Abstract
Let A be a symmetric linear relation in the Hilbert space ℌ with unequal deficiency indices n−A < n+(A). A self-adjoint linear relation \( \tilde{A}\supset A \)in some Hilbert space \( \tilde{\mathrm{\mathfrak{H}}}\supset \mathrm{\mathfrak{H}} \)is called an (exit space) extension of A. We study the compressions \( C\left(\tilde{A}\right)={P}_{\mathrm{\mathfrak{H}}}\tilde{A}\upharpoonright \mathrm{\mathfrak{H}} \) of extensions \( \tilde{A}=\tilde{A^{\ast }}. \) Our main result is a description of compressions \( C\left(\tilde{A}\right) \) by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter τ(λ) from the Krein formula for generalized resolvents. We describe also all extensions \( \tilde{A}=\tilde{A^{\ast }}. \) of A with the maximal symmetric compression \( C\left(\tilde{A}\right) \) and all extensions \( \tilde{A}=\tilde{A^{\ast }}. \) of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators A with equal deficiency indices n+(A) = n−(A).
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References
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vols. I and II, Pitman, Boston–London–Melbourne, 1981.
V. M. Bruk, “Extensions of symmetric relations,” Math. Notes, 22, No. 6, 953–958 (1977).
V. A. Derkach, S. Hassi, M. M. Malamud, and H. S. V. de Snoo, “Generalized resolvents of symmetric operators and admissibility,” Meth. Funct. Anal. Topol., 6, No. 3, 24–55 (2000).
V. A. Derkach, S. Hassi, M. M. Malamud, and H. S. V. de Snoo, “Boundary relations and generalized resolvents of symmetric operators,” Russian J. Math. Ph., 16, No. 1, 17–60 (2009).
V. A. Derkach and M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gaps,” J. Funct. Anal., 95, 1–95 (1991).
V. A. Derkach and M. M. Malamud, Extension Theory of Symmetric Operators and Boundary-Value Problems, Institute of Mathematics of the NAS of Ukraine, Kyiv, 2017.
A. Dijksma and H. Langer, “Finite-dimensional self-adjoint extensions of a symmetric operator with finite defect and their compressions,” in: Advances in Complex Analysis and Operator Theory, Festschrift in Honor of Daniel Alpay, Birkhäuser, Basel, 2017, pp. 135–163.
A. Dijksma and H. Langer, Compressions of Self-Adjoint Extensions of a Symmetric Operator and M.G. Krein’s Resolvent Formula, Springer, Berlin, 2018.
A. Dijksma and H. S. V. de Snoo, “Self-adjoint extensions of symmetric subspaces,” Pacif. J. Math., 54, No. 1, 71–100 (1974).
V. I. Gorbachuk and M. L. Gorbachuk, Boundary Problems for Differential-Operator Equations, Kluwer, Dordrecht-Boston-London, 1991.
M. G. Krein and H. Langer, “Defect subspaces and generalized resolvents of a Hermitian operator in the space Πҡ,” Funct. Anal. Its Appl., 5, 136–146, 217–228 (1971/1972).
H. Langer and B. Textorious, “On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space,” Pacif. J. Math., 72, No. 1, 135–165 (1977).
M. M. Malamud, “On the formula of generalized resolvents of a nondensely defined Hermitian operator,” Ukr. Math. Zh., 44, No. 12, 1658–1688 (1992).
V. I. Mogilevskii, “Nevanlinna type families of linear relations and the dilation theorem,” Meth. Funct. Anal. Topol., 12, No. 1, 38–56 (2006).
V. I. Mogilevskii, “Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers,” Meth. Funct. Anal. Topol., 12, No. 3, 258–280 (2006).
V. I. Mogilevskii, “On exit space extensions of symmetric operators with applications to first order symmetric systems,” Meth. Funct. Anal. Topol., 19, No. 3, 268–292 (2013).
V. I. Mogilevskii, “Symmetric extensions of symmetric linear relations (operators) preserving the multivalued part,” Meth. Funct. Anal. Topol., 24, No. 2, 152–177 (2018).
V. I. Mogilevskii, “On compressions of self-adjoint extensions of a symmetric linear relation,” Integr. Equ. Oper. Theory, 91, No. 9 (2019).
M. A. Naimark, “On self-adjoint extensions of the 2-nd kind of a symmetric operator,” Izv. Akad. Nauk SSSR, Ser. Mat., 4, 53–104 (1940).
A. V. Shtraus, “On one-parameter families of extensions of a symmetric operator,” Izv. Akad. Nauk SSSR, Ser. Mat., 30, 1325–1352 (1966).
A. V. Shtraus, “Extensions and generalized resolvents of a symmetric operator which is not densely defined,” Math. USSR-Izv., 4, No. 1, 179–208 (1970).
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 16, No. 4, pp. 567–587 October–December, 2019.
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Mogilevskii, V.I. On Compressions of Self-Adjoint Extensions of a Symmetric Linear Relation with Unequal Deficiency Indices. J Math Sci 246, 671–686 (2020). https://doi.org/10.1007/s10958-020-04772-7
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DOI: https://doi.org/10.1007/s10958-020-04772-7