Topologized Hilbert spaces (Q489048)

The starting point of this article is the question when a collection of densely defined linear operators on a Hilbert space forms a \(*\)-algebra. Inspired by \textit{E. Nelson} [J. Funct. Anal. 15, 103--116 (1974; Zbl 0292.46030)], the author equips the given Hilbert space \(\mathcal{H}\) with a Hausdorff vector topology \(\mathcal{P}\) coarser than the original one. Introducing the concepts of \(\mathcal{P}\)-maximal and \(\mathcal{P}\)-adjointable operator, the author shows that the collection \(\mathfrak{M}^*_\mathcal{P}(\mathcal{H})\) consisting of operators having these properties can be organized as a \(*\)-algebra, provided that \(\mathcal{P}\) is locally convex. Other related properties, sometimes related to concrete spaces, are also presented.