Multidimensional spectral order for selfadjoint operators

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 $\alpha\in\mathbb{Z}_+^\kappa$.