# Journal of Operator Theory

Volume 73, Issue 2, Spring 2015  pp. 361-384.

Weak type estimates for the absolute value mapping

Authors:  M. Caspers (1) D. Potapov (2) F. Sukochev (3) D. Zanin (4)
Author institution:(1) Fachbereich Mathematik und Informatik der Universitaet Muenster, Einsteinstrasse 62, 48149 Muenster, Germany
(2) School of Mathematics and Statistics, UNSW, Kensington 2052, NSW, Australia
(3) School of Mathematics and Statistics, UNSW, Kensington 2052, NSW, Australia
(4) School of Mathematics and Statistics, UNSW, Kensington 2052, NSW, Australia

Summary: 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 $\mathcal{L}_{1,\infty}$ in the algebra $\mathcal{L}(H)$ of all bounded operators on an infinite-dimensional Hilbert space generated by an operator whose sequence of eigenvalues is $\{1, \frac12,\frac13,\dots\}$. Moreover, $\mu(j;|A| -|B|)\leqslant \mathrm{const}(1 + j )^{-1}\|A-B\|_1$. We also obtain a semifinite version of this result, as well as the corresponding commutator estimates.

DOI: http://dx.doi.org/10.7900/jot.2013dec20.2021
Keywords: operator ideals, commutator estimates, Lipschitz estimates, Schatten classes, weak type inequalities, absolute value mapping