Functions of noncommuting selfadjoint operators under perturbation and estimates of triple operator integrals
Abstract
We define functions of noncommuting selfadjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function $f$ on ${\Bbb R}^2$, for which the map $(A,B)\mapsto f(A,B)$ is Lipschitz in the operator norm and in Schattenvon Neumann norms $\boldsymbol{S}_p$. It turns out that for functions $f$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$, the above map is Lipschitz in the $\boldsymbol{S}_p$ norm for $p\in[1,2]$. However, it is not Lipschitz in the operator norm, nor in the $\boldsymbol{S}_p$ norm for $p>2$. The main tool is triple operator integrals. To obtain the results, we introduce new Haageruplike tensor products of $L^\infty$ spaces and obtain Schattenvon Neumann norm estimates of triple operator integrals. We also obtain similar results for functions of noncommuting unitary operators.
 Publication:

arXiv eprints
 Pub Date:
 May 2015
 DOI:
 10.48550/arXiv.1505.07173
 arXiv:
 arXiv:1505.07173
 Bibcode:
 2015arXiv150507173A
 Keywords:

 Mathematics  Functional Analysis
 EPrint:
 43 pages