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Two variables functionals and inequalities related to measurable operators.

Jingjing Shao1

  • 1College of Mathematics and Statistics Science, Ludong University, Yantai, 264000 China.

Journal of Inequalities and Applications
|December 26, 2017
PubMed
Summary

This study introduces two-variable norm functionals for τ-measurable operators, proving their joint log-convexity. This finding enables new interpolated inequalities, including Young, Heinz, and Trace inequalities, and explores their monotonicity.

Keywords:
Heinz inequalityYoung inequalityinterpolated inequalityvon Neumann algebraτ-measurable operators

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Area of Science:

  • Functional Analysis
  • Operator Theory
  • Convexity Theory

Background:

  • τ-measurable operators are a key class of operators in functional analysis.
  • Log-convexity is a crucial property with significant implications in inequalities and analysis.
  • Existing inequalities often lack extensions to more general operator classes.

Purpose of the Study:

  • To introduce and analyze two-variable norm functionals for τ-measurable operators.
  • To establish the joint log-convexity property of these newly defined functionals.
  • To demonstrate the utility of this log-convexity through applications to important inequalities.

Main Methods:

  • Definition of two-variable norm functionals tailored for τ-measurable operators.
  • Application of convexity and functional analysis techniques to prove joint log-convexity.
  • Derivation of interpolated inequalities using the established log-convexity property.

Main Results:

  • Successful introduction of novel two-variable norm functionals for τ-measurable operators.
  • Rigorous proof of the joint log-convexity of these functionals.
  • Establishment of interpolated Young, Heinz, and Trace inequalities for τ-measurable operators.

Conclusions:

  • The joint log-convexity of the introduced functionals provides a powerful tool for deriving operator inequalities.
  • The findings extend existing results to the broader class of τ-measurable operators.
  • The study opens avenues for further research into the monotonicity and applications of these interpolated inequalities.