S.I.: ICWAA-2018 Perturbations of discrete spectra of holomorphic operator- valued functions Rafikul Alam 1 • Jibrail Ali 1 Received: 26 April 2019 / Accepted: 2 November 2019 Ó Forum D’Analystes, Chennai 2019 Abstract Let X be a complex Banach space and L(X) be the Banach space of all bounded linear operators on X. Let X  C be open and connected. Let T ; V : X ! LðXÞ be holomorphic operator-valued functions. We consider the one parameter family of operator-valued functions Wðk; tÞ :¼ T ðkÞþ tV ðkÞ, for t 2 C, and analyze evolu- tion of the discrete eigenvalues of Wðk; tÞ when t varies in C: We provide a brief review of the discrete spectrum of T ðkÞ and present several equivalent characteri- zations for discrete eigenvalues of T ðkÞ: We also prove Rouche’s theorem for operator-valued functions under a weaker assumption, which we utilize to derive perturbation bounds for the discrete eigenvalues of Wðk; tÞ when |t| is small. Keywords Banach space  Fredholm operator  Operator-valued function  Spectrum  Discrete spectrum  Discrete eigenvalues  Linearization Mathematics Subject Classifications 47A75  47A55  47A10  47A53 1 Introduction Let X be a complex Banach space and L(X) be the Banach space of all bounded linear operators on X. Let X  C be open and connected. Let T : X ! LðXÞ be holomorphic and regular, that is, T ðkÞ is invertible for some k 2 X: The nonlinear eigenvalue problem is to solve T ðlÞv ¼ 0 for l 2 X and a nonzero vector v 2 X: Dedicated to Professor S. H. Kulkarni on the occasion of his 65th birthday. & Rafikul Alam rafik@iitg.ernet.in; rafikul68@gmail.com Jibrail Ali jibrail@iitg.ernet.in 1 Department of Mathematics, IIT Guwahati, Guwahati 781039, India 123 The Journal of Analysis https://doi.org/10.1007/s41478-019-00212-1