Closed embeddings of Hilbert spaces
The authors extend the notion of continuous embedding of Hilbert spaces to that of closed embedding in such a way that unbounded kernel operators are allowed, and investigate its connection with operator ranges, its properties and especially uniqueness properties. They exemplify certain constructions by pointing out closed embeddings that appear when comparing some Hilbert spaces of holomorphic functions on the disk (Hardy, Dirichlet, Bergman). They prove the closed embedding for certain homogeneous Sobolev spaces on the semi-axis and the Euclidean space \(\mathbb{R}^n\) into the ambient \(L_2\) space and obtain an equivalent characterization of absolutely continuous bounded kernel operators in terms of closed embeddings, see \textit{P.\,A.\thinspace Cojuhari} and \textit{A.\,Gheondea} [J.~Math.\ Anal.\ Appl.\ 304, No.\,2, 584--598 (2005; Zbl 1092.46004)]. The authors show that closed embedding is a special representation of a Hilbert space induced by a positive selfadjoint operator within the ambient Hilbert space.
scientific article; zbMATH DE number 5731664
scientific article