Operator integration, perturbations and commutators
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 34-66
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Under mild conditions integral representations of the following kind are justified: $$ f(A_1)\cdot J-J\cdot f(A_0)=\iint\frac{f(\mu)-f(\lambda)}{\mu-\lambda}d \,E_1(\mu)(A_1J-JA_0)d \,E_0(\mu). \qquad{(*)} $$ Here $A_k$, $k=0,1$, is a self-adjoint operator on a Hilbert space $\mathcal{H}_k$, $J$ is an operator acting from $\mathcal{H}_0$ to $\mathcal{H}_1$; all operators are, in general, unbounded; $E_k$ is the spectral measure for $A_k$. On the basis of the representation ($*$) estimates of s-numbers of the operator $f(A_1)\cdot J-J\cdot f(A_0)$ in terms of the $s$-numbers of $A_1J-JA_0$ are given. Analogous results are obtained for commutators and anticommutators.
@article{ZNSL_1989_170_a2,
author = {M. S. Birman and M. Z. Solomjak},
title = {Operator integration, perturbations and commutators},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {34--66},
publisher = {mathdoc},
volume = {170},
year = {1989},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_170_a2/}
}
M. S. Birman; M. Z. Solomjak. Operator integration, perturbations and commutators. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 34-66. http://geodesic.mathdoc.fr/item/ZNSL_1989_170_a2/