On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the Infinite-Dimensional Case
DOI:
https://doi.org/10.15407/mag10.02.163Анотація
Визначено сімейства максимальних і мінімальних відношень, що породжені інтегральним рівнянням з неванліннівською операторною мірою в нескінченновимірному випадку й доведено, що ці сімейства голоморфні. Показано, що якщо звуження максимальних відношень неперервно оборотні, то оператори, обернені до таких звужень, є інтегральними. Використовуючи ці результати, доводимо існування характеристичного оператора й описуємо сімейства лінійних відношень, що породжують характеристичний оператор.
Mathematics Subject Classification: 47A06, 47A10, 34B27.
Ключові слова:
гільбертів простір, лінійне відношення, інтегральне рівняння, характеристичний оператор, неванліннівська міраПосилання
A.M. Savchuk and A.A. Shkalikov, Sturm–Liouville Operators with Singular Potentials. — Mat. Zametki 66 (1999), No. 6, 897–912. (Russian) (Engl. transl.: Math. Notes 66 (1999), No. 6, 741–753.) https://doi.org/10.1007/BF02674332
F.S. Rofe-Beketov, Square-Integrable Solutions, Self-Adjoint Extensions and Spectrum of Differential Systems. Differential Equations. Proc. from the Uppsala 1977 Intern. Conf. on Differ. Equations. Uppsala, 1977, 169–178.
Yu.V. Pokornyi, M.B. Zvereva, and S.A. Shabrov, Sturm–Liouville Oscillation Theory for Impulsive Problems. — Usp. Mat. Nauk 68 (2008), No. 1, 111–154. (Russian) (Engl. transl.: Russian Mathem. Surv. 63 (2008), No. 1, 109–153.) https://doi.org/10.1070/RM2008v063n01ABEH004502
V.I. Khrabustovsky, On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. 1. General case. 2. Abstract Theory. 3. Separated Boundary Conditions. — J. Math. Phys., Anal., Geom. 2 (2006), No. 2, 149–175, No. 3, 299–317, No. 4, 449–473.
V.I. Khrabustovsky, On Characteristic Matrix of Weil-Titchmarsh Type for Differential-Operator Equations which Contains the Spectral Parameter in Linearly or Nevanlinna’s Manner. — Mat. Fiz., Anal., Geom. 10 (2003), No. 2, 205–227.
V.M. Bruk, On Linear Relations Generated by a Differential Expression and by a Nevanlinna Operator Function. — J. Math. Phys., Anal., Geom. 7 (2011), No. 2, 115–140.
V.M. Bruk, On Linear Relations Generated by an Integral Equation with a Nevanlinna Measure. — Izv. VUZ. Mathem.(2012), No. 10, 3–19. (Russian) (Engl. transl.: Rus. Mathem. 56 (2012), No. 10, 1–14.) https://doi.org/10.3103/S1066369X12100015
V.M. Bruk, On Invertible Linear Relations Generated by an Integral Equation with a Nevanlinna Measure. — Izv. VUZ. Mathem.(2013), No. 2, 16–29. (Russian) (Engl. transl.: Rus. Mathem. 57 (2013), No. 2, 13–24.) https://doi.org/10.3103/S1066369X13020023
Yu.M. Berezanski, Expansions in Eigenfunctions of Selfadjoint Operators. Amer. Math. Soc., Providence, RI, 1968. (Russian edition: Naukova Dumka, Kiev, 1965.)
V.M. Bruk, On Linear Relations in a Space of Vector Functions. — Mat. Zametki 24 (1978), No. 4, 499–511. (Russian) (Engl. transl.: Math. Notes 24 (1979), No. 4, 767–773.)
V.I. Gorbatchuk and M.L. Gorbatchuk, Boundary Value Problems for Differential Operator Equations. Kluwer Acad. Publ., Dordrecht, Boston, London, 1991. (Russian edition: Naukova Dumka, Kiev, 1984.) https://doi.org/10.1007/978-94-011-3714-0
J.-L. Lions and E. Magenes, Problemes aux Limites non Homogenes et Applications. Dunod, Paris, 1968.
M.M. Malamud and S.M. Malamud, On the Spectral Theory of Operator Measures. — Funk. Anal. 36 (2002), No. 2, 83–89. (Russian) (Engl. transl.: Funct. Anal. and Appl. 36 (2002), No. 2, 154–158.) https://doi.org/10.1023/A:1015630909658
A.G. Baskakov, Spectral Analysis of Differential Operators with Unbounded Operator-valued Coefficients, Difference Relations and Semigroups of Difference Relation. — Izv. RAN. Ser. Mat. 73 (2009), No. 2, 3–68. (Russian) (Engl. transl.: Izv. Math. 73 (2009), No. 2, 215–278.) https://doi.org/10.1070/IM2009v073n02ABEH002445
T. Kato, Perturbation Theory for Linear Operators. Springer–Verlag, Berlin, Heidelberg, New York, 1966. https://doi.org/10.1007/978-3-642-53393-8
V.M. Bruk, On Invertible Restrictions of Closed Operators in Banach Spaces. — Funct. Anal., Uljanovsk 28 (1988), 17–22. (Russian)
H. Schaefer, Topological Vector Spaces. The Macmillan Company, New York; Collier–Macmillan Limited, London, 1966.
V.M. Bruk, Generalized Resolvents of Symmetric Relations Generated on SemiAxis by a Differential Expression and a Nonnegative Operator Function. — J. Math. Phys., Anal., Geom. 2 (2006), No. 4, 372–387.
A.V. Strauss, Generalized Resolvents of Symmetric Operators. — Izv. Akad. Nauk SSSR, Ser. Mat. 18 (1954), 51–86. (Russian)
A. Dijksma and H.S.V. de Snoo, Self-adjoint Extensions of Symmetric Subspaces. — Pacific J. Math. 54 (1974), No. 1, 71–100.
A.N. Kochubei, On Extensions of Symmetric Operators and Symmetric Binary Relations. — Mat. Zametki 17 (1975), No. 1, 41–48. (Russian) (Engl. transl.: Math. Notes 17 (1975), No. 1, 25–28.) https://doi.org/10.1007/BF01093837
V.M. Bruk, On One Class of Boundary Value Problems with a Spectral Parameter in the Boundary Condition. — Mat. Sb. 100 (1976), No. 2, 210–216. (Russian) (Engl. transl.: Math. USSR-Sb. 29 (1976), No. 2, 186–192.) https://doi.org/10.1070/SM1976v029n02ABEH003662
F.S. Rofe-Beketov, Self-adjoint Extensions of Differential Operators in a Space of Vector Functions. — Dokl. Akad. Nauk SSSR 184 (1969), No. 5, 1034–1037. (Russian) (Engl. transl.: Soviet. Math. Dokl. 10 (1969), No. 1, 188–192.)
F.S. Rofe-Beketov and A.M. Khol’kin, Spectral Analysis of Differential Operators.Interplay between Spectral and Oscillatory Properties. World Sci. Monogr. Ser.Math. 7, Singapore (2005).
