A metric for unbounded linear operators in a Hilbert space (Q645103)

Let \(M\) be a subspace of an infinite-dimensional complex Hilbert space \((H; \langle\cdot,\cdot\rangle)\). Then \(M\) is called a semiclosed subspace of \(H\) if there exists an inner product \(\langle\cdot,\cdot\rangle_M\) on \(M\) such that \(M\) is a complete inner product space with respect to \(\langle\cdot,\cdot\rangle_M\) and the inclusion mapping \(J: (M,\|\cdot\|_M) \to H\) is continuous with respect to the Hilbert norm \(\|\cdot\|_M\) induced by \(\langle\cdot,\cdot\rangle_M\). It is known that there is a bijective mapping between the set of Hilbert norms \(\{\|\cdot\|_M: (M,\|\cdot\|_M) \hookrightarrow H\}\) and the set of positive bounded operators \(\{T\geq 0: M=TH\}\); see [\textit{T. Ando}, De Branges spaces and analytic operator functions. Sapporo: Hokkaido University (1990; Zbl 0717.47003)]. If \(s: \text{dom}(s)\subseteq H\to H\) is an operator, then \(s\) is said to be semiclosed if its graph is a semiclosed subspace in the product Hilbert space \(H \times H\). The author defines a metric on the set \(S(H)\) of all semiclosed operators in \(H\), investigates its topological structures and proves that the set of all closed and densely defined operators in \(H\) is an open subset of \(S(H)\) under the topology induced by the introduced metric.