Journal of Operator Theory

Volume 55, Issue 1, Winter 2006 pp. 49-90.

Directional operator differentiability of non-smooth functions

Authors: Jonathan Arazy (1) and Leonid Zelenko (2)
Author institution: (1) Department of Mathematics, University of Haifa, Haifa, 31905, Israel
(2) Department of Mathematics, University of Haifa, Haifa, 31905, Israel

Summary: We obtain (very close) sufficient conditions and necessary conditions on the spectral measure of a self-adjoint operator $A$, under which any continuous function $\phi$ (without any additional smoothness properties) has a directional operator-derivative $$ \phi^{\prime}(A)(B):= \frac{\partial}{\partial \gamma} \; \phi(A+\gamma B)_{|\gamma = 0} $$ in the direction of a quite general bounded, self-adjoint operator $B$. Our sharp-est 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 $\phi$.

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