Double Derivations, Higher Double Derivations

Let be a banach algebra. let be linear mappings on . first we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple banach algebras, in certain case. afterwards we define a new vocabulary called “ -higher double derivation” and present a relation between this subject and derivations and finally give some theorems and corollaries about the automatic continuity of this notion.