Monatshefte für Mathematik https://doi.org/10.1007/s00605-018-1257-9 Drazin inverse of multivalued operators and its applications Ayoub Ghorbel 1 · Maher Mnif 1 Received: 3 April 2018 / Accepted: 17 December 2018 © Springer-Verlag GmbH Austria, part of Springer Nature 2019 Abstract In this paper, the notion of Drazin invertibility in the case of multivalued operators is introduced. Many results from operator theory are covered. Applications of some obtained results allow to study the Drazin invertibility of a multivalued operator matrix M C :=  A C 0 B  acting in the product of Banach or Hilbert spaces X × Y . Keywords Drazin invertible multivalued operators · Left Drazin invertible multivalued operators · Right Drazin invertible multivalued operators · Upper triangular multivalued operator matrices Mathematics Subject Classification 47A06 · 47A53 1 Introduction Let L ( X , Y ) denotes the Banach algebra of all bounded operators acting between Banach spaces X and Y . Drazin invertible operators were introduced and investigated in the case of the Banach algebra L ( X ) by several authors [1,8,22,23] among others. A bounded operator T ∈ L ( X ) is said to be Drazin invertible, if there exists an operator T D ∈ L ( X ), called the Drazin inverse of T , such that TT D = T D T , T D TT D = T D and T k +1 T D = T k . Communicated by G. Teschl. B Maher Mnif maher.mnif@gmail.com Ayoub Ghorbel ghorbel.agr@gmail.com 1 Département de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, B.P. 1171, 3000 Sfax, Tunisia 123