Norms Of Generalized Derivations On Norm Ideals

Abstract

The norms of inner and generalized derivations on dierent kinds of al-

gebras 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. Specically, we have de-

termined 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 opti-

mal 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 denition in the con-

text 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 spec-

tral 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 eld of derivations and provided motivations

for further research to pure mathematicians in this area of study.