Abstract
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 language | English |
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Pages (from-to) | 49-90 |
Number of pages | 42 |
Journal | Journal of Operator Theory |
Volume | 55 |
Issue number | 1 |
State | Published - Dec 2006 |
Keywords
- Borel transform
- Directional operator differentiability
- Functional calculus
- Hankel operators
- Rank-one perturbations
- Riesz projections
ASJC Scopus subject areas
- Algebra and Number Theory