Additive Maps Preserving The Nullity Or The Deficiency Of Operators

The nullity and deficiency of linear operators defined by kernel and range are important functions, so the research about maps preserving the nullity or the defi-ciency becomes an interesting problem. Let B(X) and B(Y) be the Banach algebras of all bounded linear operators on infinite dimensional complex Banach spaces X and Y, respectively. Let a(T) be the nullity of T, and let β(T) be the deficiency of T. Noticing that, let A,B∈B(X), form α(A)=α(B)(orβ(A)=β(B)), we can define an equivalence relation of B(X). In this paper, we discuss the character-izations of maps who preserve such a relationship, that is the characterizations of additive surjective maps preserving the nullity or the deficiency in both directions.For the study of additive surjective map Φ from B(X) to B(Y) preserving the nullity in both directions, firstly, we prove that such a map Φ can also preserve the rank one operators in both directions. Secondly, by the proof of invertibility of Φ(I), we define another additive surjective map Ψ, which preserves the nullity and unit in both directions, and Ψ must be isomorphisms or anti-isomorphisms on the operator spaces. Finally, we obtain Φ must have the following form:Φ(T)=UTV,(?)T∈B(X), where U:Xâ†'Y and V:Yâ†'X are invertible bounded linear or conjugate linear operators.For the study of additive surjective mapΦ from B(X) to B(Y) preserving the deficiency in both directions, we obtain that such a map Φ must have the following form:there exists invertible bounded linear or conjugate linear operators U:Xâ†'Y and V:Yâ†'X, such that Φ(T)=UTV for all T∈B(X). Or there exists invertible bounded linear or conjugate linear operators U’:X*â†'Y and V:Yâ†'X*such that Φ(T)=U’T*V’,VT E B(X). And in this case, X and Y are reflexive.