Geodesic
Estimates for the difference of the fractional powers of self-adjoint operators under unbounded perturbations
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 18, Tome 178 (1989), pp. 120-145 
M. Sh. Birman; M. Z. Solomyak. Estimates for the difference of the fractional powers of self-adjoint operators under unbounded perturbations. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 18, Tome 178 (1989), pp. 120-145. http://geodesic.mathdoc.fr/item/ZNSL_1989_178_a4/
@article{ZNSL_1989_178_a4,
     author = {M. Sh. Birman and M. Z. Solomyak},
     title = {Estimates for the difference of the fractional powers of self-adjoint operators under unbounded perturbations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {120--145},
     year = {1989},
     volume = {178},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_178_a4/}
}
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For positive self-adjoint operators $A_0$, $A_1$ on Hilbert spaces $\mathcal{H}_0$, $\mathcal{H}_1$ and for an operator $\mathcal{J}: \mathcal{H}_0\to\mathcal{H}_1$, the following estimate is obtained: $$ |\alpha^{-1}(A_1^\alpha\mathcal{J}-\mathcal{J}A_0^\alpha)|_\gamma\leqslant A_1^{-\delta}(A_1\mathcal{J}-\mathcal{J}A_0)A_0^{-\delta},\quad 2\delta=1-\alpha,\quad-1<\alpha<1. $$ Here $|\cdot|_\gamma$ denotes the norm in some symmetric-normed operator ideal $\gamma$. Some generalizations of this estimate are presented too. Applications to the differential operators are discussed.