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 is preserved by transformations represented by spectral integrals of separately increasing Borel functions on R κ . In particular, the κ-dimensional spectral order is the restriction of product of κ spectral orders for selfadjoint operators. In the context of positive families of pairwise commuting selfadjoint operator, the relation A B holds if and only if A α B α for every α ∈ Z κ + .Key words and phrases. spectral order, joint spectral measure, joint bounded vectors, integral inequalities, separately increasing function.The author was supported by the Ministry of Science and Higher Education of the Republic of Poland.