On The Algebraic Numerical Range of the Basic Elementary Operator and Norm of a Generalized Derivation

Abstract

An elementary operator is a bounded linear map defined on the set of bounded linear operators acting on an infinite dimensional complex Hilbert space H. Various forms of elementary operators have been studied in the past including generalized elementary operator, left and right operators, inner derivation, generalized derivationandbasicelementaryoperatorsamongothers. Variousaspectssuchasspectra, compactness, norm properties, numerical ranges among others have been used to study the properties of these operators in the recent past and good results have been obtained. Over time, the relationship between the various generalizations of numericalrangehavebeeninvestigatedanditemergesthatforthebasicelementary operator, an exact description of the numerical range has not been exhaustively explored as the operator acts on various algebras. In this study, we have investigated boththealgebraicnumericalrangeofthebasicelementaryoperatorandthenormof a generalized derivation on the operator algebra B(H). Specifically, by application of set inclusion approach and convexity properties of sets, it has been shown that the algebraic numerical range of the basic elementary operator is convex, contains the closure of its classical numerical range and that the algebraic numerical range of the basic elementary operator is equal to the closure of the classical numerical range of the implementing operators. Finally, we have shown that the norm of a generalized derivation is equal to the sum of the norms of implementing operators if the operators are finite rank. These results are of great importance to quantum physics, for solving force closure in robotic grasping and provision of basis of solution to optimization and duality problems.