On self-adjoint operators connected by inequalities and on their applications to problems of mathematical physics
Let \(H\) be a Hilbert space with a scalar product \((\cdot, \cdot)\) and norm \(|\cdot |\), and let \(A\) be a closed symmetric operator with a dense domain \(D(A)\), \(|A|\) its module. We denote by \(D[A]\) the completion of \(D(A)\) with respect to the seminorm \((|A|x, x)^{1/2}\) \((x\in D(A))\). Let \(A_0\) and \(A_1\) be closed symmetric operators with \(D[A_0]\subset H\) and \(D[A_1]\subset H\), and let \([\cdot, \cdot]_0\) and \([\cdot, \cdot]_1\) be extensions by continuity of the forms \((A_0\cdot, \cdot)\) and \((A_1\cdot, \cdot)\) on \(D[A_0]\) and \(D[A_1]\), respectively. We say that \(A_0\) precedes \(A_1\) and denote \(A_0\prec A_1\), if \([x, x]_0\leq [x, x]_1\) for \(x\in D[A_0]\cap D[A_1]\). In the sequel, \(A_0\) and \(A_1\) are assumed to be partially invertible selfadjoint operators. Our purpose is to study properties of the inverse operators \(A^{- 1}_0\) and \(A^{- 1}_1\) and also ``intermediate'' operators \(A: A_0\prec A\prec A_1\).
partially invertible selfadjoint operators
scientific article; zbMATH DE number 797868