Multidimensional spectral order for selfadjoint operators

The notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in Hilbert space. It is shown that the multidimensional spectral order (sic) is preserved by transformations represented by spectral integrals of separately increasing Borel functions on R-kappa. In particular, the kappa-dimensional spectral order is the restriction of product of kappa spectral orders for selfadjoint operators. In the context of positive kappa-tuples of pairwise commuting selfadjoint operator, the relation A (sic) B holds if and only if A(alpha) <= B-alpha for every kappa-tuples of nonnegative integer numbers alpha. (C) 2020 The Author(s). Published by Elsevier Inc.