Abstract:
Let B(X) denote the Banach algebra of all bounded linear operator, on the infinite dimensional Banach space X. We say that infinite dimensional Banach spaces X and Form a dual system, denoted (X,Y), if there is defined on X x Y a no degenerate bounded bilinear form <•,•>. Let T є B(X), if there exists T* єB(Y) such that = for all x є X and y є Y, then Т* is called a conjugate operator to T relative to the dual system (X,Y). Let (X, Y) be the algebra of all T є B(X) that have, with respect to the dual system (X,Y), a conjugate T* є B(Y). It is a Banach algebra with respect to the norm given by ||T|| = max {||T||, ||T*||}. The Freedom operator in A(X,Y) is defined as one which is invertible modulo the ideal of finite rank operators in A(X, Y). The aim of this thesis is to study the connection between some properties of operators T € A(X,Y) and their conjugates T* and to study how the classical theory of Freedom operators in B(X) can be generalized to the setting of A(X, Y). We give a survey of known results, we supply the details of the proofs for almost all these results, we prove the results that appeared without proofs and we also prove some well known results in more general setting to Freedom operators in Ad(X,Y), Moreover, we add some results that seem to be new to the best of our knowledge. Among these results are the following: Let T be an operator in A(X, Y). We show that if the dimension of the null space of T is finite or the co dimension of the image of T* is finite, then R(T*) = N(T)1 and β(T*} = a(T) are equivalent. We prove that if the co dimension of the image of T is finite, then the dimension of the null space of the conjugate operator T is also finite and in fact, is less or equal to it.
