Range Kernel Orthogonality and Finite Operators

Abstract

Let H be a separable infinite dimensional complex Hilbert space, and let denote the algebra of all bounded linear operators on H into itself. Let we define the generalized derivation by , we note . If the inequality holds for all and for all , then we say that the range of is orthogonal to the kernel of in the sense of Birkhoff. The operator is said to be finite [22] if for all , where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.