Opuscula Math. 31, no. 4 (2011), 501-517
http://dx.doi.org/10.7494/OpMath.2011.31.4.501

Opuscula Mathematica

# Operators in divergence form and their Friedrichs and Kreĭn extensions

Yury Arlinskiĭ
Yury Kovalev

Abstract. For a densely defined nonnegative symmetric operator $$\mathcal{A} = L_2^*L_1$$ in a Hilbert space, constructed from a pair $$L_1 \subset L_2$$ of closed operators, we give expressions for the Friedrichs and Kreĭn nonnegative selfadjoint extensions. Some conditions for the equality $$(L_2^* L_1)^* = L_1^* L_2$$ are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given.

Keywords: symmetric operator, divergence form, Friedrichs extension, Kreĭn extension.

Mathematics Subject Classification: 47A20, 47B25, 47E05, 34L40, 81Q10.

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• Yury Arlinskiĭ
• East Ukrainian National University, Department of Mathematical Analysis, Kvartal Molodyozhny 20-A, Lugansk 91034, Ukraine
• Yury Kovalev
• East Ukrainian National University, Department of Mathematical Analysis, Kvartal Molodyozhny 20-A, Lugansk 91034, Ukraine
• Revised: 2011-02-21.
• Accepted: 2011-02-22.