Some strong limit theorems for weighted sums of measurable operators

The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.

Further generalized refinement of Young’s inequalities for τ -mesurable operators

In this paper, we prove that if a, b > 0 and 0 ≤ v ≤ 1. Then for all positive integer m(1) - For v ∈ v \in \left[ {0,{1 \over {{2^n}}}} \right], we have {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{v^m}{{\left( {\sqrt {{b^m}} - \root {{2^k}} \of {\left( {a{b^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m}.}(2) - For v ∈ v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right], we have {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{{\left( {1 - v} \right)}^m}{{\left( {\sqrt {{a^m}} - \root {{2^k}} \of {\left( {b{a^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m},} we also prove two similar inequalities for the cases v ∈ v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right] and v ∈ v \in \left[ {{1 \over 2},{{{2^n} + 1} \over {{2^n}}}} \right]. These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and p-norms of positive τ-measurable operators.

A new generalization of refined young inequalities for τ-measurable operators

In this paper, by the arithmetic-geometric mean inequality, we give a new generalization of refined Young?s inequality. As applications we present some new generalizations of refinements of Young inequalities for the determinants, traces and p-norms of ?-measurable operators.