Accretive Canonical Type Quasi-Differential Operators for First Order

Year 2018, Volume: 10, 43 - 49, 29.12.2018

Pembe Ipek Al , Zameddin Ismaılov

Abstract

It is known that a linear closed densely defined operator in any Hilbert space   is called accretive if its real part is non-negative and maximal accretive if it is accretive and it does not have any proper accretive extension [1].

Note that the study of abstract extension problems for operators on Hilbert spaces goes at least back to J.von Neumann [2] such that in [2] a full characterization of all selfadjoint extensions of a given closed symmetric operator with equal deficiency indices was investigated.  

Class of accretive operators is an important class of non-selfadjoint operators in the operator theory. Note that spectrum set of the accretive operators lies in right half-plane.

The maximal accretive extensions of the minimal operator generated by regular differential-operator expression in Hilbert space of vector-functions defined in one finite interval case and their spectral analysis have been studied by V. V. Levchuk [3].

In this work, using the method Calkin-Gorbachuk all maximal accretive extensions of the minimal operator generated by linear canonical type quasi-differential operator expression in the weighted Hilbert space of the vector functions defined at right semi-axis are described. Lastly, geometry of spectrum set of these type extensions will be investigated.

Keywords

Accretive operator, Quasi-differential operator, Spectrum

References

• Arlinskii, Yu. M., On proper accretive extensions of positive linear relations , Ukrainian Mat. Zh. 47(6) (1995), 723-730.
• Arlinskii, Yu. M., Abstract boundary conditions for maximal sectorial extensions of sectorial operators , Math. Nachr. 209 (2000), 5-36.
• Arlinskii, Yu. M., Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser., 404, Cambridge Univ. Press, Londan, 2012.
• Arlinskii, Yu. M., Kovalev, Yu., Tsekanovskii, E., Accretive and sectorial extensions of nonnegative symmetric operators , Complex Anal. Oper. Theory 6 (2012), 677-718.
• Arlinskii, Yu. M., Popov, A. B., m-Accretive extensions of a sectorial operator , Sbornik: Mathematics 204 (2013), 1085-1121.
• Evans, W. D., On the extension problem for accretive di_erential operators, Journal of Functional Analysis 63 (1985), 276-298.
• Fischbacher, C., The nonproper dissipative extensions of a dual pair , Trans. Amer. Math. Soc. 370 (2018), 8895-8920.
• Gorbachuk, V. I., Gorbachuk, M. L., Boundary Value Problems for Operator Di_erential Equations, Kluwer Academic Publisher, Dordrecht, 1991.
• Hörmander, L., On the theory of general partial di_erential operators, Acta Mathematica 94(1955), 161-248.
• Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag Inc., New York, 1966.
• Levchuk, V. V., Smooth maximally dissipative boundary-value problems for a parabolic equation in a Hilbert space, Ukrainian Mathematic Journal 35(4) (1983), 502-507.
• Von Neumann, J., Allgemeine eigenwerttheorie hermitescher funktionaloperatoren, Math. Ann. 102 (1929-1931), 49-131.