Sesquitransitive and localizing operator algebras on Banach spaces

In this study, the conditions of the transitive and localizing operator algebras on Banach spaces are investigated. By the minimal vectors method, which is defined by Vladimir G. Troitsky, the connection between being a localizing operator algebra of the commutant set of the operator T, and having a hyperinvariant subspace of T is searched for. Also required situations for the adjoint operator of T are dealed. In addition to this, for the topological dual space Y of a Banach space X, some circumstances of a subalgebra A of B(X), that contains a compact operator and its dual algebra, which is a subalgebra of B(Y) and to satisfy to be dense according to the weak operator topology are investigated. Also, similar conditions are investigated for the algebra A which contains a weakly compact operator.