Geodesic
Invertible linear relations generated by an integral equation with a~Nevanlinna measure
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2013), pp. 16-29.

Voir la notice de l'article provenant de la source Math-Net.Ru

We define families of maximal and minimal relations generated by integral equations with a Nevanlinna operator measure and a non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the operator inverse to this restriction is integral. We establish a sufficient condition ensuring that the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations. The obtained results are applicable to differential equations with singular coefficients.
Keywords: Hilbert space, linear relation, integral equation, holomorphic family of relations, resolvent convergence. 
@article{IVM_2013_2_a1,
     author = {V. M. Bruk},
     title = {Invertible linear relations generated by an integral equation with {a~Nevanlinna} measure},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {16--29},
     publisher = {mathdoc},
     number = {2},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2013_2_a1/}
}
TY  - JOUR
AU  - V. M. Bruk
TI  - Invertible linear relations generated by an integral equation with a~Nevanlinna measure
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2013
SP  - 16
EP  - 29
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2013_2_a1/
LA  - ru
ID  - IVM_2013_2_a1
ER  - 
%0 Journal Article
%A V. M. Bruk
%T Invertible linear relations generated by an integral equation with a~Nevanlinna measure
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2013
%P 16-29
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2013_2_a1/
%G ru
%F IVM_2013_2_a1
V. M. Bruk. Invertible linear relations generated by an integral equation with a~Nevanlinna measure. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2013), pp. 16-29. http://geodesic.mathdoc.fr/item/IVM_2013_2_a1/

[1] Bruk V. M., “O lineinykh otnosheniyakh, porozhdennykh integralnym uravneniem s nevanlinnovskoi meroi”, Izv. vuzov. Matem., 2012, no. 10, 3–19

[2] Rofe-Beketov F. S., “Square-integrable solutions, self-adjoint extensions and spectrum of differential systems”, Differential Equations, Proc. Intern. Conf. Differ. Eq., Uppsala, 1977, 169–178 | MR | Zbl

[3] Savchuk A. M., Shkalikov A. A., “Operatory Shturma–Liuvillya s singulyarnymi potentsialami”, Matem. zametki, 66:6 (1999), 897–912 | DOI | MR | Zbl

[4] Savchuk A. M., Shkalikov A. A., “Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami”, Tr. MMO, 64, 2003, 159–212 | MR | Zbl

[5] Pokornyi Yu. V., Zvereva M. B., Shabrov S. A., “Ostsillyatsionnaya teoriya Shturma–Liuvillya dlya impulsnykh zadach”, UMN, 63:1 (2008), 111–154 | DOI | MR | Zbl

[6] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Nauk. dumka, Kiev, 1965 | MR | Zbl

[7] Bruk V. M., “O lineinykh otnosheniyakh v prostranstve vektor-funktsii”, Matem. zametki, 24:4 (1978), 499–511 | MR | Zbl

[8] Baskakov A. G., “Spektralnyi analiz differentsialnykh operatorov s neogranichennymi operatornymi koeffitsientami, raznostnye otnosheniya i polugruppy raznostnykh otnoshenii”, Izv. RAN, Ser. matem., 73:2 (2009), 3–68 | DOI | MR | Zbl

[9] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[10] Khrabustovsky V. I., “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., 2:2 (2006), 149–175 | MR | Zbl

[11] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[12] Bruk V. M., “O kraevykh zadachakh, svyazannykh s golomorfnymi semeistvami operatorov”, Funktsionalnyi analiz, 29, izd-vo Ulyanovskogo pedinstituta, Ulyanovsk, 1989, 32–42 | MR

[13] Bruk V. M., “On linear relations generated by a differential expression and by a Nevanlinna operator function”, J. Math. Phys., Anal., Geom., 7:2 (2011), 115–140 | MR | Zbl

[14] Orlov S. A., “Gnezdyaschiesya matrichnye krugi, analiticheski zavisyaschie ot parametra, i teoremy o invariantnosti rangov radiusov predelnykh krugov”, Izv. AN SSSR, Ser. matem., 40:3 (1976), 593–644 | MR | Zbl