# Multidimensional spectral order for selfadjoint operators

## TL;DR

This paper extends the spectral order concept to multidimensional settings for commuting selfadjoint operators, showing its preservation under spectral integral transformations and characterizing the order via operator inequalities.

## Contribution

It introduces a multidimensional spectral order for families of commuting selfadjoint operators and explores its properties and characterizations.

## Key findings

- Multidimensional spectral order is preserved by spectral integral transformations.
- The spectral order for positive tuples is characterized by operator inequalities.
- The multidimensional spectral order generalizes the product of one-dimensional spectral orders.

## Abstract

The aim of this paper is to extend the notion of the spectral order for finite families of pairwise commuting bounded and unbounded selfadjoint operators in Hilbert space. It is shown that the multidimensional spectral order $\preccurlyeq$ is preserved by transformations represented by spectral integrals of separately increasing Borel functions on $\mathbb{R}^\kappa$. In particular, the $\kappa$-dimensional spectral order is the restriction of product of $\kappa$ spectral orders for selfadjoint operators. If $\mathbf{A}$ and $\mathbf{B}$ are positive $\kappa$-tuples of pairwise commuting selfadjoint operators, then relation $\mathbf{A}\preccurlyeq\mathbf{B}$ holds if and only if $\mathbf{A}^\alpha\leqslant \mathbf{B}^\alpha$ for every $\alpha\in\mathbb{Z}_+^\kappa$.