Translation-invariant maps and applications
A general framework for deriving upper bounds for some quantities induced by translation-invariant maps is given. Vectorial intervals induced by two preorders on a linear space and radii of families of maps at a point are used. Section 2 introduces the relevant notation and terminology and it includes the main general result, Theorem 2.2. At the end of Section 2 the study is confined to GIC preorders; an extension of a result given by \textit{C.-K. Li} and \textit{R. Mathias} [J. Inequal. Appl. 3, 137--142 (1999; Zbl 0958.15017)] is proved. Section 3 is devoted to bounds for norms of the difference of two operators. In particular, norms of commutators are considered. For instance, recent results given by \textit{Y.-Q. Wang} and \textit{H.-K. Du} [J. Math. Anal. Appl. 342, 747--751 (2008; Zbl 1139.47010)] and \textit{R. Bhatia} and \textit{F. Kittaneh} [Oper. Matrices 2, 143--151 (2008; Zbl 1147.15019)] are corollaries of general results in this section. The last section deals with the Grüss inequality. Again, a general theorem is proved which has some known results as corollaries. The paper is well written. It seems that the study of general results and their applications may give some results that are not included in the paper.
scientific article; zbMATH DE number 5553657
scientific article