When do quasinilpotents lie in the Jacobson radical?

Abstract

In this paper, we give some spectral characterizations of the Jacobson radical; that is, we will show thatsome conditions with -multiplicativity imply that the set of all quasinilpotent elements equals the Jacobson radical.We also give some conditions to make sure the quasinilpotents lie in the Jacobson radical, using the set of elements withsingleton spectra.