Opuscula Math. 41, no. 3 (2021), 381-393
https://doi.org/10.7494/OpMath.2021.41.3.381

Opuscula Mathematica

# Extensions of dissipative operators with closable imaginary part

Christoph Fischbacher

Abstract. Given a dissipative operator $$A$$ on a complex Hilbert space $$\mathcal{H}$$ such that the quadratic form $$f \mapsto \text{Im}\langle f, Af \rangle$$ is closable, we give a necessary and sufficient condition for an extension of $$A$$ to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

Keywords: extension theory, dissipative operators, ordinary differential operators.

Mathematics Subject Classification: 34L99, 47H06.

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1. A. Alonso, B. Simon, The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators, J. Operator Theory 4 (1980), 251-270.
2. T. Ando, K. Nishio, Positive selfadjoint extensions of positive symmetric operators, Tôhoku Math. J. 22 (1970), 65-75.
3. Yu. Arlinskiĭ, Boundary triplets and maximal accretive extensions of sectorial operators, [in:] S. Hassi, H.S.V. de Snoo, F.H. Szafraniec (eds.), Operator Methods for Boundary Value Problems, 1st ed. Cambridge: Cambridge University Press, 2012, 35-72.
4. Yu. Arlinskiĭ, E. Tsekanovskiĭ, M. Kreĭn's research on semi-bounded operators, its contemporary developments and applications, Oper. Theory Adv. Appl. 190 (2009), 65-112.
5. Gr. Arsene, A. Gheondea, Completing matrix contractions, J. Operator Theory 7 (1982), 179-189.
6. J. Behrndt, S. Hassi, H. de Snoo, Boundary Value Problems, Weyl Functions and Differential Operators, Monographs in Mathematics, vol. 108, Springer, Berlin, 2020.
7. M. Crandall, Norm preserving extensions of linear transformations on Hilbert spaces, Proc. Amer. Math. Soc. 21 (1969), 335-340.
8. M. Crandall, R. Phillips, On the extension problem for dissipative operators, J. Funct. Anal. 2 (1968), 147-176.
9. C. Fischbacher, On the Theory of Dissipative Extensions, PhD Thesis, University of Kent, 2017.
10. C. Fischbacher, The nonproper dissipative extensions of a dual pair, Trans. Amer. Math. Soc. 380 (2018), 8895-8920.
11. C. Fischbacher, A Birman-Kreĭn-Vishik-Grubb theory for sectorial operators, Complex Anal. Oper. Theory 13 (2019), 3623-3658.
12. C. Fischbacher, S. Naboko, I. Wood, The proper dissipative extensions of a dual pair, Integr. Equ. Oper. Theory 85 (2016), 573-599.
13. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
14. R. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 192-254.
15. G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, Graduate Studies in Mathematics, vol. 99, Amer. Math Soc., Providence, RI, 2009.
• Communicated by Andrei Shkalikov.
• Accepted: 2021-01-23.
• Published online: 2021-04-19.