class of non-selfadjoint reflexive operator algebras, which is the natural generalization of upper triangular matrix algebra in infinite dimensional space. Simliar to isomorphism and derivation, centralizer is an important transformation on algebra or ring. This thesis is dovoted to the investigation of centralzer on nest algebra.Chapter 1 introduces the background and preliminary, and summary the main results of this thesis.Chapter 2 proves that every bijective Jordan centralizer on a prime ring is automatically additive, and we construct a counterexample which shows that this result is not necessarily true for nest algebras. However, it is proved that a special class of bijective Jordan centralizers on nest algebras have the automatic additivity.In Chapter 3, we first show that two class of maps on nest algebras satisfying some conditions are centralizers, which are the generalizations of the well-known results of prime rings or semiprime rings. Taking into account that nest algebras are not prime and semiprime, these generalizations are not trivial. Further, we describe the forms of left (right) centralizers on nest algebras.It turns out that a norm continuous left (right) centralizer on a nest algebra has the form: A→TA( A→AT), and a centralizer has the form: A→λA. Here, T is a fixed operator in the nest algebra andλis a fixed scalar.