Perturbations of Operator Functions: A Survey

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


The chapter is a survey of the recent results of the author on the perturbations of operator-valued functions. A part of the results presented in this chapter is new. Let A and à be bounded linear operators in a Banach space X and f(.) be a function analytic on neighborhoods of spectra of A and Ã. The chapter is devoted to norm estimates for ΔA= f(A) − f(Ã) under various assumptions on functions and operators. In particular, we consider perturbations of entire operator-valued functions and Taylor series whose arguments are bounded operators in a Banach space. In the case of the separable Hilbert space, we derive a sharp perturbation bound for the Hilbert–Schmidt norm of Δf, provided A− à is a Hilbert–Schmidt operator and the function is regular on the convex hull of the spectra A and Ã. In addition, operator functions in a Hilbert lattice are explored. Besides, two-sided estimates for f(A) are established. These estimates enable us to obtain positivity conditions for functions of a given operator and of the perturbed one. As examples of concrete functions, we consider the operator fractional powers and operator logarithm. Moreover, applications of our results to infinite matrices and integral operators are discussed.

Original languageEnglish
Title of host publicationSpringer Optimization and Its Applications
Number of pages46
StatePublished - 1 Jan 2021

Publication series

NameSpringer Optimization and Its Applications
ISSN (Print)1931-6828
ISSN (Electronic)1931-6836


  • Hilbert lattice
  • Infinite matrices
  • Integral operators
  • Operator logarithm
  • Operator-valued functions
  • Perturbations
  • Positivity

ASJC Scopus subject areas

  • Control and Optimization


Dive into the research topics of 'Perturbations of Operator Functions: A Survey'. Together they form a unique fingerprint.

Cite this