On linear relations generated by nonnegative operator function and degenerate elliptic differential-operator expression
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 5 (2009) no. 2, pp. 123-144 Cet article a éte moissonné depuis la source Math-Net.Ru

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In terms of boundary values, we describe a spectrum of the linear relations indicated in the paper title. We study the invertible restrictions of maximal relation and show that the operators inverse to these restrictions are integral. The criterion of holomorphicity of the family of these operators is determined. Using the results obtained, we show that the minimal relation is symmetric in Hilbert space and describe all generalized resolvents of this relation.
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V. M. Bruk. On linear relations generated by nonnegative operator function and degenerate elliptic differential-operator expression. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 5 (2009) no. 2, pp. 123-144. http://geodesic.mathdoc.fr/item/JMAG_2009_5_2_a1/

[1] M. L. Gorbatchuk, “Selfadjoint Boundary Problems for a Differential Equation of Second Order with an Unbounded Operator Coefficient”, Funct. Anal. Appl., 5:1 (1971), 10–21 (Russian) | MR

[2] L. I. Vainerman, “Selfadjoint Boundary Problems for Strongly Elliptic and Hyperbolic Equations of Second Order in a Hilbert Space”, Dokl. Akad. Nauk USSR, 218 (1974), 345–348 (Russian) | MR

[3] V. M. Bruk, “Dissipative Extensions of a Differential Elliptic-type Operator”, Funct. Anal., Ulyanovsk, 3 (1974), 35–43 (Russian) | MR

[4] L. I. Vainerman, “A Degenerating Elliptic Equation of Second Order in a Hilbert Space”, Diff. Eq., 14 (1978), 482–491 (Russian) | MR

[5] L. I. Vainerman, “A Degenerating Elliptic Equation with a Variable Operator Coefficient”, Ukr. Mat. Zh., 31 (1979), 247–255 (Russian) | MR

[6] V. I Gorbatchuk, M. L. Gorbatchuk, Boundary Value Problems for Differential-Operator Equations, Kluwer Acad. Publ., Dordrecht–Boston–London, 1991

[7] V. I. Khrabustovsky, “On the Characteristic Operators and Projections and on the Solutions of Weil Type of Dissipative and Accumulative Operator Systems. 1. General Case”, J. Math. Phys., Anal., Geom., 2 (2006), 149–175 ; “2. Abstract Theory”, 299–317 ; “3. Separated Boundary Conditions”, 449–473 | MR | Zbl | MR | Zbl | MR | Zbl

[8] V. M. Bruk, “On Invertible Restrictions of Relations Generated by a Differential Expression and by a Nonnegative Operator Function”, Mat. Zametki, 82 (2007), 652–664 (Russian) | DOI | MR | Zbl

[9] Yu. M. Berezanski, Expansions in Eigenfunctions of Selfadjoint Operators, Naukova Dumka, Kiev, 1965 (Russian) | MR

[10] V. M. Bruk, “On Linear Relations in a Space of Vector Functions”, Mat. Zametki, 24 (1978), 499–511 (Russian) | MR | Zbl

[11] V. I Gorbatchuk, M. L. Gorbatchuk, “Some Questions of Spectral Theory of a Linear Differential Equation of Second Order with Unbounded Operator Coefficients”, Ukr. Mat. Zh., 23 (1971), 3–14 (Russian) | MR

[12] V. M. Bruk, “On Generalized Resolvents of Linear Relations Generated by a Nonnegative Operator Function and a Differential Elliptic-Type Expression”, Izv. Vuz. Mat., 11 (2008), 12–26 (Russian) | MR | Zbl

[13] G. I. Laptev, “Strong Elliptic Differential Equations of Second Order in a Hilbert Space”, Lit. Mat. Sb., 8:1 (1968), 87–99 (Russian) | MR | Zbl

[14] F. S. Rofe-Beketov, A. M. Khol'kin, Spectral Analysis of Differential Operators, Interplay between Spectral and Oscillatory Properties, World Sci. Monogr. Ser. Math., 7, 2005, 438 pp. | DOI | MR | Zbl

[15] A. G. Baskakov, K. I. Chernyshov, “Spectral Analysis of Linear Relations, and Degenerate Semigroups of Operators”, Mat. Sb., 193 (2002), 1573–1610 (Russian) | DOI | MR | Zbl

[16] V. M. Bruk, “On Invertible Restrictions of Closed Operators in Banach Spaces”, Funct. Anal., Ulyanovsk, 28 (1988), 17–22 (Russian) | MR | Zbl

[17] V. M. Bruk, “On the Spectrum of Linear Relations Associated with Uniformly Well-Posed1 Problems”, Diff. Eq., 43 (2007), 21–27 (Russian) | DOI | MR | Zbl

[18] A. N. Kochubei, “On the Spectrum of Selfadjoint Extensions of the Symmetric Operator”, Mat. Zametki, 19 (1976), 429–434 (Russian) | MR

[19] V. A. Derkach, M. M. Malamud, “Generalized Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps”, J. Funct. Anal., 95:1 (1991), 1–95 | DOI | MR | Zbl

[20] A. N. Kochubei, “On Extensions of Symmetric Operators and Symmetric Binary Relations”, Mat. Zametki, 17 (1975), 41–48 (Russian) | MR

[21] V. M. Bruk, “On One Class of Boundary Value Problems with a Spectral Parameter in the Boundary Condition”, Mat. Sb., 100 (1976), 210–216 (Russian) | MR | Zbl

[22] L. I. Vainerman, “On Extensions of Closed Operators in a Hilbert Space”, Mat. Zametki, 28 (1980), 833–842 (Russian) | MR

[23] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin–Heidelberg–New York, 1966 | MR

[24] V. M. Bruk, “On Holomorphic Families of Linear Relations”, Funct. Anal., Ulyanovsk, 33 (1992), 24–28 (Russian) | MR | Zbl

[25] F. S. Rofe-Beketov, “Selfadjoint Extensions of Differential Operators in a Space of Vector Functions”, Dokl. Akad. Nauk USSR, 164 (1969), 1034–1037 (Russian) | MR