Some inequalities for generalized commutators
type: Item
instance of: scholarly article
title: Some inequalities for generalized commutators
zbMATH Open document ID: 0731.47036
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 4208892
zbMATH Keywords: operator inequality
zbMATH Keywords: inequalities for generalized commutators
zbMATH Keywords: spectra
zbMATH Keywords: monotonic function
zbMATH Keywords: Schatten norm
author: Khristo N. Boyadzhiev
reviewed by: Rho Jae-Chul
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195170960
OpenAlex ID: W2016003964
cites work: Notes on extensions of \(C^*\)-algebras
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DOI: 10.2977/PRIMS/1195170960
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description: scientific article; zbMATH DE number 4208892