Central Power Values of Generalized Derivation and Structure of Unital Banach Algebra

The structure of a prime Banach algebra \(\mathfrak {A}\) containing the identity element e with the help of a nonzero continuous linear generalized derivation \(\mathfrak {D}\) satisfying one of the following conditions, is investigated in this paper : (i) \(x\Big (\mathfrak {D}((ab)^m)\pm a^mb^m\Big )\in \mathcal {Z}_\mathfrak {A}\) ; (ii) \(x\Big (\mathfrak {D}((ab)^m)\pm b^ma^m\Big )\in \mathcal {Z}_\mathfrak {A}\) ; (iii) \(x\Big (\mathfrak {D}((ab)^m)\pm \mathfrak {D}(a^mb^m)\Big )\in \mathcal {Z}_\mathfrak {A}\) ; (iv) \(x\Big (\mathfrak {D}((ab)^m)\pm \mathfrak {D}(b^ma^m)\Big )\in \mathcal {Z}_\mathfrak {A}\) ; (v) \(x\Big (\mathfrak {D}((ab)^m)\pm [a^m, b^m]\Big )\in \mathcal {Z}_\mathfrak {A}\) ; (vi) \(x\Big (\mathfrak {D}((ab)^m)\pm [b^m, a^m]\Big )\in \mathcal {Z}_\mathfrak {A}\) ; (vii) \(x\Big (\mathfrak {D}((ab)^m)\pm (ab)^m\Big )\in \mathcal {Z}_\mathfrak {A}\) ; (viii) \(x\Big (\mathfrak {D}((ab)^m)\Big )\in \mathcal {Z}_\mathfrak {A}\) for all a, b in some suitable non empty subset of \(\mathfrak {A}\) , where m is a positive integer such that \(m=m(a, b)>1\) . Moreover, we give a characterization of the map \(\mathfrak {D}\) . Finally it is shown that our theorems do not hold in case \(\mathfrak {A}\) is not prime.