Perturbations of Operator Functions: A Survey

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 $$\tilde A$$ be bounded linear operators in a Banach space X $$\mathcal {X}$$ and f(.) be a function analytic on neighborhoods of spectra of A and à $$\tilde A$$ . The chapter is devoted to norm estimates for Δ A = f ( A ) − f ( à ) $$\varDelta A=f(A)-f(\tilde A)$$ 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 $$A-\tilde A$$ is a Hilbert–Schmidt operator and the function is regular on the convex hull of the spectra A and à $$\tilde A$$ . 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.