## The nonproper dissipative extensions of a dual pair

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## Abstract:

We consider dissipative operators $A$ of the form $A=S+iV$, where both $S$ and $V\geq 0$ are assumed to be symmetric, but neither of them needs to be (essentially) self-adjoint. After a brief discussion of the relation of the operators $S\pm iV$ to dual pairs with the so-called common core property, we present a necessary and sufficient condition for any extension of $A$ with domain contained in $\mathcal {D}((S-iV)^*)$ to be dissipative. We will discuss several special situations in which this condition can be expressed in a particularly nice form—accessible to direct computations. Examples involving ordinary differential operators are given.## References

- Alberto Alonso and Barry Simon,
*The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators*, J. Operator Theory**4**(1980), no. 2, 251–270. MR**595414** - A. Alonzo and B. Simon,
*Addenda to: “The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators” [J. Operator Theory 4 (1980), no. 2, 251–270; MR 81m:47038]*, J. Operator Theory**6**(1981), no. 2, 407. MR**643699** - Tsuyoshi Ando and Katsuyoshi Nishio,
*Positive selfadjoint extensions of positive symmetric operators*, Tohoku Math. J. (2)**22**(1970), 65–75. MR**264422**, DOI 10.2748/tmj/1178242861 - Yu. M. Arlinskiĭ,
*On proper accretive extensions of positive linear relations*, Ukraïn. Mat. Zh.**47**(1995), no. 6, 723–730 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**47**(1995), no. 6, 831–840 (1996). MR**1355979**, DOI 10.1007/BF01058773 - Yury Arlinskiĭ,
*Boundary triplets and maximal accretive extensions of sectorial operators*, Operator methods for boundary value problems, London Math. Soc. Lecture Note Ser., vol. 404, Cambridge Univ. Press, Cambridge, 2012, pp. 35–72. MR**3050303** - Yuri Arlinskii, Sergey Belyi, and Eduard Tsekanovskii,
*Conservative realizations of Herglotz-Nevanlinna functions*, Operator Theory: Advances and Applications, vol. 217, Birkhäuser/Springer Basel AG, Basel, 2011. MR**2828331**, DOI 10.1007/978-3-7643-9996-2 - Yury Arlinskiĭ, Yury Kovalev, and Eduard Tsekanovskiĭ,
*Accretive and sectorial extensions of nonnegative symmetric operators*, Complex Anal. Oper. Theory**6**(2012), no. 3, 677–718. MR**2944079**, DOI 10.1007/s11785-011-0169-7 - Yury Arlinskiĭ and Eduard Tsekanovskiĭ,
*The von Neumann problem for nonnegative symmetric operators*, Integral Equations Operator Theory**51**(2005), no. 3, 319–356. MR**2126815**, DOI 10.1007/s00020-003-1260-x - Yu. Arlinskiĭ and E. Tsekanovskiĭ,
*M. Kreĭn’s research on semi-bounded operators, its contemporary developments, and applications*, Modern analysis and applications. The Mark Krein Centenary Conference. Vol. 1: Operator theory and related topics, Oper. Theory Adv. Appl., vol. 190, Birkhäuser Verlag, Basel, 2009, pp. 65–112. MR**2568624**, DOI 10.1007/978-3-7643-9919-1_{5} - Gr. Arsene and A. Gheondea,
*Completing matrix contractions*, J. Operator Theory**7**(1982), no. 1, 179–189. MR**650203** - M. Š. Birman,
*On the theory of self-adjoint extensions of positive definite operators*, Mat. Sb. N.S.**38(80)**(1956), 431–450 (Russian). MR**0080271** - M. S. Brodskiĭ and M. S. Livšic,
*Spectral analysis of non-selfadjoint operators and intermediate systems*, Amer. Math. Soc. Transl. (2)**13**(1960), 265–346. MR**0113144** - B. M. Brown and W. D. Evans,
*Selfadjoint and $m$ sectorial extensions of Sturm-Liouville operators*, Integral Equations Operator Theory**85**(2016), no. 2, 151–166. MR**3511362**, DOI 10.1007/s00020-016-2296-z - Laurent Bruneau, Jan Dereziński, and Vladimir Georgescu,
*Homogeneous Schrödinger operators on half-line*, Ann. Henri Poincaré**12**(2011), no. 3, 547–590. MR**2785138**, DOI 10.1007/s00023-011-0078-3 - Michael G. Crandall,
*Norm preserving extensions of linear transformations on Hilbert spaces*, Proc. Amer. Math. Soc.**21**(1969), 335–340. MR**238092**, DOI 10.1090/S0002-9939-1969-0238092-8 - Michael G. Crandall and Ralph S. Phillips,
*On the extension problem for dissipative operators*, J. Functional Analysis**2**(1968), 147–176. MR**0231220**, DOI 10.1016/0022-1236(68)90015-3 - V. A. Derkach and M. M. Malamud,
*Generalized resolvents and the boundary value problems for Hermitian operators with gaps*, J. Funct. Anal.**95**(1991), no. 1, 1–95. MR**1087947**, DOI 10.1016/0022-1236(91)90024-Y - V. A. Derkach, M. M. Malamud, and È. R. Tsekanovskiĭ,
*Sectorial extensions of a positive operator, and the characteristic function*, Dokl. Akad. Nauk SSSR**298**(1988), no. 3, 537–541 (Russian); English transl., Soviet Math. Dokl.**37**(1988), no. 1, 106–110. MR**925955** - D. E. Edmunds and W. D. Evans,
*Spectral theory and differential operators*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR**929030** - W. Desmond Evans and Ian Knowles,
*On the extension problem for accretive differential operators*, J. Funct. Anal.**63**(1985), no. 3, 276–298. MR**808264**, DOI 10.1016/0022-1236(85)90089-8 - W. Desmond Evans and Ian Knowles,
*On the extension problem for singular accretive differential operators*, J. Differential Equations**63**(1986), no. 2, 264–288. MR**848270**, DOI 10.1016/0022-0396(86)90050-1 - C. Fischbacher: On the Theory of Dissipative Extensions, PhD Thesis, University of Kent (2017), https://kar.kent.ac.uk/61093.
- Christoph Fischbacher, Sergey Naboko, and Ian Wood,
*The proper dissipative extensions of a dual pair*, Integral Equations Operator Theory**85**(2016), no. 4, 573–599. MR**3551233**, DOI 10.1007/s00020-016-2310-5 - I. C. Gohberg and M. G. Kreĭn,
*Introduction to the theory of linear nonselfadjoint operators*, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR**0246142** - Gerd Grubb,
*A characterization of the non-local boundary value problems associated with an elliptic operator*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**22**(1968), 425–513. MR**239269** - M. Krein,
*The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I*, Rec. Math. [Mat. Sbornik] N.S.**20(62)**(1947), 431–495 (Russian, with English summary). MR**0024574** - M. S. Livšic,
*On a class of linear operators in Hilbert space*, Amer. Math. Soc. Transl. (2)**13**(1960), 61–83. MR**0113142**, DOI 10.1090/trans2/013/04 - M. S. Livšic,
*On spectral decomposition of linear nonself-adjoint operators*, Mat. Sbornik N.S.**34(76)**(1954), 145–199 (Russian). MR**0062955** - V. È. Lyantse and O. G. Storozh,
*Metody teorii neogranichennykh operatorov*, “Naukova Dumka”, Kiev, 1983 (Russian). MR**757535** - M. M. Malamud and V. I. Mogilevskii,
*On extensions of dual pairs of operators*, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki**1**(1997), 30–37 (English, with Ukrainian summary). MR**1490700** - M. M. Malamud and V. I. Mogilevskii,
*On Weyl functions and $Q$-functions of dual pairs of linear relations*, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki**4**(1999), 32–37 (English, with Ukrainian summary). MR**1708283** - M. M. Malamud and V. I. Mogilevskii,
*Kreĭn type formula for canonical resolvents of dual pairs of linear relations*, Methods Funct. Anal. Topology**8**(2002), no. 4, 72–100. MR**1942823** - J. v. Neumann,
*Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren*, Math. Ann.**102**(1930), no. 1, 49–131 (German). MR**1512569**, DOI 10.1007/BF01782338 - R. S. Phillips,
*Dissipative operators and hyperbolic systems of partial differential equations*, Trans. Amer. Math. Soc.**90**(1959), 193–254. MR**104919**, DOI 10.1090/S0002-9947-1959-0104919-1 - Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy,
*Harmonic analysis of operators on Hilbert space*, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR**2760647**, DOI 10.1007/978-1-4419-6094-8 - E. R. Tsekanovskiĭ: Non-self-adjoint accretive extensions of positive operators and theorems of Friedrichs-Krein-Phillips, Func. Anal. Appl.
**14**, 156-157 (1980). - E. R. Tsekanovskiĭ,
*The Friedrichs and Kreĭn extensions of positive operators, and holomorphic semigroups of contractions*, Funktsional. Anal. i Prilozhen.**15**(1981), no. 4, 91–92 (Russian). MR**639213** - M. I. Višik,
*On general boundary problems for elliptic differential equations*, Trudy Moskov. Mat. Obšč.**1**(1952), 187–246 (Russian). MR**0051404**

## Additional Information

**Christoph Fischbacher**- Affiliation: Department of Mathematics, The University of Alabama, Birmingham, Alabama 35294
- MR Author ID: 1080234
- Email: cfischb@uab.edu
- Received by editor(s): June 13, 2017
- Received by editor(s) in revised form: October 26, 2017
- Published electronically: September 5, 2018
- Additional Notes: The author is indebted to the UK Engineering and Physical Sciences Research Council (Doctoral Training Grant Ref. EP/K50306X/1).
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8895-8920 - MSC (2010): Primary 47B44, 47A20; Secondary 47E05
- DOI: https://doi.org/10.1090/tran/7511
- MathSciNet review: 3864399