# Subadditive inequalities for operators

## TL;DR

This paper explores new subadditivity inequalities for convex and concave functions applied to operators in Hilbert spaces, generalizing previous results and extending the subadditivity concept through various operator inequalities.

## Contribution

It introduces novel subadditivity inequalities for operators, broadening the scope of earlier results and extending the subadditivity framework in operator theory.

## Key findings

- Established inequalities for operator powers with spectral assumptions
- Generalized previous subadditivity results by Aujla and Silva
- Extended subadditivity concepts inspired by Ando, Zhan, Bourin, and Uchiyama

## Abstract

In this article, we present a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators. For example, under suitable assumptions on the spectrum of the positive operators $A$ and $B$, we prove that \[{{2}^{1-r}}{{\left( A+B \right)}^{r}}\le {{A}^{r}}+{{B}^{r}}\quad\text{ for }r>1\text{ and }r<0,\] and \[{{A}^{r}}+{{B}^{r}}\le {{2}^{1-r}}{{\left( A+B \right)}^{r}}\quad\text{ for }r\in \left[ 0,1 \right].\] These results provide considerable generalization of earlier results by Aujla and Silva. Further, we present several extensions of the subadditivity idea initiated by Ando and Zhan, then extended by Bourin and Uchiyama.