On linear operators with closed range (Q2906761)

Let \(X\) be an F-space (that is, a complete metrizable topological vector space) and \(Y\) be a Fréchet space (that is, a locally convex F-space) and \(T,S\in B(X,Y)\). The main theorem of this article states that \(k_1|f(Tx)|\leq|f(Sx)|\leq k_2|f(Tx)|\) for all \(f\in X^*\), for all \(x\in X\) and some \(k_1,k_2>0\) if and only if \(S\) is a constant multiple of \(T\). As a conclusion of this theorem, one can show that \(S\) has closed range if and only if \(T\) does. The authors also point to some special cases, for example, when \(X\) is a Hilbert space, or some results in Banach spaces.