Quantities related to the openness constant of linear operators (Q1989148)

Given a linear operator \(T\) between two normed vector spaces \(X\) and \(Y\), with domain \(D(T)\subseteq X\) and range \(R(T)\subseteq Y\), the relative openness constant of \(T\) is defined as the supremum over all \(\tau\ge0\) such that \(\tau U_{R(T)}\subseteq T(U_{D(T)})\), where \(U_{D(T)},U_{R(T)}\) denote the closed unit balls of \(D(T)\) and \(R(T)\), respectively. The operator \(T\) is said to be relatively open if its relative openness constant is strictly positive. The authors introduce various quantities related to the relative openness constant of a linear operator between normed vector spaces and prove interesting relationships between them (some of which simplify in the case of closed linear operators on Banach spaces). As an example of a consequence of the authors' general results, it is shown that the dual \(T^*\) of any non-trivial, densely defined, closed linear operator \(T\) is relatively open if and only if \(T^*\) has closed range. In the final section of this relatively short paper, the authors present an estimate for the relative openness constant of the product of two linear operators.