On the range and the kernel of elementary operators

Abstract

The first chapter is essentially a survey and synthesis of what is known about the properties of P-Symmetric operators and Finite operators. In the second chapter, we establish the orthogonality of the range and the kernel of a derivation A induced by a cyclic subnormal operator A, in the usual operator norm. We provide another proof of a principal result of F.Wening and J.Guo Xing. We give a characterization of the class of PSymmetric operators. We characterize also operators A such that the pair (A,A) satisfy the Putnam-Fuglede property in Cp(H), where Cp(H) denotes the Von Newmann-Schatten class for p > 1. In the third chapter, we wish to consider the class of Finite operators. We use new techniques and approachs to generalize and develop some properties of Finite operators. In the following chapter, we give some properties concerning the class of PSymmetric operators. We turn our attention to commutant and derivation ranges. We obtain the new results concerning the intersection of the kernel and the closure of the range of an inner derivation. We obtain new classes of operators A such that I 62 R( A), where R( A) is the norm closure of the range of A, ( A(X) = AX − XA). The last chapter represents some properties which enjoy the range of an elementary operator. We initiate the study of the class of Quasi-adjoint operators, i.e. operators A for which R( A) = R( A ), where R( A) denotes the norm closure of the range of the elementary operator A(X) = AXA −X. We give a characterization and some basic properties concerning this class of operators.