Locating the range of an operator with an adjoint
The authors consider the following question: given a linear operator on a Hilbert space, can one compute the projection on the closure of its range? To answer this question affirmatively, it is enough to show that the range \({\text{ran}}(T)\) of the operator \(T\) on the Hilbert space \(H\) is \textit{located} -- that is, the distance \(\rho(x, \text{ran}(T)) = {\text{inf}} \{\| x - Ty\| : y \in H\}\) exists (is computable) for each \(x \in H.\) The paper presents necessary and sufficient conditions under which the locatedness of \({\text{ker}}(T^*)\) (the kernel of the adjoint \(T^*\) of \(T\)) ensures that of \({\text{ran}}(T)\). Namely, it is shown that for any operator \(T\) with an adjoint \(T^*\), the following four statements are equivalent: (i) \(\text{ran}(T)\) is located, (ii) \(\text{ran}(T^*)\) is located, (iii) \(\text{ker}(T)\) is located and \(T\) is decent, (iii) \(\text{ker}(T^*)\) is located and \(T^*\) is decent. (\(T\) is \textit{decent} if for any bounded sequence \(\langle x_n\rangle\) such that \(Tx_n \rightarrow 0\), there exists a sequence \(\langle y_n\rangle\) in \(\text{ker}(T)\) such that \(\langle x_n + \,y_n, z \rangle \rightarrow 0\) for all \(z \in H\).) All the results of the paper are acceptable both in classical and Bishop-stile mathematics.
scientific article; zbMATH DE number 1961118
scientific article