Some inequalities for generalized commutators
In this theses the author obtained some inequalities for generalized commutators. The main part is the following. Let A, B be two (unbounded) linear operators on a Banach space \({\mathbb{X}}\) such that the spectra sp(A), sp(B) are subsets of \({\mathbb{C}}\setminus (- \infty,0)\), and that \(\| t(A+t)^{-1}\| \leq M\), \(\| t(B+t)^{- 1}\| \leq N\) for all \(t>0\). Then for a monotonic function f of the form \[ f(z)=kz+\int^{\infty}_{0}\frac{z}{z+t}d\mu (t),\quad k\geq 0,\quad f(0)=\lim_{s\to 0^+}f(s)=0,\quad \int^{\infty}_{1}d\mu (t)/t<\infty,\quad \mu (0)=0, \] \[ \| f(A)X-Xf(B)\| \leq af(b\| AX-XB\|) \] holds, where \(X\in B({\mathbb{X}})\), \(\| X\| <1\); \(a=2(M+N)\), \(b=MN/M+N\), MN\(\geq 1/2\) or \(k=0.\) If \({\mathbb{X}}\) is a Hilbert space, \(\| \cdot \|_ p\) is the Schatten norm (1\(\leq p\leq \infty)\), then the inequality \[ \| f(A)X- Xf(B)\|_ p\leq af(b\| AX-XB\|_ p) \] holds.
inequalities for generalized commutators
scientific article; zbMATH DE number 4208892