Closedness of ranges of upper-triangular operators
The authors consider block upper-triangular operators of the form \(M_C = \left( \begin{smallmatrix} A & C \\ 0 & B \\ \end{smallmatrix} \right) \) on separable Hilbert spaces. Results concerning the closure of the range of \(M_C\) are obtained in terms of properties of the operators \(A\) and \(B\). In particular, it is shown that if the ranges of both \(A\) and \(B\) are closed, then the closure of the range of \(M_C\) depends solely on whether or not either the dimension of the nullspace of \(A^*\) or the nullspace of \(B\) is finite. An operator \(T\) is Kato non-singular if the range of \(T\) is closed and the nullspace of \(T\) is contained in the range of \(T^n\) for all \(n \in \mathbb N\). The authors also prove that if \(A\) and \(B\) are Kato non-singular operators, then \(M_C\) is Kato non-singular if and only if either \(A\) is surjective or \(B\) is injective.
scientific article; zbMATH DE number 5563843