Closed embeddings of Hilbert spaces |
| |
Authors: | Petru Cojuhari |
| |
Institution: | a Department of Applied Mathematics, AGH University of Science and Technology, Al. Mickievicza 30, 30-059 Cracow, Poland b Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey c Institutul de Matematic? al Academiei Române, C.P. 1-764, 014700 Bucure?ti, Romania |
| |
Abstract: | Motivated by questions related to embeddings of homogeneous Sobolev spaces and to comparison of function spaces and operator ranges, we introduce the notion of closely embedded Hilbert spaces as an extension of that of continuous embedding of Hilbert spaces. We show that this notion is a special case of that of Hilbert spaces induced by unbounded positive selfadjoint operators that corresponds to kernel operators in the sense of L. Schwartz. Certain canonical representations and characterizations of uniqueness of closed embeddings are obtained. We exemplify these constructions by closed, but not continuous, embeddings of Hilbert spaces of holomorphic functions. An application to the closed embedding of a homogeneous Sobolev space on Rn in L2(Rn), based on the singular integral operator associated to the Riesz potential, and a comparison to the case of the singular integral operator associated to the Bessel potential are also presented. As a second application we show that a closed embedding of two operator ranges corresponds to absolute continuity, in the sense of T. Ando, of the corresponding kernel operators. |
| |
Keywords: | Hilbert space Continuous embedding Operator range Closed embedding Kernel operator Homogeneous Sobolev space Absolute continuity |
本文献已被 ScienceDirect 等数据库收录! |
|