Submajorization inequalities for matrices of τ-measurable operators

Abstract

Let $(\mathcal{M},\tau )$ be a semi-finite von Neumann algebra, $L_0\left(\mathcal{M}\right)$ be the set of all τ-measurable operators, ${\mu }_t\left(x\right)$ be the generalized singular number of $x\in L_0\left(\mathcal{M}\right)$ and $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a concave function. We proved that if $x_1,x_2,\dots ,x_n$ are normal operators in $L_0\left(\mathcal{M}\right)$, then $\mu \left(f\right(|\sum _{k=1}^n{x}_k|\left)\right)$ is submajorized by $\mu \left(f\right(\sum _{k=1}^n|x_k|\left)\right)$. As an application, we obtained that if x is a matrix of normal operators $x_{ij}$ in $L_0\left(\mathcal{M}\right)$, then $\mu \left(f\right(|x|\left)\right)$ is submajorized by $\mu \left(\sum _{i,j=1}^{n}f\right(|x_{ij}|\left)\right)$.

Keywords:

• Normal operator
• submajorization
• τ-measurable operator
• semifinite von Neumann algebra

2010 Mathematics Subject Classifications:

• 46L52
• 47L05

Acknowledgments

We thank the referee for very useful comments, which improved the paper. R. Ahat was partially supported by NSFC grant No. 11771372, and M. Raikhan was partially supported by project AP05131557 of the Science Committee of Ministry of Education and Science of the Republic of Kazakhstan.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

R. Ahat was partially supported by the National Natural Science Foundation of China (NSFC) [grant number 11771372], and M. Raikhan was partially supported by project AP05131557 of the Science Committee of Ministry of Education and Science of the Republic of Kazakhstan.