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›› 2014, Vol. 31 ›› Issue (5): 590-594.DOI: 10.7523/j.issn.2095-6134.2014.05.003

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Additive maps preserving the nullity or deficiency of operators

LI Chenxin, JI Guoxing   

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China
  • Received:2013-06-21 Revised:2013-09-09 Online:2014-09-15

Abstract:

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. It is proved that an additive surjective map φ from B(X) to B(Y) preserves the nullity in both directions if and only if there exists invertible bounded linear or conjugate linear operators U:XY and V:YX, such that φ(T)=UTV for all TB(X). A form for additive surjective maps preserving the deficiency in both directions is also obtained.

Key words: nullity, deficiency, additive maps

CLC Number: