On products of unbounded operators

A symmetric operator (X) over cap is attached to each operator X that leaves the domain of a given positive operator A invariant and makes the product AX symmetric. Some spectral properties of (X) over cap are derived from those of X and, as a consequence, various conditions ensuring positivity of products of the form AX(1)... X(n) are proved. The question of Omega-complete positivity of the mapping p bar right arrow Ap(X(1),..., X(n)) defined on complex polynomials in n variables is investigated. It is shown that the set Omega is related to the McIntosh-Pryde joint spectrum of (X(1),...,X(n)) in case all the operators A, X(1),...,X(n) are bounded. Examples illustrating the theme of the paper are included.