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      Closed embeddings of Hilbert spaces

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      Author
      Cojuhari, P.
      Gheondea, A.
      Date
      2010
      Source Title
      Journal of Mathematical Analysis and Applications
      Print ISSN
      0022247X
      Volume
      369
      Issue
      1
      Pages
      60 - 75
      Language
      English
      Type
      Article
      Item Usage Stats
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      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. © 2010 Elsevier Inc.
      Keywords
      Absolute continuity
      Closed embedding
      Continuous embedding
      Hilbert space
      Homogeneous Sobolev space
      Kernel operator
      Operator range
      Permalink
      http://hdl.handle.net/11693/22231
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/j.jmaa.2010.02.027
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