CONVEXITY AND CONCAVITY IN SUBMAJORIZATION INEQUALITIES FOR τ-MEASURABLE OPERATORS

τ–өлшемдi операторлар үшiн субмажорланған теңсiздiктердегi ойыс және дөңес функциялар
Выпуклость и вогнутость в субмажоризационных неравенствах для τ–измеримых операторов
Dauitbek D.
25 December 2025 al-Farabi Kazakh State National University

KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2025 #128 Issue 4

We proved the following result. Consider a semi-finite von Neumann algebra equipped with a trace, and let there be several τ-measurable operators together with a nonnegative function defined on the nonnegative real line. Suppose also that we are given positive weights whose total sum equals one. If the function obtained by applying f to the square root of its argument is convex and if f vanishes at zero, then the weighted sum of the values of f applied to the absolute values of the operators is at least as large as a certain expression involving f evaluated at both the average of all the operators and their pairwise differences. If the same function of the square root is concave, the inequality reverses: the mentioned expression becomes no smaller than the weighted sum of the transformed absolute values. This theorem yields a significant generalization of Clarkson-type inequalities in the noncommutative setting and extends the result previously established by Alrimawi, Hirzallah, and Kittaneh.

Clarkson inequality , concave function , convex function , generalized singular value function , submajorization inequality , von Neumann algebra , τ −measurable operator

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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

Institute of Mathematics and Mathematical Modeling

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