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arXiv · 1309.3378

Weak type estimates for the absolute value mapping

M. Caspers·D. Potapov·F. Sukochev·D. Zanin
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Abstract

We prove that if A and B are bounded self-adjoint operators such that A-B belongs to the trace class, then |A| -|B| belongs to the principal ideal L_{1,\infty} in the algebra L(H) of all bounded operators on an infinite-dimensional Hilbert space generated by an operator whose sequence of eigenvalues is {1, 1/2, 1/3, 1/4, ...}. Moreover, μ(j;|A| -|B|)\leq const(1 + j)^{-1}\|A-B\|_1. We also obtain a semifinite version of this result, as well as the corresponding commutator estimates.

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M. Caspers, D. Potapov, F. Sukochev, D. Zanin. 2014-06-18. Weak type estimates for the absolute value mapping. https://arxiv.org/abs/1309.3378

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