A mathematical interpretation of Dirac's formalism; Part B: Generalized eigenfunctions in trajectory spaces
Starting with a Hilbert spce \(L_ 2({\mathbb{R}},\mu)\) the authors introduce the dense subspace \({\mathcal R}(L_ 2({\mathbb{R}},\mu))\) where \({\mathcal R}\) is a positive self-adjoint Hilbert-Schmidt operator on \(L_ 2({\mathbb{R}},\mu)\). For the space \({\mathcal R}(L_ 2({\mathbb{R}},\mu))\) a measure-theoretical Sobolev lemma is proved. The results for the spaces of type \({\mathcal R}(L_ 2({\mathbb{R}},\mu))\) are applied to nuclear analyticity spaces \(S_{X,{\mathcal A}}=\cup_{t>0}e^{-t{\mathcal A}}(X)\) where \(e^{-t{\mathcal A}}\) is a Hilbert-Schmidt operator on the Hilbert space X for each \(t>0\). The so-called generalized eigenvalue problem for a general self-adjoint operator \({\mathcal P}\) in X is solved completely. Meanwhile a detailed account of Dirac's formalism has appeared in our book: A mathematical introduction to Dirac's formalism. North-Holland Math. Libr. 36 (1986). [For part A see the review above.]
positive self-adjoint Hilbert-Schmidt operator
scientific article; zbMATH DE number 3994292