Norms of generalized derivations on norm ideals

Abstract

The norms of inner and generalized derivations on different kinds of algebras have been determined. However, the norms of their restrictions to norm ideals have not been fully explored. For instance, the concept of S−universality having been introduced by Fialkow in 1979, has not been fully characterized yet it plays a critical role in the study of norms of derivations. In this study, we have investigated both the algebraic and the norm properties of a generalized derivation. Specifically, we have determined the norm of generalized derivation on a norm ideal, extended the concept of S−universality to the setting of a generalized derivation, and established the necessary conditions for the attainment of the optimal value of the circumdiameters of numerical ranges and spectra of two bounded linear operators in a Hilbert space. It turns out that for a pair of S−universal, normaloid or spectraloid operators, the circumdiameter of the numerical ranges or the spectra is the sum of the numerical radii or the spectral radii respectively. We have characterized the antidistance from an operator to its similarity orbit in terms of the circumdiameter, norms, numerical and spectral radii. Based on the definition in the context of inner derivations we have extended the concept of S−universality to the generalized derivation. Using the relations between the norm of an inner derivation and the diameter of the numerical range as well as spectral inclusion theorem, we have established various relations between the norm of a generalized derivation and the circumdiameters of the numerical ranges and spectra. We hope that the results obtained in this study has greatly contributed to the field of derivations and provided motivations for further research to pure mathematicians in this area of study