Remarks on infimum and maximal lower bounds of a set of bounded self-adjoint operators

Abstract

The notions of infimum and maximal lower bounds of a set $\mathfrak M$ of bounded self-adjoint operators were mainly studied for a set $\mathfrak M$ of two elements. The present paper deals with more general sets $\mathfrak M$, where it is required that $\mathfrak M$ is nonempty and bounded from below. Kadison's theorem on the existence of the infimum of a two-element set is proved for a countable and weak-operator compact set $\mathfrak M$. Stott's recent results on the structure of the set of maximal lower bounds of a finite set of Hermitian matrices are discussed and partially generalized. We are also concerned with the greatest lower bound and maximal lower bounds under certain restrictions. It is shown that the set of all lower bounds of $\mathfrak M$ commuting with all elements of $\mathfrak M$ possesses the greatest element if $\mathfrak M$ is a set of pairwise commuting operators. The theorem of Moreland and Gudder on the existence of the greatest positive lower bound of a set of two positive matrices is extended to an arbitrary finite set of positive matrices.