Acta Mathematica Sinica, English Series
Sep., 2017, Vol. 33, No. 9, pp. 1225–1241
Published online: June 28, 2017
Acta Mathematica Sinica,
Springer-Verlag Berlin Heidelberg &
The Editorial Office of AMS 2017
Additive Preservers of Drazin Invertible Operators
with Bounded Index
Mourad OUDGHIRI Khalid SOUILAH
D´epartement Math-Info, Labo LAGA, Facult´e des Sciences, Universit´e Mohammed Premier,
60000 Oujda, Maroc
E-mail : email@example.com firstname.lastname@example.org
Abstract Let B(X) be the algebra of all bounded linear operators on an inﬁnite-dimensional complex
or real Banach space X. Given an integer n ≥ 1, we show that an additive surjective map Φ on B(X)
preserves Drazin invertible operators of index non-greater than n in both directions if and only if Φ is
either of the form Φ(T )=αAT A
or of the form Φ(T )=αBT
where α is a non-zero scalar,
A : X → X and B : X
→ X are two bounded invertible linear or conjugate linear operators.
Keywords Linear preserver problems, Drazin inverse, ascent, descent
MR(2010) Subject Classiﬁcation 47B49, 47L99, 47A55, 47B37
Throughout this paper, X denotes an inﬁnite-dimensional Banach space over K = R or C.Let
B(X) be the algebra of all bounded linear operators acting on X.
An operator T ∈B(X) is called Drazin invertible if there exist S ∈B(X) and a non-negative
integer k such that
ST = T
,STS= T and TS = ST. (1.1)
Such operator S is unique, it is called the Drazin inverse of T and denoted by T
index of T ,designatedbyi(T ), is the smallest non-negative integer k satisfying (1.1). The
concept of Drazin inverse was introduced in  and it has numerous applications in matrix
theory, iterative methods, singular diﬀerential equations, and Markov chains, see for instance
[1, 2, 4, 15] and the references therein.
Let Λ be a subset of B(X). An additive map Φ : B(X) →B(X)issaidtopreserve Λ in
both directions if for every T ∈B(X),
T ∈ Λ if and only if Φ(T ) ∈ Λ.
In the last decades, there has been a remarkable interest in the so-called linear preserver
problems which concern the question of characterizing linear, or additive, maps on Banach
algebras that leave invariant a certain subset. For excellent expositions on linear preserver
problems, the reader is referred to [7, 8, 12, 13, 18–20] and the references therein.
Received November 24, 2016, accepted February 13, 2017