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Title Krein-von~Neumann and Friedrichs extensions for second order operators on time scales

Name Petr Zem\'{a}nek
Country Czech Rep
Email zemanek@math.muni.cz
Submit Time 2010-02-24 16:23:01
Special Session 48: Differential, Difference, and Dynamic Equations
We consider an operator defined by the second order Sturm--Liouville equation on an unbounded time scale. For such an operator we give characterizations of the domains of its Krein--von~Neumann and Friedrichs extensions by using the recessive solution. This generalizes and unifies similar results obtained for operators connected with the second order Sturm--Liouville differential equations by Niessen and Zettl and for operators associated with the second order Sturm--Liouville difference equations by Brown and Christiansen. References: [1] B.~M.~Brown, J.~S.~Christiansen, On the Krein and Friedrichs extensions of a positive Jacobi operator, {\it Expo. Math.} {\bf 23} (2005), no.~2, 179-ĘC186. [2] H.~Freudenthal, \"{U}ber die Friedrichssche Fortsetzung halbbeschr\"{a}nkter Hermitescher Operatoren {\rm(German)}, {\it Proc. Akad. Wet. Amsterdam} {\bf 39} (1936), 832--833. [3] H.~D.~Niessen, A.~Zettl, Singular Sturm-Liouville problems: The Friedrichs extension and comparison of eigenvalues, {\it Proc. London Math. Soc. (3)} {\bf 64} (1992), no.~3, 545--578.