Additive Maps Preserving Quadratic Operators

Let X be an infinite dimensional complex Banach space with infinite multiplic-ity or a finite dimensional Banach space with dim X ? 3 and B(X)be the Banach algebra of all bounded linear operators on X.Let T E B(X).If there are ?1,?2? Csuch that T2 + ?1T + ?2I = 0,then T is a quadratic operator.Clearly,we known that square-zero operators and idempotent operators are special quadratic operators from the spectrum of quadratic operators.Specially,several scholars have researched the characterization of linear maps or additive maps preserving square-zero oper-ators and idempotent operators in both directions.Afterwards,Mourad Oudghiri and Khalid Souilah obtained the characterization of surjective linear maps on the algebra B(H)of all bounded linear operators on a complex Hilbert space H that p-reserve quadratic operators in both directions.We note that an infinite dimensional Hilbert space is of infinite multiplicity.Motivated by this point,this paper takes quadratic operators as the research object and focus on the characterization of sur-jective additive maps on B(X)that preserve quadratic operators in both directions.We have the following results in this paper:In the first part,we study the elementary properties for quadratic operators.we first prove that identity operator is not the sum of three square-zero operators on B(X).Then the block operator matrix technique is used to obtain the essential features of square-zero operators.In the second part,we characterize the structure of surjective additive maps on B(X)that preserve quadratic operators in both directions.First of all,based on the essential features of square-zero operators,we study the structure of surjective additive maps on Banach algebra of all bounded linear operators on a complex Banach spaces with infinite multiplicity that preserve quadratic operators in both directions.Finally,an analogue result in the setting of finite-dimensional Banach spaces with dim X ? 3 is given.