Directional operator differentiability of non-smooth functions

Research output: Contribution to journalReview articlepeer-review


We obtain (very close) sufficient conditions and necessary conditions on the spectral measure of a self-adjoint operator A, under which any continuous function Φ (without any additional smoothness properties) has a directional operator-derivative Φ′(A)(B):=∂/∂γ Φ(A + γB)|γ=0 in the direction of a quite general bounded, self-adjoint operator B. Our sharpest results are in the case where B is a rank-one operator. We pay particular attention to the case where the spectral measure of A is absolutely continuous, and its additional smoothness properties compensate the lack of smoothness of the function Φ.

Original languageEnglish
Pages (from-to)49-90
Number of pages42
JournalJournal of Operator Theory
Issue number1
StatePublished - Dec 2006


  • Borel transform
  • Directional operator differentiability
  • Functional calculus
  • Hankel operators
  • Rank-one perturbations
  • Riesz projections

ASJC Scopus subject areas

  • Algebra and Number Theory


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