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Operators with Continuous Kernels

  • W. Arendt
  • A. F. M. ter ElstEmail author
Article
  • 78 Downloads

Abstract

Let \(\Omega \subset \mathbb {R}^d\) be open. We investigate conditions under which an operator T on \(L_2(\Omega )\) has a continuous kernel \(K \in C({{\overline{\Omega }}} \times \overline{\Omega })\). In the centre of our interest is the condition \(T L_2(\Omega ) \subset C({{\overline{\Omega }}})\), which one knows for many semigroups generated by elliptic operators. This condition implies that \(T^3\) has a kernel in \(C({{\overline{\Omega }}} \times \overline{\Omega })\) if T is self-adjoint and \(\Omega \) is bounded, and the power 3 is best possible. We also analyse Mercer’s theorem in our context.

Keywords

Intergral operator Continuous kernel Mercer’s theorem 

Mathematics Subject Classification

Primary 47G10 Secondary 47B10 

Notes

Acknowledgements

The first-named author is most grateful for the hospitality extended to him during a fruitful stay at the University of Auckland and the second-named author for a wonderful stay at the University of Ulm. Both authors wish to thank the referee for the comments and corrections which improved significantly this paper.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Applied AnalysisUniversity of UlmUlmGermany
  2. 2.Department of MathematicsUniversity of AucklandAuckland 1142New Zealand

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