Difference of Gaussian measures (Q921473)

[For the entire collection of the original see Zbl 0626.00025.] Let H be a Hilbert space with the scalar product \(<, >\) and the norm \(| \cdot | =<\cdot,\cdot >^{1/2}\). Denote by \({\mathcal S}\) the set of all non-negative selfadjoint trace-class operators in H and by \(\| \cdot \|\) the operator norm in \({\mathcal S}\). Given \(V\in {\mathcal S}\), write \({\mathcal N}_ V\) for the mean zero normal measure on H with the covariance operator V. We quote the two first theorems proved in the paper: Theorem 1. Consider U,V\(\in {\mathcal S}\). There exist constants \(C_ k\), \(k=1,2,3,4\), depending only of U, V such that for any ball B with center at a point \(a\in H\) we have \[ | {\mathcal N}_ U(B)-{\mathcal N}_ V(B)| \leq C_ 1 Tr| U-V| +\| U-V\| | a|^ 2{\mathfrak g}, \] where \(m=\min \{<Ua,a>,<Va,a>\}\) and \({\mathfrak g}=C_ 2 \exp (-C_{3^ m})+\pi^{-1} \min \{m^{-1},C_ 4\}\). If U and V commute, then \(\| V-U\| | a|^ 2{\mathfrak g}\) can be substituted by \(<U-V| a,a>{\mathfrak g}.\) Theorem 2. There exists a constant C depending only on U and V such that if \({\mathfrak d}=\| U^{-1/2}VU^{1/2}-I\| <\), then for any ball B we have \(| {\mathcal N}_ U(B)-{\mathcal N}_ V(B)| \leq C{\mathfrak d}.\) The paper includes more results of a similar nature.